Physics: concepts and connections [Pearson new international edition, Fifth edition] 1292039582, 1269374508, 9781292039589, 9781269374507, 9781292053615, 1292053615

Table of contents :
Cover......Page 1
Table of Contents......Page 4
Glossary......Page 8
1. The Way of Science: Experience and Reason......Page 23
Problem Set (5/e): The Way of Science: Experience and Reason......Page 52
2. Atoms: The Nature of Things......Page 56
Problem Set (5/e): Atoms: The Nature of Things......Page 74
3. How Things Move: Galileo Asks the Right Questions......Page 79
Problem Set (5/e): How Things Move: Galileo Asks the Right Questions......Page 94
4. Why Things Move as They Do......Page 100
Problem Set (5/e): Why Things Move as They Do......Page 120
5. Newton’s Universe......Page 127
Problem Set (5/e): Newton’s Universe......Page 148
6. Conservation of Energy: You Can’t Get Ahead......Page 153
Problem Set (5/e): Conservation of Energy: You Can’t Get Ahead......Page 168
7. Second Law of Thermodynamics......Page 175
Problem Set (5/e): Second Law of Thermodynamics......Page 200
8. Electromagnetism......Page 206
Problem Set (5/e): Electromagnetism......Page 228
9. Waves, Light, and Climate Change......Page 234
Problem Set (5/e): Waves, Light, and Climate Change......Page 268
10. The Special Theory of Relativity......Page 277
Problem Set (5/e): The Special Theory of Relativity......Page 300
11. Einstein’s Universe and the New Cosmology......Page 306
Problem Set (5/e): Einstein’s Universe and the New Cosmology......Page 328
12. The Quantum Idea......Page 332
Problem Set (5/e): The Quantum Idea......Page 350
13. The Quantum Universe......Page 355
Problem Set (5/e): The Quantum Universe......Page 382
14. The Nucleus and Radioactivity: A New Force......Page 391
Problem Set (5/e): The Nucleus and Radioactivity: A New Force......Page 412
15. The Energy Challenge......Page 418
Problem Set (5/e): The Energy Challenge......Page 448
16. Fusion and Fission—and a New Energy......Page 457
Problem Set (5/e): Fusion and Fission—and a New Energy......Page 480
17. Quantum Fields: Relativity Meets the Quantum......Page 484
Problem Set (5/e): Quantum Fields: Relativity Meets the Quantum......Page 512
18. Summing Up......Page 519
Periodic Table of the Elements......Page 522
Flow Chart of Topics......Page 524
B......Page 526
D......Page 527
E......Page 528
F......Page 529
H......Page 530
M......Page 531
N......Page 532
P......Page 533
R......Page 534
S......Page 535
U......Page 536
Z......Page 537

Citation preview

Physics: Concepts and Connections Hobson

9 781292 039589

5e

ISBN 978-1-29203-958-9

Physics: Concepts and Connections Art Hobson Fifth Edition

Pearson New International Edition Physics: Concepts and Connections Art Hobson Fifth Edition

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners.

ISBN 10: 1-292-03958-2 ISBN 10: 1-269-37450-8 ISBN 13: 978-1-292-03958-9 ISBN 13: 978-1-269-37450-7

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

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Table of Contents Glossary Art Hobson

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1. The Way of Science: Experience and Reason Art Hobson

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Problem Set (5/e): The Way of Science: Experience and Reason Art Hobson

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2. Atoms: The Nature of Things Art Hobson

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Problem Set (5/e): Atoms: The Nature of Things Art Hobson

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3. How Things Move: Galileo Asks the Right Questions Art Hobson

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Problem Set (5/e): How Things Move: Galileo Asks the Right Questions Art Hobson

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4. Why Things Move as They Do Art Hobson Problem Set (5/e): Why Things Move as They Do Art Hobson

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5. Newton’s Universe Art Hobson

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Problem Set (5/e): Newton’s Universe Art Hobson

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6. Conservation of Energy: You Can’t Get Ahead Art Hobson

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Problem Set (5/e): Conservation of Energy: You Can’t Get Ahead Art Hobson

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7. Second Law of Thermodynamics Art Hobson

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Problem Set (5/e): Second Law of Thermodynamics Art Hobson

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8. Electromagnetism Art Hobson

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Problem Set (5/e): Electromagnetism Art Hobson

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9. Waves, Light, and Climate Change Art Hobson

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Problem Set (5/e): Waves, Light, and Climate Change Art Hobson

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10. The Special Theory of Relativity Art Hobson

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Problem Set (5/e): The Special Theory of Relativity Art Hobson

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11. Einstein’s Universe and the New Cosmology Art Hobson

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Problem Set (5/e): Einstein’s Universe and the New Cosmology Art Hobson

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12. The Quantum Idea Art Hobson

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Problem Set (5/e): The Quantum Idea Art Hobson

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13. The Quantum Universe Art Hobson

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Problem Set (5/e): The Quantum Universe Art Hobson

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14. The Nucleus and Radioactivity: A New Force Art Hobson

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Problem Set (5/e): The Nucleus and Radioactivity: A New Force Art Hobson

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15. The Energy Challenge

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Problem Set (5/e): The Energy Challenge Art Hobson

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16. Fusion and Fission—and a New Energy Art Hobson

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Problem Set (5/e): Fusion and Fission—and a New Energy Art Hobson

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17. Quantum Fields: Relativity Meets the Quantum Art Hobson

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Problem Set (5/e): Quantum Fields: Relativity Meets the Quantum Art Hobson

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18. Summing Up Art Hobson

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Periodic Table of the Elements Art Hobson

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Flow Chart of Topics Art Hobson

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Index

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GLOSSARY A-bomb See fission bomb. AC See alternating current. accelerating universe Observations of distant supernova explo-

sions show rather conclusively that the universe is not only expanding but is expanding at an ever-increasing speed. acceleration An accelerated object is one whose velocity is changing. Quantitatively, the acceleration is the change in velocity during a time interval divided by the duration of that time interval. It can be measured in (km/hr)/s, or in (m>s)>s = m>s2. acceleration due to gravity The acceleration of any freely falling object. On Earth this is about 10 m>s2 or, more precisely, 9.8 m>s2. action-reaction cycle A mutually reinforcing cycle of increased armaments by two or more hostile nations. active solar energy Energy provided by a solar-heated liquid or gas that is pumped to a location where it can be used for space or water heating. air A mixture of several chemical compounds: nitrogen (N2, about 80%), oxygen (O2, about 20%), argon (Ar, about 1%), and a smattering of trace gases. air resistance The resistive force that air molecules exert on an object moving through the air. alpha decay One type of radioactive decay. The spontaneous emission, by a nucleus, of an alpha particle (a helium nucleus that breaks off of a larger nucleus). alpha particle See alpha decay. alpha rays, beta rays, gamma rays The streams of particles that are emitted by a macroscopic sample of radioactive material. alternating current An electric current that reverses its direction of flow many times every second. amp or ampere The measurement unit for electric current. One amp is defined as a flow of 1 coulomb per second. amplitude The maximum disturbance in a wave; the maximum deviation from the undisturbed state of the medium. anthropic principle The idea that our universe must be organized in the way that it is because any other organization would not allow intelligent beings to be here to ask the question in the first place. antielectron See positron. antimatter Made of antiprotons, antineutrons, and positrons. Today’s universe consists overwhelmingly of matter, not antimatter. antineutron Antiparticle of the neutron. See also antiparticle. antiparticle The theory of special relativity requires that for every existing type of particle, there is an antiparticle carrying the opposite charge. Quantum uncertainties allow the creation and annihilation of particle–antiparticle pairs such as electron–positron pairs. antiproton Antiparticle of the proton. See also antiparticle. Aristarchus’s hypothesis A sun-centered theory that was rejected because it seemed to conflict with everyday observations. Aristotelian physics, difficulties Contrary to Aristotelian predictions, heavy and light objects often fall at the same acceleration, and horizontally moving objects would move forever if there were no external forces.

Aristotle’s physics

Plausible notions that have since been discarded by both Newtonian and modern physics. Aristotle believed that there were three kinds of motion: natural, violent, and celestial. artificial radiation Ionizing radiation from artificial sources such as medical sources. See also natural radiation. astrology The belief, rejected by science for over two centuries, that events on Earth are influenced by the positions and motions of the planets. astronomy The scientific study of the stars and other objects in space. atom See chemical element. atomic bomb See fission bomb. atomic number The number of protons in an atom. Also the number of electrons in a neutral atom. An atom’s atomic number determines its chemical properties and the element to which it belongs. atomic theory of matter All matter is made of tiny particles, too small to be seen. atomism The notion that nature can be reduced to the motion of tiny material particles. average speed An object’s average speed is its distance traveled during a time interval divided by the duration of that time interval. Measured in meters per second. See also instantaneous speed. beta decay

The other main type of radioactive decay. The spontaneous emission, by a nucleus, of a beta particle (an electron created in the nucleus). See also alpha decay. beta particle See beta decay. beta rays See alpha rays. big bang The event some 14 billion years ago that created time, space, matter, and the different forms of energy, and started the expansion of the universe. biofuel Biomass (organic substances) that has been processed to make transportation or other fuels. biology and the second law of thermodynamics The entropy of a growing plant decreases at the expense of a far greater entropy increase in the absorbed and reradiated solar energy that passes through the plant. A similar situation exists for all biological processes, including the evolution of species. biomass energy The chemical energy of organic substances. bit See quantum computer. black hole Any object whose matter (mass) has gravitationally collapsed into a single point. Nothing can escape from its vicinity. When very massive stars run out of fuel, they explode, and the remnant collapses to become a black hole. Giant black holes exist at the centers of most galaxies, including ours. breeder reactor A reactor that creates more than one 239Pu nucleus (from 238U) for each 235U nucleus it fissions and so creates more fuel than it uses. Brownian motion The erratic motion of a microscopic dust or pollen grain immersed in a liquid or gas, caused by numerous moving atoms or molecules colliding with the grain every second. bubble chamber See cloud chamber. burning This chemical reaction creates warmth by combining oxygen from air with a fuel such as carbon or hydrogen.

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GLOSSARY C See coulomb. Calorie The amount of work (or energy) needed to raise the

temperature of 1 kilogram of water by 1°C. cap and trade See carbon cap and trade. carbon cap and trade One suggested way to incorporate the cost of global warming into the cost of products and thus reduce CO2 emissions. An overall limit (cap) would be placed on total carbon emissions, and “tradable permits” to emit a certain amount of CO2 (adding up to the allowed cap) would be issued to carbon-emitting companies. Companies could buy and sell permits among themselves, so that less efficient companies would be able to buy additional permits from more efficient companies. carbon capture and storage The process of capturing CO2 from fossil-fuel-fired generating plants, compressing it, piping it to a storage facility, and permanently storing it underground. carbon dating A radioactive dating method in which radioactive 14C in a dead organism is measured as a fraction of the total carbon in order to determine how long the 14C has been decaying and when the organism died. carbon tax One suggested way of incorporating the cost of global warming into the cost of products and thus reducing CO2 emissions. Companies would be taxed a certain amount for each tonne of CO2 they emit. cellulosic biomass Biomass from non-edible organic materials such as grasses, paper products, wood, agricultural waste, and municipal solid waste. centimeter (cm) One-hundredth of a meter. centrifuge separation One way, widely used today, to enrich uranium. See also uranium enrichment. CFC See chlorofluorocarbons. chain reaction A series of neutron-induced fission reactions that proceed from one nucleus to the next by means of the neutrons released during each fission reaction. charge See electrically charged object. chemical compound A pure substance that can be chemically decomposed. Its smallest particle is a molecule, two or more atoms connected into a single unit. All of a compound’s molecules are identical. chemical decomposition Any process that changes a single substance into two or more other substances. chemical element One of the approximately 116 different substances that cannot be chemically decomposed. An atom whose nucleus has a specific atomic number (number of protons), regardless of how many neutrons it might have. The elements are listed in the periodic table. An atom is the smallest particle of an element. chemical energy Energy due to molecular structure. chemical origin of life See hypothesis of a chemical origin of life. chemical reaction A rearrangement of the atoms in molecules into new molecular forms. For examples, see burning, respiration, and photosynthesis. chemically inert Does not participate readily in chemical reactions. chemistry The study of the properties and transformations of substances (chemical compounds). Chernobyl Site of world’s worst nuclear power accident. The fuel melted down, the reactor suffered a “slow nuclear explosion,” and a large amount of radioactivity escaped. A few tens of people were killed from short-term effects, and about 4000 are expected to die of long-term cancers. See also meltdown.

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chlorofluorocarbons (CFCs)

Chemicals whose molecules are made of chlorine, fluorine, and carbon. Manufactured as coolants, spray propellants, foaming agents, and solvents. They destroy stratospheric ozone. climate model Computer calculations based on accepted principles of physics, chemistry, and other science that predict climatic conditions on Earth’s surface and throughout the atmosphere for many years into the future, at each point of a three-dimensional grid of points separated by several hundred kilometers. closed universe See geometry of the universe. cloud chamber A device that shows the path of a charged particle as a trail of droplets in water vapor. Its successor, the bubble chamber, shows a particle’s path as a trail of bubbles in a liquid. cogeneration The simultaneous production of electricity and useful thermal energy. collapse of a wave packet The instantaneous reduction in the size of a wave packet that occurs when a particle’s position is measured. See also wave packet. combined heat and power See cogeneration. compact fluorescent bulb See fluorescent bulb. conduction electrons The outermost one or two electrons in the atoms of any metal. These electrons move easily through the metal when subjected to even a small electric force. conductor See electrical conductor. conservation of charge See law of conservation of charge. conservation of energy See law of conservation of energy. conservation of matter The total amount of matter involved in any chemical reaction, or in any other physical process, is the same after the reaction as it was before. Although it is a useful principle, experiments have proved that it is only a useful approximation in chemical reactions, and entirely wrong in many other situations. conservation of momentum See law of conservation of momentum. constructive interference See interference. containment dome See nuclear power reactor. continuous spectrum A spectrum containing a continuous range of frequencies. See also spectrum. control rods See nuclear power reactor. coolant See nuclear power reactor. Copernican revolution The rejection of the idea that Earth is at the center of, and therefore basically different from, the rest of the universe. Copernican viewpoint The view that Earth is not a unique place in the universe, that the same principles of nature apply throughout the universe. For example, since intelligent life occurred here, this view argues that it also should have occurred elsewhere. Copernicus’s theory A sun-centered theory, similar to Aristarchus’s. The planets, including Earth, circle the sun, and Earth spins on its axis. cosmic inflation A widely respected hypothesis of the details of how the big bang occurred. According to this hypothesis, during a short period in the very early universe, the universe expanded enormously at a speed greater than lightspeed. This “inflation” stretched the universe to many times the size it had before the inflationary process began, flattening the geometry of the universe and creating the mass and energy observed today.

GLOSSARY cosmic microwave background

The faint microwave remnant of the high-energy radiation from the big bang that still fills the universe. cosmic rays High-energy particles that travel through outer space. cosmology The study of the origin, structure, and evolution of the large-scale universe. coulomb Abbreviated “C.” The metric measurement unit for electric charge. It’s the amount of charge that causes an electric force of 9 * 109 N on an identical charge at a distance of 1 m. It turns out to be the charge on 6.25 * 1018 electrons. Coulomb’s law of the electric force Between any two small charged objects there is a force that is repulsive if both objects have positive charge or if both have negative charge, and is attractive if one has positive and the other has negative charge. This force is proportional to the amount of charge on each object, and proportional to the inverse of the square of the distance between them. In symbols it is F r q1q2>d2. If charge is measured in coulombs and distance in meters, then the force in newtons is given by F = 9 * 109 q1q2>d2. creation and annihilation See antiparticle. creationism The belief that the Bible’s Old Testament can be read literally as scientific and historical truth and that Earth and the biological organisms, including humans, were created separately just a few thousand years ago. Scientists overwhelmingly reject creationism as pseudoscientific and false. critical mass The minimum amount of fissionable material that will sustain a chain reaction. curved space See gravity and warped space. dark energy

Observations show that the universe is filled with a nonmaterial form of energy that pushes outward on the fabric of space, causing it to accelerate in its outward expansion. The properties and the cause of this energy are not understood. It comprises 73% of the energy (and therefore 73% of the mass) of the universe. dark matter Also known as “exotic dark matter,” this is matter that does not interact with electromagnetic radiation and so does not emit, absorb, or reflect light. It is made of entirely new and unknown forms of matter. Dark matter comprises 23% of the universe’s mass (and therefore 23% of its energy), while other forms of matter (stars, planets, gas, black holes, neutrinos) comprise 4%. daughter nucleus The nucleus that remains after a radioactive decay has occurred. DC See direct current. decarbonization Replacement of high-carbon fossil fuels such as coal with lower-carbon fuels such as natural gas, and with zero-carbon fuels such as solar energy, nuclear power, and hydrogen gas generated from nonfossil energy. decay curve A graph of the amount of a radioactive material remaining, versus time. degrees Celsius See temperature. destructive interference See interference. difficulties with Aristotelian physics See Aristotelian physics, difficulties. direct current An electric current that maintains the same direction. dirty bomb A bomb powered with conventional explosions that does its damage primarily by the dispersal of radioactive materials. One of the possible forms of nuclear terrorism.

double-slit experiment with electrons When an electron beam passes through two narrow slits and strikes a viewing screen, an interference pattern is observed. This demonstrates that an electron beam is a wave. The same phenomenon occurs with a proton beam, neutron beam, and other beams of matter. double-slit experiment with light When single-frequency light from two synchronized sources such as two narrow slits strikes a viewing screen, an interference pattern is observed. This demonstrates that light is a wave. doubling time See exponential growth. d-quark See strong force. drift velocity The average forward speed of a typical electron along a wire in an electric current, typically less than 1 mm/s. dualism Descartes’ idea that there are two realities, physical and spiritual. In the physical realm, the real or primary qualities are objective, impersonal phenomena such as the motion of atoms. Human sense impressions are considered to be secondary qualities, caused by the primary qualities. E = mc2 See principle of mass-energy equivalence. efficiency See transportation efficiency, heat engine, and

energy efficiency. Energy due to the ability of a deformed system to snap back. electric circuit A closed loop around which electric current can flow. electric charge See electrically charged object. electric current A flow or motion of electrically charged particles. Electric currents in wires are due to electrons moving along the wire. electric discharge See excite. electric (or electromagnetic) energy The energy that an electrically charged object has due to its position in an electromagnetic field. electric field Exists wherever a charged object would feel an electric force if such an object were present. electric force The electric part of the electromagnetic force. See also electromagnetic force. electric force law Electrically charged objects exert forces on each other at a distance. Like charges repel each other, and unlike charges attract each other. electric force law (field version) An electric field surrounds every charged object. Charged objects feel electric forces whenever they are placed in an electric field. electric generator See steam-electric power plant. electric vehicle (EV) Vehicle powered by a storage battery that is recharged by plugging into a wall socket. Nonpolluting, provided that the electricity comes from an environmentally friendly source. electrical conductor A material (usually a metal) through which electric current can easily flow. The atoms of these materials have one or two outermost electrons that are only loosely attached. electrical insulator A material that doesn’t permit the easy flow of electric current. The atoms of these materials have firmly attached outermost electrons. electrical resistance See resistance. electrically charged (in quantum electrodynamics) See quantum electrodynamics. electrically charged object Any object that can exert or feel the electric force is said to “contain electric charge.” There are two elastic energy

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GLOSSARY

types of charge, positive and negative. Any process that causes an object to gain a net positive or negative charge is called charging. electromagnetic energy See electric energy. electromagnetic field The effect that electrically charged objects have on the surrounding space. An electromagnetic field fills the space around every electrically charged object and exists everywhere that a charged object would feel an electromagnetic force if such an object were present. Light is a wave in an electromagnetic field. electromagnetic force The total (electric and magnetic) force between charges. electromagnetic radiation Any electromagnetic wave. electromagnetic spectrum The complete range of electromagnetic waves. Divided into the radio, infrared, visible, ultraviolet, X-ray, and gamma-ray regions. electromagnetic wave See electromagnetic wave theory of light. All of the following are electromagnetic waves: radio, infrared, light, ultraviolet, X-rays, gamma rays. electromagnetic wave theory of light Every vibrating charged object creates a disturbance in its own electromagnetic field, which spreads outward through the field at 300,000 km/s. Light is just such an electromagnetic wave. electromagnetism The combined effects of electric and magnetic forces. electron One of the fundamental particles. A point particle (so far as we know today) having negative charge, about 2000 times less massive than a proton. electron field The quantized matter field for electrons and positrons. See also matter field. electron microscope Uses electron matter waves to form images of microscopic objects. See also wave theory of matter. electron–positron pair See antiparticle. electroweak force The combined EM and weak forces. The quanta (or exchange particles) of the electroweak force field are photons, W + , W - , and Z particles. The quanta of the electroweak matter field are electrons and electron-neutrinos. In addition, there are two more electroweak matter fields corresponding to a second and third generation of particles: the muon and its neutrino and the tau and its neutrino. Only the first generation is stable and contributes to ordinary matter. The other two generations are unstable and transmute quickly into other particles. electroweak force field See electroweak force. electroweak matter field See electroweak force. element See chemcial element. EM field See electromagnetic field. emission of radiation by an atom Atoms emit radiation when they quantum-jump to a lower energy level, creating and emitting a photon whose energy equals the difference between the two energy levels. energy The capacity to do work. The energy of a system is the amount of work the system can do. Units: joule (J), Calorie, kilowatt-hour (a power of 1000 watts operating for 1 hour). See also work. energy conservation Measures to reduce energy consumption, including both efficiency measures to save energy while providing the same services and switching to less energy-intensive lifestyles. See also energy efficiency. energy efficiency Useful energy output of a device divided by its total energy input. See also heat engine.

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energy flow diagram A diagram showing the energy transformations occurring during some process, with energy pictured as though it were a liquid flowing through pipes. energy fluctuations See vacuum. energy level The precise, predictable energy an atom has when it is in a particular quantum state. energy-level diagram A diagram showing the collection of possible energy levels for an atom. energy resource A natural resource containing useful energy. The major U.S. resources today are shown in Table 1. A resource is renewable if it can be replaced within a human lifetime; otherwise it is nonrenewable. enrichment See isotope separation. entropy A quantitative measure of a system’s microscopic disorganization. equivalence principle No experiment performed inside a closed room can tell you whether you are at rest in the presence of gravity or accelerating in the absence of gravity. ether See ether theory. ether theory The idea that a nonatomic, continuous, material medium, the ether, fills the entire universe and that light waves are waves traveling through this medium. This theory was rejected after 1905 because it contradicted Einstein’s theory. Philosophically, this amounts to a rejection of the idea that every physical thing is made of a material substance, because light waves are physical but not made of matter. exchange particle In quantum field theory, forces between two particles A and B are exerted by means of other particles, called exchange particles, that pass back and forth between A and B. For example, the electromagnetic force is exerted by exchanging photons. excite We excite one or more atoms whenever we cause them to go into an excited state; this generally causes the atom(s) to emit radiation. A gas can be excited by heating and by electric discharge—an electric current flowing through it. See also excited state and ground state. excited state Any quantum state of an atom having an energy higher than the lowest possible energy. See also ground state. exhaust See heat engine. exhaust temperature See heat engine. expansion of the universe The general theory of relativity predicts that the universe’s three-dimensional space must either expand or contract. Observations show that expansion is, in fact, occurring, as evidenced by the fact that distant galaxies are all moving away from us and from each other. experiment See observation. exponential growth Growth by a fixed percentage in each unit of time. It has a fixed doubling time, related to its fixed percentage growth rate by T L 70>P. external combustion engine See steam-electric power plant. Faraday’s law When a wire loop is placed in the vicinity of a magnet, and when either the loop or the magnet is moved, an electric current is created within the loop for as long as the motion continues. Stated in terms of fields, a changing magnetic field creates an electric field. feedback Any effect that further influences the phenomenon that caused it. Negative feedback diminishes its cause, and positive feedback enhances its cause. field A physical entity that is spread throughout a region of space. See also force field and matter field.

GLOSSARY field view of reality The view that the universe is made of fields, subject to the rules of relativity and quantum physics. fission bomb, or atomic bomb (A-bomb) A bomb that gets its energy from a fission chain reaction. The fuel can be either the uranium isotope 235U or plutonium. In the 235U bomb, the design can be as simple as bringing two subcritical masses together to equal or exceed a critical mass. If the bomb contains Pu, a subcritical mass is made critical by squeezing it to high density. fission fragment One of the two pieces that results from the fissioning of a nucleus. flat universe See geometry of the universe. fluorescent bulb In this device, an electric current flows through a gas that fills the bulb, exciting the gas to emit ultraviolet radiation, which is then absorbed by a phosphor coating inside the glass bulb causing the coating to emit light. New highfrequency compact fluorescent bulbs, in which the current oscillates much more frequently than the normal 60 Hz, have recently been developed for increased efficiency. force A body exerts a force on another body whenever the first body causes the second body to accelerate. A force is an action by one body on another; it is not a thing or a property of a body. Every force is similar to a push or a pull. force diagram A diagram that shows all of the individual forces acting on an object. Each force is shown as an arrow pointing in the direction of that force. force field The effect that the source of a force has on the surrounding space. Examples include gravitational fields and electromagnetic fields, which exist everywhere an object would feel, respectively, a gravitational or electromagnetic force if such an object were present. force of gravity The downward pull by Earth on objects in Earth’s vicinity; the pull that every material (i.e., having mass) object exerts on every other material object. See also Newton’s law of gravity. force pair The two forces that two bodies exert on each other. forms of energy See kinetic energy, gravitational energy, elastic energy, thermal energy, electromagnetic energy, radiant energy, chemical energy, and nuclear energy. fossil fuels Combustible fuels, including coal, oil, and natural gas, that store the chemical energy created by millions of years of accumulating layers of energy-rich plant and animal remains. four fundamental forces The gravitational, electromagnetic, strong, and weak forces. free fall Falling that is influenced only by gravity and not by air resistance or other influences. For an object that starts from rest and then falls freely to Earth, speed is proportional to the time, and distance is proportional to the square of the time. These proportionalities are also correct for any motion that starts from rest and maintains an unchanging acceleration in a straight line. See also weightlessness. frequency The number of vibrations that any part of a medium completes in each second as a wave passes through the medium. Also the number of complete wavelengths sent out by the wave source in each second. Higher-frequency waves have shorter wavelengths and (assuming equal amplitudes) higher energies. See also medium. friction The force that one surface exerts on another due to the roughness of the surfaces.

fuel cell See fuel cell vehicle. fuel cell vehicle Vehicle powered by hydrogen (or a hydrocar-

bon fuel such as methane) that is continuously injected into a “fuel cell,” a battery-like device that converts the chemical energy of the hydrogen directly into electricity that then runs the car. It is highly efficient and nonpolluting if the hydrogen is produced in an environmentally friendly way. fundamental forces See four fundamental forces. fusion bomb, or hydrogen bomb (H-bomb) A bomb that gets its energy from the fusion of hydrogen, triggered by a fission bomb. fusion reactor A nuclear power reactor that obtains its thermal energy from fusion rather than fission. Now under development, it might be commercially viable by the middle of the century. galaxy A large aggregation of stars. Most galaxies, such as our own Milky Way, have a disklike, pizza shape and revolve about their centers. Galilean relativity The intuitive theory of relativity, in which time and space are absolute (in other words, different observers measure the same time intervals and the same distances) and light has different speeds relative to different reference frames. See also reference frame. Galileo’s law of falling Neglecting air resistance, any two objects dropped together will fall together, regardless of their weights or shapes or substances of which they are made. gamma ray See alpha rays, beta rays, gamma rays. gamma-ray photon High-energy photons coming from nuclear and other processes. They often accompany alpha decay and beta decay. gas See three states of matter. gas pressure The outward push caused by gas molecules hitting the walls of a container. gasoline-electric hybrid vehicle Vehicle powered by a small gasoline engine that continuously runs an electric generator that energizes a small storage battery. The battery then runs the car, much as in an electric vehicle. It is highly efficient and thus creates little pollution. general theory of relativity Einstein’s theory based on the principle of equivalence. In this theory, gravity is a consequence of the warping of spacetime by masses. This theory applies to accelerated observers; the special theory of relativity applies only to nonaccelerated observers. generation See electroweak force. geological ages The major eras in Earth’s history, as determined by the differing layers of rock characterizing those eras. Some approximate ages, determined by several radioactive and other methods, are: Earth, 5 billion years; life, nearly 4 billion years; humans, 6 million years. geometry of the universe According to general relativity, the large-scale structure of the three-dimensional universe must have one of three possible shapes: A closed universe bends back on itself to form a three-dimensional space that is analogous to the two-dimensional surface of a sphere; it has a finite total volume. A flat universe has no overall large-scale curvature, and is analogous to a flat two-dimensional surface; it has infinite total volume. An open universe is analogous to a two-dimensional saddle-shaped surface; it has an infinite total volume. geothermal energy The thermal energy of hot underground steam, water, or rock.

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GLOSSARY global warming

The additional greenhouse-effect warming of Earth that is caused by fossil-fuel use, deforestation, and other human activities. gluon The exchange particle of the strong force. Gluons have zero mass and travel at lightspeed. See also exchange particle. grand unified theory A quantum field theory that would unify the standard model’s electroweak and strong forces into a single force, much as the EM force and weak force were unified into a single electroweak force. Although the parallels between the theories of the electroweak and strong forces suggest that such a theory should exist, there is as yet no agreed-on grand unified theory. gravitational collapse Dispersed matter drawing itself together because of gravity. gravitational energy Energy due to gravitational forces: GravE = weight * height. gravitational field The force field that surrounds every particle having mass and that is felt by every particle having mass. gravitational force See force of gravity. graviton The quantum of the quantized gravitational field, predicted but not yet observed. It is predicted to move at lightspeed and have zero mass. gravity See force of gravity. gravity and warped space Gravity bends light beams, so gravity must warp, or curve, spacetime. In the general theory of relativity, gravity is the warping of spacetime caused by masses. Greek atom See models of the atom. Greek model of the atom This model pictures the atom as a tiny indestructible object, like a small and rigid pea. greenhouse effect The warming created by Earth’s surrounding blanket of atmospheric gases. greenhouse gas The atmospheric gases, mostly water vapor and carbon dioxide, that cause the greenhouse effect. ground state The quantum state of the atom having the lowest possible energy. See also excited state. growth rate See exponential growth. half-life

The time during which half of a macroscopic amount of a radioactive isotope will decay. H-bomb See fusion bomb. heat engine Any cyclic device that uses thermal energy to do work. Its energy efficiency, the fraction of its input thermal energy that is converted to work, will be higher if the input temperature is higher and the exhaust temperature is lower. The portion of the input energy that is not converted to work is called the exhaust. heating The spontaneous flow of thermal energy from a highertemperature object to a lower-temperature object. Heisenberg uncertainty principle See uncertainty principle. hertz (Hz) The unit of frequency: 1 Hz = 1 vibration>second. See also frequency. Higgs field The standard model requires this field because without it all the particles of the standard model would need to have zero mass. However, there is as yet no direct evidence for this field. Its quanta, called Higgs particles, are currently sought in high-energy accelerators. The Higgs field is predicted to pervade the universe, interacting even with isolated particles. This interaction acts on accelerated particles in such a way as to resist their acceleration. Thus the Higgs field could be the reason that some of the fundamental particles have mass.

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high-level nuclear waste Used reactor fuel rods containing highly radioactive fission products. highly enriched uranium Uranium that has been enriched to about 90% 235U, which is suitable for nuclear weapons use. See also isotope separation. Hiroshima and Nagasaki The Japanese cities that were fissionbombed near the end of World War II. hybrid vehicle See gasoline-electric hybrid vehicle. hydroelectric energy The gravitational energy of raised water. hydrogen bomb See fusion bomb. hypothesis An educated suggestion or guess, a tentative theory. Hz See hertz. Industrial Revolution

The onset of the fossil-fueled industrial age, around 1750. inertia A body’s ability to stay at rest or to maintain an unchanging speed and direction of motion whenever no force is exerted on it. Quantitatively, a body’s inertia is its degree of resistance to acceleration when a force is exerted on it. infrared radiation Created by the thermal motion of molecules. Not visible to the human eye. input temperature See heat engine. instantaneous speed (speed) The average speed during a time interval that is so short that the speed hardly changes. Speedometers measure this. Measured in meters per second. See also average speed. insulator See electrical insulator. intelligent design The view that life is too complex in certain regards, such as complex cellular structures, to have evolved by Darwinian processes and that they must have therefore been created by an “intelligent designer.” Its key idea is that complex structures could not evolve through intermediate nonfunctional steps. Scientists overwhelmingly reject intelligent design as pseudoscientific and false. Intergovernmental Panel on Climate Change (IPCC) Thousands of cooperating international scientists who, on the basis of their study of the scientific literature, reported their conclusions regarding the causes and implications of global warming. Their reports, regarded as the scientific consensus on this topic, were issued in 1990, 1995, 2001, and 2007. internal combustion engine A heat engine in which burning occurs within the hot gases that push directly on mechanical parts, such as a piston, to provide useful work. ion Any atom having an excess or deficiency of electrons. ionizing radiation Radiations (including electromagnetic radiation but also material radiations from radioactive materials) having sufficient energy to ionize biological molecules. Includes higher alpha rays, beta rays, gamma rays, X-rays, and higherenergy ultraviolet rays. The biological damage is measured in a unit called the sievert. A millisievert (mSv) is one-thousandth of a sievert. The main types of damage are radiation sickness, mutation, and cancer. IPCC See Intergovernmental Panel on Climate Change. irreversibility The second law implies that most processes are irreversible, for example, that physical systems proceed spontaneously toward states of higher entropy and will not proceed spontaneously in the reverse direction. This principle is responsible for the difference between forward and backward in time. isotope A particular type of nucleus, having a particular number of protons and of neutrons. Isotopes are specified by their

GLOSSARY

atomic number (number of protons) and mass number (total protons and neutrons). A symbol like 146C represents an isotope with atomic number 6 and mass number 14. isotope separation, or enrichment, of uranium Any process that increases the percentage of 235U relative to 238U. One method is by use of spinning centrifuges. Weapons-grade uranium is highly enriched to about 90% 235U. joule (J)

The metric unit of energy. See also work.

Kepler’s theory

The theory that the planets orbit the sun in ellipses with the sun at one focus. This theory agrees with Brahe’s observations. kg See kilogram. kilo- (k) Prefix meaning one thousand. kilogram (kg) A unit of mass. The mass (or inertia) of the object known as a standard kilogram. Any object that has the same inertia as the standard kilogram has a mass of 1 kilogram. kilometer (km) One thousand meters. kiloton The amount of energy that would be released in the explosion of one thousand tons of TNT. kilowatt-hour See energy. 1 kinetic energy Energy due to motion: KinE = A 2 B mv2. Lamb shift

A small change in the energy levels of the hydrogen atom that is caused by vacuum energy fluctuations in the space surrounding the atom. Large Hadron Collider (LHC) Currently the world’s largest particle accelerator. It accelerates two narrow beams of protons in different directions around a circular ring 27 km long lying 100 m underground near Geneva, Switzerland. The beams collide to create many kinds of particles and phenomena. Each collision has an energy of 14 trillion electron-volts. law See theory. law of conservation of charge Although charge can be moved around and although charged particles can be created or destroyed, no net charge (positive minus negative) can be created or destroyed. law of conservation of energy The total energy of the participants in any process must remain unchanged throughout that process. There are no known exceptions. law of conservation of momentum The total momentum of any system remains unchanged, regardless of interactions among the system’s parts, so long as no part of the system is acted upon by forces external to that system. law of entropy See second law of thermodynamics. law of force pairs Forces always come in pairs: Whenever one body exerts a force on a second body, the second exerts a force on the first. The two forces are equal in strength but opposite in direction. law of heat engines See second law of thermodynamics. law of heating See second law of thermodynamics. law of inertia A body that is subject to no external forces maintains an unchanging velocity (or remains at rest). length contraction See relativity of space. LHC See Large Hadron Collider. light clock A clock whose timekeeping is based on the motion of a light beam.

lightspeed 300,000 km/s, or 3 * 108 m>s. light-year The distance that light travels in one year. limitations of Newtonian physics Newtonian physics gives

incorrect predictions for fast-moving objects (near lightspeed), strong gravitational forces or large distances (intergalactic), and small objects (atomic dimensions). Special relativity, general relativity, and quantum physics, respectively, do give correct predictions for each of these three classes of phenomena. line spectrum A spectrum that contains only separated precise frequencies. See also spectrum. linear growth Straight-line growth; it increases by the same amount (rather than the same percentage) in each unit of time. liquid See three states of matter. macroscopic

Big enough to be visible to the naked eye. See also

microscopic. magnetic field

Exists everywhere a moving charged object would feel an electric force if such an object were present. magnetic force See magnetic force law. magnetic force law Charged objects that are moving exert and feel an additional force, called the magnetic force, in addition to the electric force that exists when they are at rest. magnetic force law (field version) A magnetic field surrounds every moving charged object. Moving charged objects feel magnetic forces whenever they are placed in a magnetic field. magnetic poles The ends of a permanent magnet. There are two types, north and south; like poles repel, and unlike poles attract. Manhattan Project The U.S. project during World War II to build fission bombs. mass A body’s mass is its amount of inertia and also (for bodies at rest) its quantity of matter. We find a body’s mass in kilograms by comparing its inertia with the inertia of a standard kilogram placed at rest. See also inertia. mass and weight See weight and mass. mass number The mass number of a nucleus is the number of particles (protons plus neutrons) it contains. See also atomic number. materialism The philosophy that only matter is real and that everything is determined by its impersonal workings. The early Greek atomists were atomic materialists. Newtonian physics is compatible with this philosophy. matter Material substances such as wood, soil, ice, water, steam, air, and gold. Matter has nonzero rest-mass and moves at less than lightspeed, in contrast to radiation, which has zero restmass and moves at lightspeed. matter–antimatter annihilation The transformation of matter into high-energy radiation that occurs when a subatomic particle such as an electron is brought close to its antiparticle (such as a positron). matter field A new type of field that was discovered during the 1920s. It is quantized, and its quanta are called “electrons,” “protons,” “neutrons,” etc. This field is seen, for example, in the double slit experiment with electrons. Also called “psi,” “wave function,” and “electron field.” See also quantized field. matter wave A wave in a matter field. See wave theory of matter and matter field. measurement See observation and measurement (in quantum physics).

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GLOSSARY measurement (in quantum physics)

Any situation in which a microscopic particle interacts with a macroscopic device such as a viewing screen in such a way as to create a macroscopically observable mark such as a flash. A human observer need not be present. mechanical universe The philosophical view, accompanying Newtonian physics, that the clocklike workings of atoms precisely and predictably determine everything else, including the entire future of the universe. medium The material or nonmaterial substance through which a wave travels. Examples: The medium for water waves is water. The medium for electromagnetic waves is the electromagnetic field. mega- (M) Prefix meaning one million. megaton The amount of energy that would be released in the explosion of one million tons of TNT. meltdown The melting together of nuclear reactor fuel into a solid radioactive mass during a nuclear power accident. metabolic rate The rate, usually measured in Cal/s, at which an animal transforms its bodily chemical energy into other forms of energy. meter (m) The basic metric unit of distance, about 39 inches. metric system The system of measurements based on the meter, second, and kilogram. Used by all nations except the United States. metric ton (tonne) 1000 kilograms, which is about 2200 pounds. micro- Prefix meaning millionth. microscopic Too small to be seen with the unaided eye, as opposed to macroscopic, or visible to the unaided eye. microscopic disorganization and the second law The second law of thermodynamics results from the fact that microscopic disorganization is overwhelmingly likely to increase rather than decrease. microscopic interpretation of warmth Temperature is associated with disorganized, or random, microscopic motions that are not visible macroscopically. Higher temperature means greater microscopic kinetic energy. Milky Way See galaxy. milli- (m) Prefix meaning one-thousandth. millimeter (mm) One-thousandth of a meter. millisievert See ionizing radiation. model See theory. models of the atom The Greek atom is a tiny indestructible object, like a small and rigid pea. The planetary atom is made of parts, including a tiny central nucleus containing protons and neutrons and one or more electrons orbiting far outside the nucleus. The quantum theory of the atom is based on the postNewtonian quantum theory. molecule See chemical compound. momentum The momentum of an object is its mass times its velocity; its direction is the same as the direction of the velocity. The total momentum of a system is the sum of the individual momenta of all the objects in the system, added together as “vectors” (added like arrows placed tip-to-tail). mSv Abbreviation for “millisievert.” See ionizing radiation. muon, tau These two particles are identical to the electron except for the facts that they are heavier and are unstable (they have short lifetimes). Like the electron, they are point particles (so far as we know). mutation See ionizing radiation.

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Nagasaki See Hiroshima. natural motion Unassisted motion. Aristotelian physics says

that falling is one form of natural motion, and horizontal motion is always unnatural, or violent. According to Newtonian (and current) physics, motion at an unchanging speed in a straight line is natural motion, and falling is not natural motion. natural radiation Ionizing radiation from natural sources such as radon gas, cosmic rays, the ground, and internal consumption. See also artificial radiation. negative charge See electrically charged object. negative feedback See feedback. net force The total, overall force on an object. The net force due to two forces acting in the same direction is the sum of the two. The net force due to two forces acting in opposite directions is the difference between the two and acts in the direction of the stronger force. neutrino A uncharged particle that experiences only the weak and gravitational forces and hence penetrates easily through matter. At least two of the three types of neutrinos are now known to have a tiny, but nonzero, mass. neutrino transformation The spontaneous change of a neutrino’s identity among the three types of neutrinos (electron-, muon-, and tau-neutrino). See also neutrino. neutron One of the fundamental particles. A composite particle made of three quarks. neutron star A compact star a few kilometers in diameter, resembling a giant nucleus made of neutrons. It spins rapidly, sending out radio beeps and light flashes. When massive stars (precollapse masses of 10 to 20 solar masses) run out of fusion fuel, they explode, and the remaining remnant collapses to become a neutron star. newton (N) A unit of force. The amount of force that can give a 1 kg mass an acceleration of 1 m>s2. Newton’s theroy of gravity Between any two objects is an attractive force proportional to the product of the two objects’ masses and proportional to the inverse of the square of the distance between them. Newton’s law of motion An object’s acceleration is proportional to the net force exerted on it by its surroundings and is proportional to the inverse of its mass. The direction of the acceleration is the same as the direction of the net force. In the appropriate units (m>s2, newtons, kilograms) it is a = F>m. Newtonian physics The ideas about motion, force, and gravity developed by Isaac Newton and others around 1700. Newtonian physics, limitations See limitations of Newtonian physics. Newtonian physics and democracy All humans are fundamentally equal, because all are ultimately governed by the same universal natural laws. Newtonian attitudes toward natural law pervade the democratic developments of the past several centuries. Newtonian worldview The philosophical notions associated with Newtonian physics, especially the mechanical universe and the democratic ideals implied by universal natural laws. Key features: Tiny indestructible particles form the fundamental reality; the future is precisely predictable from the past; nature can be understood by analyzing it into simple individual components. See also quantum worldview. newton-meter The unit of work and of energy. Also called the joule. nonlocality See quantum nonlocality.

GLOSSARY nonlocality principle

Entangled particles cooperate in a way that can be explained only by the existence of real, nonlocal connections between the particles, so that a measurement of one particle causes a real physical instantaneous change in the other. Entanglement of this sort is predicted by quantum theory and has been confirmed by experiments. nonrenewable resource A natural resource that can be used up. Its use begins exponentially, levels off, and declines. normal force The force, perpendicular to a solid surface, that is exerted by any solid surface on any object touching it. nuclear energy The energy resulting from the structure of a material’s nuclei. It is energy due to nuclear structure. nuclear energy curve A graph showing the energies of the different nuclei versus their mass number. The graph shows a lowest point at mass number 56 (iron), indicating that nuclear energy can be released by the fusion of nuclei lighter than iron and by the fission of nuclei heavier than iron. nuclear fission A nuclear reaction in which a large, single nucleus splits into two roughly equal smaller nuclei. Nuclear energy is released whenever a heavy nucleus is fissioned into two nuclei that are both heavier than iron. nuclear fusion A nuclear reaction in which two nuclei combine to form a single, larger nucleus. Nuclear energy is released whenever two light nuclei are fused to create a nucleus that is lighter than iron. nuclear power A way to get large-scale energy from the nucleus. This energy is obtainable from uranium using the world’s present uranium-fueled nuclear reactors, from plutonium using future plutonium-fueled reactors and breeder reactors, or from hydrogen using yet-to-be-developed fusion reactors. nuclear power plant sabotage One of the four possible forms of nuclear terrorism. nuclear power reactor A device in which chain-reacting nuclei transform nuclear energy into thermal energy for electric power. Its main components are fuel to provide energy, neutron-absorbing control rods to control the reaction, and a coolant to transfer thermal energy from the fuel. Most reactors are enclosed in a thick, concrete containment dome to prevent the escape of radioactivity into the environment. nuclear reaction A change in nuclear structure. The major types of nuclear reactions are radioactive decay, fusion, and fission. nuclear reactor A device that controllably transforms nuclear energy into other energy forms. nuclear terrorism Terrorism using nuclear materials. See also seizing a bomb, seizing bomb material, nuclear power plant sabotage, and dirty bomb. nuclear waste See high-level nuclear waste. nuclear weapon An explosive device fueled by nuclear fusion or nuclear fission. See also fusion bomb and fission bomb. nuclear weapons proliferation The spread of nuclear weapons to additional nations and to terrorists. nucleus The tiny center of an atom, made of protons and neutrons. objectivity An experiment is objective if its outcome is not influenced by humans. The Newtonian worldview assumes that perfect objectivity is possible, at least in principle. observation The fact-gathering process. A measurement is a quantitative observation, and an experiment is a controlled observation.

ohm The measurement unit for electrical resistance. Ohm’s law The current in a circuit element (such as a lightbulb

or toaster) is proportional to the voltage across that element. Ohm’s law is usually written V = IR, where the proportionality constant R is called the circuit element’s “resistance.” open universe See geometry of the universe. outer space The universe outside Earth and its atmosphere and outside other astronomical bodies. ozone The O3 molecule. A dilute layer of ozone fills the stratosphere, 10 to 50 km overhead. Ozone absorbs and shields biological life from most ultraviolet radiation. Ozone Treaty Treaty that called for a nearly total phaseout by 2000 CE of CFCs and most other ozone-destroying chemicals. particle accelerator

A device to accelerate microscopic parti-

cles to high energies. passive solar energy

Energy obtained from solar radiation, natural air flows, and energy storage to provide direct solar heating. peak production See production peak. per In each. periodic table See chemical element. photoelectric effect Light and other radiation shining onto a metal surface can dislodge electrons from their parent atoms. This effect provides evidence that light is quantized and is the basis for photovoltaic electricity. photon When an EM field deposits a quantum of energy in an object such as a viewing screen, it does so all at once and at a specific point on the screen. The resulting tiny impact is called a photon. A photon can be considered to be a particle, but of a very non-Newtonian sort since it really only exists at the time of impact. Considered as a particle, each photon moves at speed c, has zero rest-mass, and carries one quantum of energy. See also quantum theory of radiation. photon exchange According to quantum field theory, charged particles exert the electromagnetic force on each other by means of exchanging photons. photosynthesis This chemical reaction in plants combines atmospheric carbon dioxide with water to form carbon-based molecules such as glucose, along with oxygen. photovoltaic cell A device made of semiconducting material such as silicone, designed to use the photoelectric effect to convert solar radiation (photons from the sun) directly into usable electricity. It’s usually disk-shaped, a few inches across, and connected to other cells in a flat array. photovoltaic electricity Electricity generated from solar radiation using the photoelectric effect. Sunlight causes electrons to flow across two thin layers of semiconducting materials— materials having properties lying midway between conductors and insulators—and then around an electric circuit. See also photoelectric effect. physics The branch of science that studies the most general principles underlying the natural world. piston The part of an internal combustion engine that is pushed on by expanding gases in order to do work. Planck energy See Planck scale. Planck length See Planck scale. Planck mass See Planck scale. Planck scale The range or scale at which physicists expect typical quantum-gravitational events to occur. Specifically, such events are expected to occur within regions about as big as the

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GLOSSARY

Planck length, with a duration of about the Planck time, and an energy about equal to the Planck energy. The Planck mass is the mass of this much energy. Planck time See Planck scale. Planck’s constant See quantum theory of radiation. planet To the Greeks, these were objects that looked like stars but that wandered, out of step with the stars. Today, we view planets as objects that orbit the sun in nearly circular orbits. planetary atom See models of the atom. planetary model of the atom The atomic model in which tiny electrons, conceived of as Newtonian particles, move in planetlike orbits around a tiny nucleus. This model cannot explain line spectra and predicts that atoms will lose energy until they collapse. plug-in hybrid car A gasoline-electric hybrid car (with a small gasoline engine that generates electricity for a battery that runs the car electrically) in which the battery is a large storage battery that can be plugged in and recharged for all-electric power for trips up to a few tens of kilometers but that must then be recharged by running the small engine. Its average gasoline consumption is even lower than a “standard” hybrid car. point particle A particle whose force field is centered on a single point and that itself takes up no volume. All of the fundamental particles described by quantum field theory appear to be point particles. See also standard model. pole See magnetic poles. positive charge See electrically charged object. positive feedback See feedback. positron The electron’s antiparticle. Identical to the electron except that it carries a positive charge. power The rate of doing work. Units: joule/second, Watt (= joule>second), horsepower; power = work>time. power of 10 10 raised to some positive or negative power. Powers of 10 are used to express huge or tiny numbers. primary qualities See dualism. principle See theory. principle of the constancy of lightspeed Light (and other electromagnetic radiation) has the same speed for all nonaccelerated observers, regardless of the motion of the light source or of the observer. principle of mass–energy equivalence All mass has energy, and all energy has mass. A system with m units of mass has mc2 units of energy. A system with E units of energy has E>c2 units of mass. principle of relativity Every nonaccelerated observer observes the same laws of nature, regardless of their reference frame. “Unless you look outside, you can’t tell how fast you’re moving.” probability An event’s fractional number of occurrences in a long series of trials. Probabilities apply to both Newtonian situations such as a coin flip where the outcome is predetermined and could in principle be predicted, and to quantum situations such as radioactive decay where the outcome is not predetermined and cannot be predicted even in principle. production peak The maximum annual production of a nonrenewable resource. Production typically follows a bell-shaped curve, rising at first, then reaching a production peak, then falling. Once the production peak is reached, prices rise and continue rising as demand increases while production levels off and then declines, causing economic dislocation and hardship. proliferation See nuclear weapons proliferation.

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proportional One quantity is proportional to a second quantity if, whenever the second is multiplied by some number, the first is multiplied by the same number. One quantity is proportional to the square of a second quantity if, whenever the second is multiplied by some number, the first is multiplied by the same number squared. proportional to the inverse A quantity is proportional to the inverse of another quantity if the first is proportional to (equal to some number times) 1 divided by the second quantity. proportional to the square See proportional. proton One of the fundamental particles. It is a composite particle made of three quarks. pseudoscience Claims presented so that they appear scientific even though they lack supporting evidence and plausibility. Ptolemy’s theory An Earth-centered theory in which the planets move in circles within circles, or loop-the-loops. It was a good theory that survived for 1500 years. quanta Discrete bundles of energy associated with the interaction of any quantized field (such as a quantized EM field) with any other system (such as a viewing screen). Examples: Photons are quanta of the EM field, and electrons are quanta of the matter field for electrons. quantitative risk estimatation The quantitative evaluation of human-made and natural risks, especially for the purpose of comparing different risks. quantization of charge Electric charge comes in discrete amounts rather than any arbitrary amount. When you charge an object, it always gains or loses some whole number multiple of the charge on one electron. quantized EM field An EM field that is allowed to have only certain specific amounts of total energy. quantized field A continuous, space-filling field that is subject to the laws of quantum physics. A quantized field’s range of possible energies is “digitized,” with only certain specific energy values allowed. Examples: quantized electromagnetic field, quantized matter field. quantized matter field See quantized field. quantum See quantum theory of radiation. quantum computer A possible future device that would exploit the quantum uncertainty and quantum entanglement of individual microscopic systems (such as ions trapped in EM fields) called qubits to make powerful calculations. The power of qubits comes from the fact that microscopic systems can be in two different quantum states at the same time, in contrast to the macroscopic devices (such as switches) called bits used in ordinary computers. quantum electrodynamics The quantum field theory of electrons and photons (i.e., of the electron matter field and the EM field). According to this theory, when we say that a particle is electrically charged, we mean that it has the ability to emit and absorb photons. Particles exert electric forces on each other in tiny quantized increments, by photon exchanges in which one particle emits a photon that is then absorbed by the other particle. quantum of energy The smallest amount of energy that a quantized field can gain or lose; the energy difference between adjoining energy levels in a quantized field. When a quantized field interacts with an object such as a viewing screen, energy is deposited in particle-like quanta, each carrying one quantum of energy.

GLOSSARY quantum entanglement

Two particles are said to be entangled when their matter fields form a single, inseparable matter field, so that any alteration of the matter field of one particle instantly alters the matter field of the other. See also quantum nonlocality. quantum field theory Everything is made of quantized fields that obey special relativity and quantum theory. All the particles of nature are field quanta. A field’s intensity represents the probability of finding the particles that are the quanta of that field. quantum jump An instantaneous change of an atom’s matter field from one quantum state to a different quantum state, during which the atom’s energy also quantum-jumps from one energy level to another. quantum model of the atom The atomic model in which an atom’s electrons are standing matter waves surrounding the nucleus. This model agrees with all experiments so far. quantum nonlocality When a quantized field (EM field or matter field) interacts with an object such as a viewing screen, the entire spread-out field instantaneously shifts to a new quantum state, even though some parts of the field might be at a great distance from the point at which the interaction occurred. See also quantum entanglement. quantum physics The physical theory of the microscopic behavior of matter and radiation. quantum states of the hydrogen atom The various possible configurations of the matter field for a hydrogen atom’s electron. Each quantum state is a standing-wave pattern that obeys Schroedinger’s equation. quantum theory of the atom See models of the atom. quantum theory of fields See quantum field theory. quantum theory of matter Like EM fields, matter fields are quantized. For example, the matter field for electrons is allowed to possess enough energy for either zero electrons, one electron, two electrons, and so on. Thus, electrons are the quanta of matter fields. This is why there are electrons and other material particles. See also matter field. quantum theory of radiation All EM fields are quantized. Their allowed total energies are 0, hf, 2hf, 3hf, and so on, where f is the frequency of the radiation carried by the field and h = 6.6 * 10 - 34 J>Hz (or J-s), called Planck’s constant. The smallest allowed energy increment, hf, is called a quantum of energy. quantum uncertainty In the microscopic world, identical conditions often produce different outcomes. The different outcomes are unpredictable, or uncertain, although the overall statistics— the likelihood of each of the various possible outcomes—is predictable. This uncertainty is inherent in the microscopic world and has no known cause. quantum worldview In contrast to the Newtonian worldview, quantum physics asserts that the universe is made of malleable (capable of being changed), nonmaterial fields, the nature of microscopic systems depends on the presence of macroscopic detectors, the future is inherently nonpredictable, and nature is deeply interconnected and indivisible. See also Newtonian worldview. quark Fundamental particle, thought to be a point particle. Protons and neutrons are each made of three quarks of two different types, known as “up” and “down.” qubit See quantum computer.

radiant energy Energy carried by an electromagnetic wave. radiation Radiation has zero rest-mass and moves at lightspeed,

in contrast to matter, which has nonzero rest-mass and moves at less than lightspeed. radiation emitted by an atom See emission of radiation by an atom. radiation sickness See ionizing radiation. radio waves Created by humans in such forms as AM and FM radio, TV, radar, and microwaves. radioactive dating Determining the ages of old objects by using radioactive methods. radioactive decay See radioactive nucleus. radioactive fallout Dust that falls to the ground carrying radioactive isotopes from a nuclear explosion or nuclear accident. radioactive isotope An isotope that is radioactive. See also isotope and radioactive nucleus. radioactive nucleus A nucleus that is not stable and thus will eventually change its structure even if left undisturbed. Such a spontaneous change in structure is called radioactive decay. radon gas A radioactive gas that can seep into buildings from underground. See also natural radiation. range of possibilities The range of possible positions and velocities that a microscopic particle can have at any particular time, as determined by the particle’s matter field. This range cannot be smaller than is permitted by the uncertainty principle. See also uncertainty principle. reactor See nuclear power reactor. redshift The stretching of the wavelength of radiation caused by the stretching of space that results from the expansion of the universe. As radiation travels through expanding space, its wavelength lengthens, or shifts, toward the red end of the spectrum. reference frame The laboratory or other surroundings within which an observer makes measurements. Measurements made in a particular reference frame are said to be relative to that frame. relative motion Two objects are in relative motion whenever they have different velocities. relativity of mass An object’s inertia (in other words, its mass) increases with its speed, so its mass is different for different observers. relativity of space Moving objects are contracted along their direction of motion, so an object’s length is different for different observers. Also called length contraction. relativity of time The elapsed time (the number of seconds) between two particular events, such as two ticks on a particular clock or the birth and death of a person, is different for two observers who are in relative motion. The duration of one clock tick is longer for observers who are moving relative to the clock than it is for observers for whom the clock is at rest. Thus, moving clocks run slowly. See also time dilation. release of nuclear energy Any transformation of nuclear energy into other forms of energy. renewable resource A natural resource, such as solar energy, that is continually replaced by natural processes. Its use begins exponentially, then levels off at some sustainable level (provided it is not overconsumed). reprocessing Extraction of the plutonium from used nuclear reactor fuel rods in order to make fuel for a reactor or for a bomb.

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GLOSSARY resistance

The tendency of any circuit element (such as a lightbulb or toaster) to cause a reduction in electric current around the circuit. Quantitatively, it’s defined as the “R” (measured in units called “ohms”) in Ohm’s law, V = IR, where V is the voltage across the element and I is the current through the element. resistive force Any force that acts on a moving body in a direction opposite to the body’s motion. resource See nonrenewable resource and renewable resource. respiration This chemical reaction in animals combines oxygen with carbon-based molecules such as glucose to generate useful energy, along with carbon dioxide and water. rest-mass The mass of an object when it is at rest. Rest-mass represents quantity of matter. retrograde motion A temporary change in the direction that a planet moves relative to the stars, as seen from Earth. rocket propulsion When material is ejected from a vehicle, it exerts a reaction force back on the vehicle because of the law of force pairs. This force accelerates the vehicle, which is then said to be “rocket propelled.” See also law of force pairs. rolling resistance The resistive force by a surface on a rolling object. satellite

A body in orbit around a larger astronomical body. Inertia keeps satellites moving, and the gravitational force exerted by the central body holds satellites in their orbits. Schroedinger’s equation An equation that predicts the matter wave for material particles in a wide variety of situations. science The observation and theoretical understanding of the natural world. See also scientific process. scientific process The dynamic interplay between experience (experiments and observations) and ideas (theories and hypotheses). See also science. second law of thermodynamics Describes the tendency of nonthermal energy to end up as thermal energy. This law can be stated in three logically equivalent forms: The law of heating states that thermal energy flows spontaneously from higher to lower temperatures. The law of heat engines states that any cyclic process that uses thermal energy to do work must have a thermal energy exhaust. The law of entropy states that the total entropy of all the participants in any physical process must either increase or remain unchanged. See also biology and the second law. secondary qualities See dualism. seizing a bomb One of the four possible ways that nuclear terrorism could occur. seizing bomb fuel One of the four possible ways that nuclear terrorism could occur. semiconducting materials Materials having properties lying midway between conductors and insulators. short circuit An electrical circuit in which a low-resistance circuit element (such as a piece of metal wire) is placed across a battery or wall socket, resulting in a huge current in the circuit. sievert See ionizing radiation. solar heating Using sunlight for warmth. solar radiation Electromagnetic radiation from the sun. It is concentrated mainly in the infrared, visible, and ultraviolet. solar system The sun and the objects that orbit the sun, including the nine planets and their moons. solar-thermal electricity Energy generated from thermal energy created by the sun.

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solid See three states of matter. space See outer space. spacetime Space and time together, thought of as a single

entity instead of two different entities. special theory of relativity Einstein’s theory based on the principle of relativity and the principle of the constancy of lightspeed. In this theory, time and space are not absolute, and light has the same speed in all nonaccelerated reference frames. This theory applies only to nonaccelerated observers, whereas the general theory of relativity applies also to accelerated observers. spectroscope A device that measures the spectrum, or set of frequencies, emitted by a radiation source. spectrum The set of frequencies emitted by a radiation source. speed See instantaneous speed. stable nucleus A nucleus that, if left undisturbed, will remain unchanged forever. standard kilogram See kilogram. standard model The theory of the electroweak and strong forces. See also electroweak force and strong force. standing wave A wave in which the medium vibrates in a wave pattern but the pattern does not move. star birth Stars form from collapsing gas clouds. Gravitational collapse heats the gas, which initiates nuclear fusion in the center, which stops the collapse. steam–electric power plant Use of thermal energy from an external source such as burning coal to turn water into steam that pushes on a steam turbine that provides work to generate electricity. The device that converts the rotational motion of the turbine into electricity is called an electric generator. Since the steam is heated by fuel that burns outside of the boiler, the plant is an external combustion engine. steam turbine See steam–electric power plant. stratosphere The upper atmosphere, 10 to 50 km overhead. string See string hypothesis. string hypothesis A promising hypothesis that unifies general relativity with quantum theory but that has as yet no direct experimental verification. Its key idea is that a fundamental particle such as an electron is not concentrated at one infinitely small point but is instead a tiny loop called a string. This spreading out of the point-particle model smoothes its effects on the space around it enough so that strings can fit into general relativity. Strings are comparable in size to the Planck distance. One odd thing about strings is that they exist in 10 spatial dimensions, 7 of which are “rolled up” so that we do not observe them in the macroscopic world. Although all strings are identical, they can vibrate in a variety of ways, and each different mode of vibration is a different elementary particle. strong force One of nature’s fundamental forces. It holds the nucleus together, acts between nuclear particles (protons and neutrons), and is strongly attractive at separations of around 10 - 15 and negligible at larger distances. The quanta (or exchange particles) of the strong force field are gluons. The quanta of the strong matter field are the up quark (u) and the down quark (d). In addition, there are two more strong matter fields corresponding to a second and third generation of particles: the c-quark and s-quark, and the t-quark and b-quark. Only the first generation is stable and contributes to ordinary matter: Protons are made of u-u-d, and neutrons are made of u-d-d. The other two generations are unstable and transmute quickly into other particles. Quarks are not found in isolation because any attempt to isolate them creates more quarks.

GLOSSARY strong force field See strong force. strong matter field See strong force. supernova explosion The explosion of a giant star. Supernovae

spread the chemical elements into space and so are the source of the elements heavier than helium in our solar system. sustainability A practice or policy is sustainable if it meets the needs of people today without endangering the prospects of future generations. tau See muon. technological imperative

The tendency to build whatever tech-

nology is possible. technological momentum

The tendency to continue a technological project once it is started. temperature A quantitative measure of warmth. The units normally used to measure temperature are called degrees Celsius. Any device that measures temperature is called a thermometer. theory A well-confirmed idea or group of ideas that explains or unifies a range of observations. A model is a theory that can be visualized. A principle or law is a single idea, often within a larger theory. theory of relativity Any theory that provides answers to questions about observers in relative motion. See also relative motion. thermal energy Energy due to temperature. Equivalently, thermal energy is microscopic energy, the kinetic (and other) energy of molecules that cannot be directly observed macroscopically. This microscopic motion is called thermal motion. thermal motion The disorganized microscopic motion of molecules that is associated with temperature. thermodynamics The study of the general properties of energy. Thermal energy plays a central role in understanding these properties. thermometer See temperature. thermonuclear reaction A self-sustaining fusion reaction that creates the thermal energy needed to sustain itself. Three Mile Island Site of the most significant U.S. nuclear power plant accident. Little radioactivity escaped even though the fuel suffered a meltdown—the fuel melted together into a solid radioactive mass that slumped downward inside the reactor. three states of matter Nearly every substance can exist in any of the three states. The molecules of a solid are locked closely together in a regular pattern, the molecules of a liquid are close together but not fixed in position, and gas molecules are far apart and move around rapidly. time Time is defined by clocks, in other words, by the operations we perform to measure time. The light clock, based on the motion of light beams, is a simple instrument to define time. time dilation See relativity of time. time travel An observer who accelerates to a high speed and then returns to the initial reference frame experiences a shorter elapsed time than does an observer who remains in the initial frame. So objects in the initial reference frame have aged more than the traveler has. This effect makes it possible to travel to stars that are many light-years distant in only a few years’ travel time. It also makes one-way travel to the future possible, by going on a fast trip and returning. tonne See metric ton. trace gases Gases that form only a minute fraction of the atmosphere. Examples: ozone, carbon dioxide, and water vapor.

transportation efficiency Useful output (such as distance traveled, passengers moved, freight moved, or mass moved) per unit of fuel input. Tycho Brahe’s observations Highly accurate data on planetary positions that disproved both Ptolemy’s and Copernicus’s theories. UFO Unidentified object in the sky. See also UFO beliefs. UFO beliefs Two popular beliefs about unidentified flying

objects are that (1) UFOs are visitations by aliens today and (2) aliens visited Earth within the past few thousand years. There is no evidence to support either belief; scientists overwhelmingly reject them as pseudoscientific and false. ultraviolet radiation Radiation created by higher-energy motions of electrons in atoms that have sufficient energy to damage biological matter, can cause mutations and cancers, and is not visible to the human eye. Higher-energy ultraviolet is one type of ionizing radiation. uncertainty See quantum uncertainty. uncertainty principle Every material particle has an inherent uncertainty in position, ¢x, and in velocity. Although either ¢x or ¢y can take on any value, the two are related through the fact that their product must approximately equal h/m. See also quantum uncertainty. uniform circular motion Motion in a circle at an unchanging or uniform speed. unit A standard, relative to which a quantity such as the length of a table or the weight of a rock, is measured. For example, you might measure length in inches, feet, or miles, or in the metric system in centimeters, meters, or kilometers. u-quark See strong force. uranium enrichment The process of increasing the proportion 235 U of to 238U in natural uranium. vacuum

A region that contains no matter (no material particles). According to quantum field theory, fields exist even in vacuum. Since these fields are quantized, there is some probability that field quanta—photons, or particle-antiparticle pairs—will pop into and out of existence, even in vacuum. Furthermore, quantum uncertainties allow the energy present at any point in vacuum to undergo random energy fluctuations around its longterm average value. velocity The combined instantaneous speed and direction of motion. visible light Detectable by the human eye. Created by lowerenergy motions of electrons in atoms. Colors are due to different wavelengths, ranging from red (longest) to violet (shortest). volt The measurement unit for voltage. voltage A battery’s voltage is a measure of the amount of electrical energy the battery gives to each electron as the electron passes through the battery. Measured in units called “volts.” W + and W - particles See electroweak force. warmth, microscopic interpretation of See microscopic inter-

pretation of warmth. warped space See gravity and warped space. watt See power. wave A disturbance that travels through a medium and that

transfers energy without transferring matter.

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GLOSSARY wave interference

The effects that occur when two waves of the same type are present at the same time and place. Interference can be either constructive or destructive, depending on whether the two waves reinforce or cancel each other. wave packet A matter field for a single material particle, moving through space and spread out over only a limited distance, ¢x. wave theory of matter Every type of material particle, such as electrons or protons, has a wave associated with it. It’s wavelength is h/mv, where m and v are the mass and velocity of the particle. These waves are called matter waves. See also quantum theory of matter. wavelength The distance from any point to the next similar point along a continuous wave. wavespeed The speed at which a disturbance (a wave) moves through a medium. weak force One of the four fundamental forces; a nuclear force that plays a role in radioactive beta decay. weapons of mass destruction Nuclear, chemical, or biological weapons. weight The weight of an object is the net gravitational force exerted on it by all other objects. weight versus mass An object’s weight is the force on it due to gravity, whereas its mass is its quantity of inertia. Weight is measured in newtons (or pounds); mass is measured in kilograms. An object’s weight depends on its environment, but an object’s mass

14

is the same everywhere. For example, a kilogram has a mass of 1 kilogram regardless of whether it is on Earth or on the moon, but its weight is about 10 N (or 2.2 pounds) on Earth and only 1.6 N (or 0.36 pounds) on the moon. weightlessness Bodies are (nearly) weightless only when they are far from all other bodies. Bodies in orbit around Earth are not weightless, but they seem weightless because they are falling freely (i.e., gravity is the only force acting on them) around Earth. white dwarf An Earth-sized, compact star. When stars having about the sun’s mass run out of fuel, they flare up and then collapse to become white dwarfs. wind energy The kinetic energy of moving air. Wind turbines capture this energy and convert it to electricity. wind turbine Device for capturing wind energy and converting it to electricity. work Object A does work on object B if A exerts a force on B while B moves in the direction of that force. Unit: newton-meter (N # m) = joule (J). work–energy principle Work is an energy transfer. X-rays Created by the highest-energy motions of electrons in atoms. X-rays are one type of ionizing radiation. Z particle

See electroweak force.

The Way of Science Experience and Reason

From Chapter 1 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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The Way of Science Experience and Reason

I believe that ideas such as absolute certitude, absolute exactness, final truth, etc., are figments of the imagination which should not be admissible in any field of science.... This loosening of thinking seems to me to be the greatest blessing which modern science has given us. For the belief in a single truth and in being the possessor thereof is the root cause of all evil in the world. Max Born, Physicist

1 STARDUST: AN INVITATION TO SCIENCE

W

e came from the stars. We are made of atoms created and blown into space by ancient stars, a fact that’s only one strand in a network connecting us with the rest of the universe. Science—observing and understanding the natural world—is a path toward embracing that network. An expanding awareness of nature is discernible in the long history of life on Earth. There is good reason to believe that our planet formed about 5 billion years ago and that the earliest simple living organisms formed nearly 4 billion years ago. Since then, organisms have evolved biologically to interact with their environment in increasingly complex ways. Looked at from the human perspective (an amoeba might look at it differently), humankind is the latest in a sequence of increasingly aware biological organisms. We could even say that through biological evolution, the universe has become more aware of itself. Education and science can be viewed as an extension of this process. And you, as you learn about the universe, are part of that process of expanding awareness. Albert Einstein spoke of this widening circle of awareness when he wrote: A human being is a part of the whole, called by us “Universe,” a part limited in time and space. He experiences himself, his thoughts and feelings, as something separated from the rest—a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.

I hope that Physics: Concepts & Connections will help you discover many links between you and the universe. In writing this text, my constant criterion has been “Is this material relevant to readers who want to participate fully in our

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The Way of Science

science-based culture but who won’t necessarily use science in their professional lives?” I’ve tried to use language that’s meaningful to literate nonscientists. There are no extraneous technical terms and no extraneous mathematics—in particular, no algebra. The text does, however, make wide use of numbers, proportionalities, graphs, and numerical estimates because quantitative tools are essential to meaningful communication today. Literate people must also be numerate. I’ve discussed one reason for learning science: expanded awareness. A second reason is to develop social values appropriate to the scientific age. Take a moment to list a dozen problems that are important to the nation or the world. A typical list might include population growth, poverty, crime, species destruction, illiteracy, global warming, urban decay, drugs, war, air pollution, AIDS, and famine. Every one of these problems has a science component. Now try listing solutions to these problems. A typical list might include birth control, economic growth, education, sustainable farming, democracy, international law, environmental protection, disease control, better government, rational use of energy, more understanding among people, and concern for the environment. All of these have a science component. The problems and the solutions of our times are bound up with science and its close relative, technology.1 That’s why we call this the scientific age. To solve these problems, the world needs your help. We dare not simply entrust these critical issues entirely to experts or governments. In his book Of a Fire on the Moon, about humankind’s first venture to the moon, novelist and journalist Norman Mailer wrote pessimistically:

For any man to abdicate an interest in science is to walk with open eyes toward slavery. Jacob Bronowski, Philosopher-Scientist

Scientific activity is one of the main features of the contemporary world and, perhaps more than any other, distinguishes our times from earlier centuries. Science for all Americans, A Report to the American Association for the Advancement of Science

The [twentieth] century would create death, devastation and pollution as never before. Yet the century was now attached to the idea that man must take his conception of life out to the stars.... A century devoted to the rationality of technique was also a century so irrational as to open in every mind the real possibility of global destruction.... So it was a century which moved with the most magnificent display of power into directions it could not comprehend. The itch was to accelerate—the metaphysical direction unknown.

If we are to resolve today’s problems, we must find our metaphysical direction in this scientific age. You use the power of science daily when you switch on a light, a television set, an automobile, or a computer. Such devices have powerful effects on the world, both good (light to read by) and bad (pollution from electric-generating plants). The classic moral dilemma of the scientific age—a dilemma symbolized, for example, in Mary Shelley’s nineteenth-century novel Frankenstein—is the problem of understanding and dealing responsibly with these powerful technologies. To accept technology’s power without also accepting the responsibility to use that power wisely is to invite death, devastation, and pollution—the monster’s retaliation against its maker. I will focus on that part of science called physics, along with its human connections. You have heard of most of the major sciences: biology, geology, chemistry, astronomy, physics, and others. When people ask me “What is physics?” I like to pick up something and drop it. Most things fall when you

The dangers that face the world can, every one of them, be traced back to science. The salvations that may save the world will, every one of them, be traced back to science. Isaac Asimov, Scientist and Writer

We need people who can see straight ahead and deep into the problems. Those are the experts. But we also need peripheral vision and experts are generally not very good at providing peripheral vision. Alvin Toffler, Writer and Futurist

1

Technology is the application of science to achieve useful human goals. This text often uses the single word science to refer to science and technology.

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The Way of Science

drop them.2 You can drop a rock, a frog, a cabbage, or a king and they all fall. Physics is the study of phenomena that, like falling, are universal. Geologists study rocks and Earth’s structure, biologists study frogs and other living organisms, while physicists study the general principles obeyed by rocks and frogs and everything else. How does science operate? This is the crucial question for us to answer if we are to understand and cope with our scientific age. In most of the remainder of this chapter, I’ll answer this question by means of a significant case study: the early history of astronomy. This begins in Section 2 with commonsense conclusions about the night sky. The next three sections present three theories about the way the heavens are organized: the ancient Earth-centered theory, Copernicus’s sun-centered theory, and Kepler’s sun-focused theory. Section 6 discusses what this history teaches us about science, and Section 7 looks at the cultural implications of all of this. Finally, Section 8 studies fake science or “pseudoscience,” focusing on three important examples.

2 OBSERVING THE NIGHT SKY Teach me your mood, O patient stars! Who climb each night the ancient sky, Leaving no space, no shade, no scars, No trace of age, no fear to die. Ralph Waldo Emerson, Poet

It’s hard to see a forest when you’re standing in the middle of the trees. In the same way, it’s hard to fit science and technology into perspective because our culture is so immersed in them. This text’s most important theme is the study of the nature of science itself.3 How does it operate? What are its values? How valid are its conclusions? You will see that science is more a path, a way of learning, than it is a body of knowledge. The “scientific method” or, as I will call it, the scientific process, is often described as several activities that scientists sometimes practice: observing, hypothesizing, testing, and so forth. But such a cookbook prescription doesn’t capture how science works in real life. In fact, you use aspects of the scientific process whenever you use your own experience to reason through a problem. Science is simply a careful application of experience (often called observation and experimentation) and reason (often called hypothesis, theory, principle, and scientific law) to answer questions. For perspective on how science really operates, we’ll study a historical example: the early history of astronomy. Astronomy, the scientific study of the stars and other objects in space, has usually been closely associated with physics. The starry sky seems a more perfect place than our daily world. Life is full of clatter, the stars are serene: life is brief, the stars are forever. It is not surprising that ancient priests looked to the stars. Here was timeless knowledge. And so for at least as long as there have been records to tell the story, we have looked to the sky for the time to plant and to reap, for omens of war and peace, for life’s meaning, and for our gods (Figure 1). Although astrology, the belief that the positions of the stars significantly influence human affairs, has been a discredited superstition for two centuries, human fascination with the stars may be greater today than ever (Figure 2). In these high-technology times, we sometimes fail to see the stars. On some clear night, get away from city lights and take an hour or two to track the stars across the sky. If you’re in the Northern Hemisphere, find the moon, the Big Dipper, the North Star, any group of stars on the eastern horizon, and a group of stars on the western horizon 2 3

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There are exceptions: helium balloons, for example. Besides this theme, three others reappear throughout the text: modern physics and its significance, the social impacts of physics, and energy.

UPI/Corbis-Bettmann

The Way of Science

Figure 1

Four thousand-year-old testimony to our reverence for the stars: the remains of Stonehenge, in England. Humans hauled the huge stones for more than 200 miles to make these monuments. These stones are the remains of a much larger structure used for religious purposes and to predict astronomical events, particularly solstices (longest and shortest days) and equinoxes (equal-length days and nights). Stonehenge perhaps also predicted eclipses of the moon, an impressive feat for people who did not use writing. Eclipses occur in an irregular and apparently random pattern that repeats itself only over a 56-year cycle. Even to have been aware that a repeated pattern exists required enormous dedication and attention to detail.

(Figure 3). Observe all of these every 15 minutes for one or more hours. What happens? You should be able to see that the moon and stars move westward, that stars rise in the east and set in the west, that different stars maintain their positions relative to one another while moving as a group across the sky, that the North Star remains fixed, and that stars near the North Star move in circles around the North Star (Figure 4). There are several small and unusually bright starlike objects that do not keep pace with the stars. If you observe them for a week or more, you’ll see that they slowly shift their positions relative to the stars. These objects are called planets (wanderers in Greek). Five planets are visible without a telescope. The moon and the sun also move at a different pace from the stars. From such observations, most people would conclude that the stars, sun, moon, and planets travel in circles around Earth, with their axis of rotation fixed in the direction of the North Star. Figure 4 is rather convincing evidence of this notion. This is the conclusion most observers drew centuries ago, and it’s surely the conclusion that observers draw today unless they learn differently in school. Such observations and conclusions are typical of science’s two main processes: observation and rational thought. Science is not really different from a lot of other human endeavors. Whenever you observe your surroundings and develop ideas based on what you observe, you are acting scientifically.

North Star

Little Dipper

Big Dipper

Figure 3

Look for these two constellations and the North Star in the northern night sky.

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The Way of Science Figure 2

NASA/Johnson Space Center

Our fascination with the stars seems greater than ever. (a) The Hubble Space Telescope, launched into space in 1990. For some of its astonishing photographs, see Figure 24.

European Southern Observatory

(b) The European Southern Observatory’s four-telescope array, known as the “Very Large Telescope,” in Chile. Each telescope contains a near-perfect mirror eight meters in diameter. The telescopes are connected by “light pipes” that allow them to coordinate their four different views of a single object and use waveinterference effects to greatly improve the resolution of the image. The array saw “first light” in 1998. (b)

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The Way of Science Figure 2 (continued)

ICRR Institute for Cosmic Ray Research

NASA Headquarters

(c) Many telescopes receive nonoptical signals from space. This large radio telescope in Arecibo, Puerto Rico, receives radio signals emitted by objects that pass overhead as Earth rotates.

(d) The Super-Kamiokande underground neutrino detector, or neutrino telescope, being filled up with water. Thousands of photomultiplier tubes surround the inside of this tank of pure water, ready to record light when neutrinos from the sun or from distant exploding stars interact with atoms in the water. Neutrinos are subatomic particles that travel through space and also travel nearly uninhibited through objects such as Earth at nearly the speed of light. Neutrinos can enter the detector and be recorded from any direction: top, bottom (after coming through the entire Earth), or sides.

(d)

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National Optical Astronomy Observatories

The Way of Science

Figure 4

Time-exposed photograph showing the “star trails” near the North Star. Observations such as this appear to provide convincing evidence that the stars move in circles around Earth.

3 ANCIENT GREEK THEORIES: AN EARTH-CENTERED UNIVERSE

All things are numbers. Pythagoras, Sixth Century BCE

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At least as early as 3000 BCE, people were aware of the differing motions of the stars, sun, moon, and the five planets then known. Beginning around 500 BCE, a few Greeks sought a new kind of understanding of these motions. They desired to go beyond the observed facts, to grasp how the system worked. Figure 5 indicates the early Greek concept of the cosmic architecture. In agreement with the observations described above, the figure shows the heavenly objects circling a motionless Earth. Because the stars all keep pace with one another, the Greeks supposed that they all were attached to the inside surface of a single transparent (invisible) spherical shell centered at Earth’s center and rotating around Earth once a day, carrying the stars with it. The Greeks imagined that each of the other seven objects—sun, moon, and five visible planets—was attached to this transparent spherical shell centered on Earth, one sphere for each object. Each of the seven spheres rotated at an unchanging, or uniform, rate around Earth roughly once each day. These spheres rotated at slightly different rates about the same axis through Earth’s center. A philosophical–mathematical–religious group led by Pythagoras developed this tentative theory or hypothesis. These Pythagoreans formed a secretive cult that believed passionately in the importance of abstract ideas. An idea is, in a sense, eternal. A real table, for example, eventually rots and turns to dust, but the idea of

The Way of Science Sphere of the stars

Figure 5

The earliest Greek conception, around 500 BCE, of the layout of the universe.

Jupiter

Mars

Moon Earth

Mercury

Venus

Sun Saturn

“table” or “tableness” seems eternal. Pythagoras believed that the most perfect ideas were mathematical because they could be stated so precisely yet abstractly. The idea of a table is rather imprecise—a flat rock might be considered a table, or it might be just a rock. But mathematical ideas like a straight line, a circle, or the number 5 were precise, pure. For example, a circle is all of the points on a flat surface that are at the same distance from some fixed point on the surface. CONCEPT CHECK 1 To check your understanding of the preceding definition of a circle, try answering this multiple-choice question: The “fixed point” and the “distance” referred to in the definition are known, respectively, as the (a) center and diameter; (b) axis and radius; (c) sine and cosine; (d) center and radius; (e) center and hypotenuse.

Although this definition is precise, if you draw a circle that follows this definition (Figure 6), there will always be imperfections. Indeed, the Pythagoreans Figure 6

If you try to draw a circle, there will always be imperfections.

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The Way of Science

Let no one without geometry enter here. Inscription over the Entrance to Plato’s Academy, Fourth Century BCE

believed that it was the idea of a circle, rather than any particular representation of it, that was pure and eternal. These mathematical mystics discovered how to describe many features of the natural world by mathematical ideas. The famous “Pythagorean theorem” is one example, and the simple numerical relationships between tones in the common musical intervals is another.4 They believed that the universe is based on mathematical principles or “harmonies” analogous to the numerical relationships between the common musical intervals, and that when one studies mathematics one studies the mind of God. It isn’t surprising, then, that the Pythagoreans sought a beautiful geometric scheme for the heavens. And what geometrical forms could be more fitting for the stars than the sphere and the circle? After all, the sphere is the only perfectly symmetric (the same from all vantage points) shape in space, and the circle is the only perfectly symmetric shape on a flat surface. And as befits the timeless stars, circular paths have no beginning or end. Other Greeks regarded all of this suspiciously. The Pythagoreans were persecuted and eventually banished. But their thinking had a deep influence on subsequent Greek philosophers such as Plato and Aristotle, and on Western civilization. In line with their picture of a universe made of eight transparent Earth-centered spheres within spheres, these early Greeks had the startling notion that Earth itself was a sphere residing motionless at the center of the transparent spheres. The Pythagorean concept of a spherical Earth, although not their idea of a motionless Earth, survives to this day. How do we know that Earth is round? Science is based on observable evidence. So scientists are always skeptical, always asking, “How do we know?” I will frequently ask this question. How do we know that Earth is spherical rather than flat? The evidence is fairly direct today (Figure 7), but what evidence might the ancient Greeks have had? Take a minute, to think about this. (This is a minute, for thinking.) The Greek philosopher Aristotle, living two centuries after Pythagoras, stressed the importance of evidence. He gave many good observational reasons to believe that Earth is spherical rather than flat. For one thing, ships sink little by little below the horizon as they go out to sea (Figure 8). For a second thing, Greek travelers reported that in northern lands the noontime sun is lower in the sky. For a third, the shadow cast by Earth on the moon, as observed during an eclipse of the moon, is the shape that would be expected if both Earth and the moon were spherical.

But there was a problem. Because the spheres rotated uniformly, the transparent spheres hypothesis predicted that each planet moved at a uniform rate around Earth. But careful observation showed that they do not. Instead, their rate of rotation, as seen from Earth, changes. Figure 9 diagrams this effect for a single planet such as Mars. The diagram is drawn relative to the background stars, so it does not show the nightly rotation of Mars and the stars. Relative to the stars, Mars generally moves from west to east, but at a variable rate. Occasionally, Mars even changes directions and moves east to west relative to the stars, a phenomenon known as retrograde motion. 4

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The Pythagorean theorem states that in any triangle having a 90-degree or right angle, if you draw three squares, each one based on one of the triangle’s three sides, the sum of the areas of the two smaller squares will equal the area of the larger square. As an example of the relationships between musical tones, if you create a musical tone by plucking a string and then precisely halve the string’s length and pluck it again, the two notes you create will be exactly one octave apart. Other simple string ratios, such as 3 to 2 or 4 to 3, produce the other musical intervals that sound harmonious.

The Way of Science Figure 7

NASA Headquarters

A whole-world view showing Africa and Saudi Arabia taken 7 December 1972 as Apollo 17 left Earth’s orbit for the moon. The cultural impact of photos like this, showing Earth as a single, freely moving ball in space, may be among the space program’s most important benefits.

Horizon

The Greek philosopher Plato, convinced that an elegant mathematical reality lay behind the heavenly motions, challenged his students with the problem of finding a geometric scheme that would explain the observed motions. They constructed a hypothesis similar to Pythagoras’s but far more elaborate, involving multiple transparent spheres for each planet. One Greek thinker, Aristarchus, proposed that the sun and not Earth was at rest at the center of the universe, that Earth and the five planets circled the sun, and that Earth spun on its axis. It was a radical hypothesis, and few astronomers took it seriously because it seemed absurd for several reasons: Earth seems nothing like the heavens, so how could Earth be a planet like the heavenly planets? It seems absurd to believe that Earth moves. It’s too big! What immense force could be pushing it to keep it moving? If it does move, it seems that objects such as birds and clouds that are not attached to the ground should be left behind. If Earth spins on its axis, objects should be hurled off, just as a stone is hurled from a rotating sling. These things were not observed, and so for reasons that made sense at the time, Greeks rejected Aristarchus’s hypothesis. It would be 2000 years before a sun-centered hypothesis would again be considered. Another problem arose. The Greeks noticed that during a planet’s retrograde motion it appeared brighter than at other times, as though it were closer to Earth during this time. Yet Plato’s hypothesis, with each planet on an Earth-centered sphere, implied that each planet maintained a fixed distance from Earth. *

* *

Jan Oct

Feb East

*

Sep

*

*

*

Aug Dec

Nov *

*

Jul

Jun *

*

*

* * West

Earth

Figure 8

Evidence that Earth’s surface is spherical. As a ship sails out to sea, an observer on shore sees it sink little by little below the horizon.

Figure 9

The motion of a planet such as Mars, relative to the background stars. Relative to the stars, Mars usually moves from west to east. In this illustration, Mars moves more slowly during July–August than it does during June–July. It slows to a stop by October and then reverses direction during October–December and regains its normal direction during December–February.

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The Way of Science For its subtlety, flexibility, complexity, and power the epicycle–deferent technique... has no parallel in the history of science until quite recent times. In its most developed form the system of compounded circles was an astounding achievement. Thomas Kuhn, Historian and Philosopher of Science

To explain the varying brightness of the planets, the Greeks tried something rather different. Instead of moving on multiple spheres, each planet now moved around Earth in a circle within a circle. As shown in Figure 10, a planet such as Mars moved uniformly around a circle whose center was on another circle that was centered on Earth. The small outer circle was called the planet’s “epicycle,” and the inner circle centered on Earth was called the planet’s “deferent.” The center of the epicycle moved uniformly along the deferent, so that Mars moved in two circles at the same time. This produced a loop-the-loop orbit for each planet (Figure 10). In agreement with observation, the theory predicted that there would be occasional periods of retrograde motion (on the inside of the loops) and that the planet would be closest to Earth during retrograde motion and so should appear brightest. It was a satisfying picture, and it explained the observations. It was a good theory. You’ve probably noticed that I’m using the word theory here rather than hypothesis. Whereas a hypothesis is a tentative scientific idea without a lot of evidence, a theory is a scientific idea that is well-confirmed by evidence. Figure 11 pictures this theory greatly simplified. This theory was finally refined and summarized around 100 CE by Ptolemy, antiquity’s greatest astronomer (Figure 12). In order to agree with the known observations, Ptolemy introduced two new ideas: the displacement or “eccentricity” of the centers and the “equant point” from which the motion appears uniform.5 The details of these are not crucial here. To agree with the observations, each planet needed lots of epicycles—more than 80. Thirteenth-century Spanish king Alfonso X commented that “if the Lord Almighty had consulted me before embarking upon the creation, I should have recommended something simpler.”

Path followed by Mars

Earth Deferent Epicycle

Mars

Figure 10

The orbit of Mars around Earth, according to the epicycle theory.

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None of these ideas were original with Ptolemy, but he was the first to put them all together in a consistent and quantitatively correct theory.

The Way of Science Figure 11 Sphere of the stars

Ptolemy’s Earth-centered epicycle theory (around 100 CE) of the layout of the universe, according to which the five visible planets move on epicycles around Earth. The epicycles of the two innermost planets, Mercury and Venus, are centered on the line joining Earth to the sun.

Saturn Mars

Sun

Jupiter Venus Mercury

Moon Earth

How did we know planetary positions before there were telescopes? Ptolemy checked this elaborate theory with many quantitative (numerical) measurements of the heavens. Telescopes hadn’t been invented yet, so the measuring devices were long sighting rods with a scale to measure the angular position of a planet. The sighting devices were accurate to within about 0.2 degrees (recall that there are 360 angular degrees in a complete circle). To within this accuracy, Ptolemy’s theory agreed with all observations of the stars, sun, moon, and five known planets. It survived, with modifications, for 15 centuries and was used by navigators, astronomers, and mystics such as astrologers. Not bad.

CONCEPT CHECK 3 Venus often appears as the morning star (the last-seen “star” near the rising sun) or the evening star. Ptolemy’s explanation for this observation would be that (a) Venus’s orbit around the sun lies close to the sun; (b) the center of Venus’s epicycle lies on the line between Earth and the sun; (c) Venus and Mercury orbit the sun while the other planets orbit Earth; (d) Venus is attracted to the sun’s manly appearance. (Hint: See Figure 11.)

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CONCEPT CHECK 2 The ancient Greeks believed that the stars and other astronomical objects shine by means of their own light. Can they have believed this of every astronomical object that can be seen with the naked eye? (a) Yes. (b) No.

Figure 12

The ancient astronomer Ptolemy, 85–165 CE. Using epicycles and many other theoretical devices, he perfected the Earth-centered theory of the layout of the universe.

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4 COPERNICUS’S THEORY: A SUN-CENTERED UNIVERSE

American Institute of Physics/ Emilio Segre Visual Archives Figure 13

Polish astronomer Nicolaus Copernicus, 1473–1543. Finding Ptolemy’s system to be “neither sufficiently absolute nor sufficiently pleasing to the mind,” he devised a simpler theory. Copernicus’s theory placed the sun at the center of the universe, with Earth moving around it. The odd idea that Earth moved and was a planet like the other planets met with much resistance because it conflicted with the intuitive notion that Earth is at rest at the center of things and with prevailing philosophies. In the centre of everything rules the sun; for who in this most beautiful temple could place this luminary at another or better place whence it can light up the whole at once?... In this arrangement we thus find an admirable harmony of the world, and a constant harmonious connection between the motion and the size of the orbits as could not be found otherwise. Copernicus

We live on a great round wonder rolling through space. Walt Whitman, American Poet

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The birth year of modern science is often taken as 1543 CE. In that year, an old man in Poland, on his deathbed, signed the first printed copy of his life’s work, On the Revolutions of the Heavenly Spheres. Nicolaus Copernicus (Figure 13)— astronomer, mathematician, linguist, physician, lawyer, politician, economist, and canon in a Catholic cathedral—had kept the manuscript locked up for 30 years, fearing the criticism it would unleash. With this book, the sun was setting on the medieval world. During the Middle Ages (about 500 to 1500 CE in Europe), philosophers such as St. Augustine and Thomas Aquinas had linked Greek thought, including Ptolemy’s astronomy, to Christian theology. But in 1543 the times were changing. During the past century the intellectual and artistic flowering known as the Renaissance had germinated in Italy and spread to all of Europe. Martin Luther had led a frontal assault on the Catholic church’s authority. Christopher Columbus had made a memorable voyage. These trends had a liberating effect on thoughtful minds. Copernicus and others were enthused by the new art, new religious thought, and new explorations. As a result, Copernicus was uncomfortable with Ptolemy’s theory. Not that Copernicus was a revolutionary. Quite the contrary: Copernicus objected to Ptolemy’s theory on the grounds that with its many epicycles, eccentrics, and equants, Ptolemy had strayed far from ancient Pythagorean ideals. Ptolemy’s system lacked the simple elegance that scientists have always sought. Here’s how Copernicus put it: The planetary theories of Ptolemy and most other astronomers, although consistent with the data, seemed to present no small difficulty. For these theories were not adequate unless certain equants were also conceived; it then appeared that a planet moved with uniform motion neither on its deferent nor about the center of its epicycle. Hence a system of this sort seemed neither sufficiently absolute nor sufficiently pleasing to the mind. Having become aware of these defects, I often considered whether there could perhaps be found a more reasonable arrangement of circles, from which every apparent inequality would be derived and in which everything would move uniformly about its proper center, as the rule of absolute motion requires.

Note that like all thinkers since Pythagoras, Copernicus believed that the heavenly motions were circular and uniform. He stated adamantly that “it would be unworthy to suppose such a thing [as noncircular motion] in a Creation constituted in the best possible way.” As a Renaissance man. Copernicus adopted a broad outlook. Just as Renaissance artists looked beyond Christian art and as Columbus looked beyond Europe, Copernicus looked beyond Earth itself to imagine it as an object in space, an object that he believed to be similar to other objects in space. To Copernicus, and to science since Copernicus, the ancient idea that the universe is centered on Earth seemed narrow-minded, provincial. So Copernicus asked a broader question than had been asked before: What is the most elegant geometric scheme for the motion of the stars, sun, moon, five observed planets, and Earth that will fit the known measurements of the heavens? Given this change of focus, Copernicus soon discovered “a more reasonable arrangement of circles” in which the planets and Earth move in uniform circular motion around the sun and only the moon circles Earth. Figure 14 shows Copernicus’s theory. Copernicus obtained the east-to-west daily motion of the

The Way of Science Sphere of the stars

Figure 14

Copernicus’s sun-centered theory of the layout of the universe, 1543 CE. The diagram is simplified; the planets all move on epicycles, similar to those in Ptolemy’s theory, but here there are far fewer epicycles, and no equants.

Jupiter Mars

Mercury

Venus Sun

Moon Earth

Saturn

stars, sun, moon, and planets by allowing Earth to spin from west to east, rather than by allowing the sphere of the stars to rotate from east to west. How do we know that Earth and the other planets go around the sun? There were still no telescopes in Copernicus’s day, and data were gathered with star-sighting devices. With properly chosen radii, rotation rates, and eccentrics for the planetary orbits, Copernicus obtained quantitative agreement with the data. His theory explained many things, such as retrograde motion (Figure 15 and Concept Check 5). But as Copernicus admitted, Ptolemy’s theory was also “consistent with the data.” Both theories agreed with the data. There were good objections to the new theory, like those that had earlier confronted Aristarchus’s hypothesis. Being so large, how can Earth move? What keeps it moving? Why aren’t birds and clouds left behind? Why aren’t objects hurled off Earth? Copernicus didn’t have an answer. Instead, he pointed out that such problems loomed even larger for Ptolemy’s great spinning sphere of stars than for Copernicus’s smaller spinning Earth. In making this argument, Copernicus was assuming that the stars were subject to natural laws like those operating on Earth. Nobody had looked at it in this way before. The objections to Copernicus’s theory were not answered for more than a century, when Isaac Newton and others devised a radically new view of motion. In fact, Newton’s physics arose partly because of these questions.

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The Way of Science Figure 15

The Copernican theory’s explanation of retrograde motion. As Earth passes another planet, such as Mars, the other planet appears to move backward as seen against the background stars, because of the rotation of the Earth-based observer’s line of sight. Using this figure, you can demonstrate this by following the instructions in Concept Check 5. A similar effect occurs when you pass a car moving down a straight highway. Viewed against distant background trees and houses, the slower car appears for a few seconds to move backward, because of the rotation of your line of sight.

There is perhaps no other example in the history of thought of such dogged, obsessional persistence in error, as the circular fallacy which bedeviled astronomy for two millennia. Arthur Koestler, Twentieth-Century Writer and Historian of Science

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Mars 1 Earth

A decisive blow against Ptolemy’s theory and for a sun-centered theory did not come until Galileo introduced the telescope into astronomy, some 70 years after Copernicus’s death.6 Among other things, Galileo observed that Venus goes through phases similar to the moon’s phases (new moon, quarter moon, full moon, and the like). This means that Venus shines not by its own light but by light reflected from the sun. In Ptolemy’s theory, the center of Venus’s epicycle must be fixed on the line joining Earth to the sun (Figure 11), in order to explain the fact that Venus is never seen far from the sun. As shown in Figure 16, this means that we should never see a “full Venus” phase from Earth. On the other hand, the sun-centered theory predicts that we should see a full Venus whenever Earth and Venus are on opposite sides of the sun. Galileo observed that the phases of Venus included a full Venus.

CONCEPT CHECK 4 When you say that “the sun rises in the east,” you really mean (from the Copernican point of view) that (a) due to the sun circling around Earth, the sun begins to appear above the eastern horizon; (b) due to Earth circling around the sun, the sun begins to appear above the eastern horizon; (c) due to the sun rotating (or spinning) on its axis, the sun rotates into view above the eastern horizon; (d) Earth rotates eastward around its axis to bring the sun into view; (e) Earth rotates westward around its axis to bring the sun into view. CONCEPT CHECK 5 Figure 15 shows the positions (numbered from 1 to 9) of Earth and Mars at nine different times. As you can see, Earth is passing Mars during this time. Draw lines of sight from Earth through Mars to the background stars at each of these nine times. Based on this drawing, Mars is in retrograde motion during (a) times 4 to 6; (b) times 3 to 5; (c) times 1 to 5; (d) times 4 to 7; (e) none of the time; (f) lunchtime. CONCEPT CHECK 6 Following up on the preceding question, if Mars were viewed from the sun it would appear to move in (a) retrograde motion during times 4 to 6; (b) retrograde motion during times 1 to 5; (c) retrograde motion during times 5 to 9; (d) retrograde motion the entire time; (e) normal (forward) motion the entire time. 6

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Although Galileo did not invent the telescope, he was the first to make significant scientific use of it and the first to use it to study the heavens.

The Way of Science Figure 16

Sun

Ptolemy’s theory predicted that an Earth-based observer would never see a “full” phase of Venus because Venus’s epicycle lay between Earth and the sun. Copernicus’s theory predicted that a nearly full Venus could be seen whenever Venus was on the far side of the sun in its orbit around the sun, as it is in Figure 14. Galileo observed that the phases of Venus included a full Venus, thereby disproving Ptolemy’s theory.

Sun's orbit

Venus, in different phases

Venus's epicycle Earth

5 KEPLER’S THEORY: A SUN-FOCUSED UNIVERSE

How do we know more accurate planetary positions? Brahe’s elegant sighting devices (Figure 18) were so accurate that his data are sometimes used today. Before Brahe, the best measurements had inaccuracies (possible errors) of at least 10 arcminutes (an arc-minute is 1/60th of 1 degree). Brahe’s measurements had inaccuracies of only 2 arc-minutes. When Brahe began his project, there were two competing theories of the universe: Ptolemy’s and Copernicus’s. Despite their great dissimilarity, both theories agreed with the data known at that time. Would Brahe’s measurements be able to distinguish between them and so determine which one was correct? For the next 20 years, Brahe cataloged accurate data on the positions of the sun, moon, and planets. It soon became obvious that both theories disagreed with Brahe’s observations by several arc-minutes!

Just 18 months before Brahe died, the 29-year-old Johannes Kepler (Figure 19) managed to gain employment with the famous astronomer. Kepler was born to a ne’er-do-well father who abandoned his family and to a mother who was later tried for being a witch. Furthermore, “The boy was precocious above all in illness, being beset by small pox, headaches, boils, rashes, worms, piles, the mange, and worst of all for an aspiring astronomer, defective eyesight. His visual problems included double vision in one eye and myopia in both eyes.”7 The hardships of his youth seem to have toughened Kepler for the challenges to come. A philosopher, mathematician, astronomer, and astrologer, Kepler was devoted to the Pythagorean notion

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Tycho Brahe (Figure 17), born three years after the death of Copernicus, loved the night sky. A skillful fund-raiser, he obtained financing from the king of Denmark to build a large astronomical observatory.

Figure 17

Tycho Brahe, 1546–1601. By making measurements of the planetary positions that were five times more accurate than were previous measurements, he overthrew two theories of the architecture of the heavens.

Michael J. Crowe, Theories of the World from Antiquity to the Copernican Revolution (New York: Dover Publications, Inc., 1990).

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American Institute of Physics/Emilio Segre Visual Archives

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Figure 18

Figure 19

An instrument that Brahe used for measuring the angular altitudes of the planets. The two wooden arms are joined together. The lower arm is placed horizontally, as determined by the string and weight hanging vertically from the upper end of the scale. The upper arm is raised until the arm points toward the planet. The planet’s angular position above the horizontal is then read from the graduated scale.

Johannes Kepler, 1571–1630. He contemplated the Copernican theory with “incredible and ravishing delight,” although the scientific facts compiled by Brahe forced him to alter that theory in ways that would have displeased Copernicus. This scientist/mystic has been described by philosopher and novelist Arthur Koestler as the watershed between medieval science and modern science.

of elegant mathematical order, and he harbored another devotion that could only be described as sun worship. Regarding mathematical order, Kepler proclaimed: Why waste words? Geometry existed before the Creation, is coeternal with the mind of God, is God Himself; geometry provided God with a model for the Creation.

Regarding the sun: The sun in the middle of the moving stars, himself at rest and yet the source of motion, carries the image of God the Father and Creator. He distributes his motive force through a medium which contains the moving bodies even as the Father creates through the Holy Ghost.

Given these beliefs, it’s not surprising that Kepler was the first astronomer to openly support the Copernican system, a theory whose beauty he contemplated with “incredible and ravishing delight.” Kepler’s words and thoughts convey the scientist’s passion to understand the universe. Although he was a convinced Copernican, Kepler found that Brahe’s data for Mars were impossible to fit to Copernicus’s theory, even though Kepler tried reintroducing the equant device that Copernicus had so despised. The calculations were tedious. Kepler spent four years on this project, filling 900 notebook pages with finely handwritten calculations. But the Copernican orbit coming closest to Brahe’s data for Mars was still off by 8 arc-minutes. Before Brahe, this could have been ascribed to observational error. But Kepler, toughened by the confrontation with his master’s hard-won data, knew that neither observational error nor further tinkering

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would make uniform circular motion agree with the observed facts. Kepler rejected the Copernican theory. A less passionate person would have given up. Worse yet, a less tough-minded person would have found a way to fudge the data to get them to agree with the Copernican preconceptions that Kepler had believed most of his life. But the everfervent Kepler, writing “on this 8-minute discrepancy, I will yet build a theory of the universe,” began anew. He began studying planetary motions that, for the first time in history, were not based on combinations of uniform circular motions. Copernicus, and all previous astronomers, would have been horrified. Sixteen years later, Kepler finally had his answer: The planets don’t move in circles. They move, instead, in ellipses. He was able to fit an ellipse to Brahe’s data and thus resolve the 8-minute discrepancy that had plagued him for so long. And the data for all the planets fit into elliptical patterns. Kepler’s theory states that rather than moving in sun-centered circles, each planet moves in a sun-focused ellipse: an ellipse having the sun at one of its two “foci.” There is nothing at the other focus. Figure 20 shows how to draw an ellipse.8 You could describe it as a squashed circle. The planetary orbits are only slightly elliptical, which is why sun-centered circles come so close to fitting the observations. The ellipse has just the kind of elegance Kepler had sought. He was elated: What sixteen years ago I urged as a thing to be sought, that for which I joined Tycho Brahe... at last I have brought to light and recognize its truth beyond my fondest expectations.... The die is cast, the book is written, to be read either now or by posterity, I care not which. It may well wait a century for a reader, as God has waited six thousand years for an observer.

Figure 20

You can draw an ellipse with the help of a loop of string and two thumbtacks. The thumbtacks represent the two foci.

CONCEPT CHECK 7 You could use the tack-and-string construction of Figure 20 to construct a circle (a) by moving the two thumbtacks far apart; (b) by adding a third thumbtack, midway between the two thumbtacks shown; (c) by placing the two thumbtacks together.

6 SCIENCE: A DIALOGUE BETWEEN NATURE AND MIND Let’s draw some conclusions from all this history. The most important conclusion is that science is based on direct experience—observation and experiment—and on rational thought to organize and understand this experience. Science’s foundation in experience and reason distinguishes it from other forms of knowledge based on belief, intuition, personal authority, or authoritative books. Although observation is the beginning of the scientific process, a catalog of observed facts does not add up to an understanding of nature, any more than a telephone book adds up to an understanding of a city. To understand—literally, to “stand beneath”—means to perceive a framework. A framework of scientific ideas is called a theory. In the development of astronomy, observations stimulated speculations that led to theories, and these theories in turn suggested new observations to check the theories and suggest new speculations. This interplay between observations and theories is the essence of science. Figure 21 illustrates this dialogue with nature. 8

I measured the skies, now the shadows I measure. Skybound was the mind, earth-bound the body rests. Kepler’s Epitaph, Composed by Kepler

Here is the exact definition: An ellipse is all the points on a flat surface for which the sum of the distances of each point on the ellipse from two fixed points (the two “foci”) is constant. The construction shown in Figure 20 follows this definition.

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The Way of Science Figure 21

How we began to learn where we are in the universe. The figure illustrates the dynamic interplay between observations and theory that is the essence of science. In science it often happens that scientists say, “You know that’s a really good argument; my position was mistaken,” and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn’t happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. Carl Sagan, Astronomer and Science Writer

A Summary of the Early History of Astronomy Observations

Typical Dates

Stars, sun, moon, and planets are moving overhead.

3000 BCE

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Pythagorean hypothesis: Earthcentered transparent spheres.

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Plato’s hypothesis: multiple transparent spheres.

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Aristarchus’s hypothesis: sun-centered circles.

Each planet moves at a varying rate; retrograde motion.

Heaven and Earth seem different; Earth seems motionless, apparently contradicting Aristarchus’s hypothesis.

200 Planets are brighter during retrograde motion. 100

Physical theory without experiment is empty. Experiment without theory is blind. Heinz Pagels, Physicist

The whole of science is nothing more than a refinement of everyday thinking.... The scientific way of forming concepts differs from that which we use in our daily life, not basically, but merely in the more precise definition of concepts and conclusions, more painstaking and systematic choice of experimental material, and greater logical economy. Albert Einstein

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Theories

Detailed quantitative measurements show need for small corrections.

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Hypothesis of Earth-centered epicycles.

Ptolemy’s theory: Earthcentered epicycles, equants.

100 CE 1500 Copernicus’s theory: suncentered circles. Brahe’s accurate measurements disprove Ptolemy’s and Copernicus’s theories. 1600

Kepler’s theory: sun-focused ellipses.

Galileo’s telescopic observations disprove Earth-centered theories.

Observation refers to the data-gathering process. A measurement is a quantitative observation, and an experiment is an observation that is designed and controlled by humans, perhaps in a laboratory. A scientific theory is a well-confirmed framework of ideas that explains what we observe. A model is a theory that can be visualized, and a principle or law is one idea within a more general theory. The word law can be misleading because it sounds so certain. As you will see, scientific ideas are never absolutely certain. Note that a theory is a well-confirmed framework of ideas. It’s a misconception to think that a scientific theory is mere guesswork, as nonscientists occasionally do when they refer to some scientific idea as “only a theory.” Theories—wellconfirmed explanations for what we observe—are what science is all about and are as certain as any idea can be in science. The correct word for a reasonable but unconfirmed scientific suggestion (or guess) is hypothesis. For example, Kepler’s first unconfirmed suggestion that the

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planets might move in elliptical orbits was a hypothesis. Once the data of Brahe and others confirmed Kepler’s suggestion, elliptical orbits took on the status of theory rather than mere hypothesis. Figure 22 shows the general form of Kepler’s theory of the solar system (the sun and its planets), extended to include all eight planets known today. This theory explains all of Brahe’s data and all preceding observations and unifies these data into a few principles such as the principle of elliptical orbits. As you can see, a theory represents an enormous simplification or reduction of many observations into a few simple ideas. But Kepler’s theory does more than describe known data. It also predicts new observations. For example, when the new planets Uranus and Neptune were discovered, Kepler’s theory predicted, correctly, that they too would move in elliptical orbits. A theory having no predictive value, which needs to be patched up to account for every new observation, isn’t worth much. For example, Ptolemy’s theory could doubtlessly be amended with enough new epicycles to make it agree with all of Brahe’s data, but the result would be a confusing mess with little predictive ability. Most importantly, Kepler’s theory suggested further developments. Isaac Newton, born a few years after Kepler’s death, built on Kepler’s theory in developing his own theories of motion and gravity. Another misconception about theories, especially if they happen to be called “laws,” is that they are absolutely certain and hence that scientific knowledge is absolute. Let’s look at history. Ptolemy’s theory correctly predicted the planetary observations, and so did Copernicus’s theory. Both were, and are, good theories for many purposes. But new, more accurate observations by Brahe contradicted both theories, opening the way for Kepler’s theory. Did Kepler, then, discover the true motion of the planets? Not necessarily. In the future, astronomers might discover that the planets have begun severely deviating from their elliptical paths, as could happen if, for example, another star passed close to our sun. It is always possible that new data will contradict any general theory. Good science is always provisional, nondogmatic. All theories dangle by the slender thread of evidence.

The great intellectual division of mankind is not along geographical or racial lines, but between those who understand and practice the experimental method and those who do not understand and do not practice it. George Sarton, Historian of Science

The aim of science is not to open the door to everlasting wisdom, but to set a limit on everlasting error. Bertolt Brecht, Playwright, in The Life of Galileo

Figure 22

The arrangement of the solar system as it is now known. Uranus and Neptune are visible only with a telescope. The orbits are elliptical, although their ellipticity is too small to be visible in this diagram. Saturn

Sun

Uranus

Mars Earth Venus Mercury

Jupiter

Neptune

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[The scientific process is] designed to counter human selfdeception. People always think they’re right, and powerful people will tend to use their authority to bolster their prestige and suppress inconvenient opposition. You try to set up the game of science so that the truth will out despite this ugly side of human nature. Steven Pinker, Harvard, Cognitive Psychologist, Author of The Modern Denial of Human Nature

In fact, today’s highly accurate observations show that the planets move along orbits that actually do deviate slightly from precise ellipses. According to Isaac Newton’s theories, Kepler’s elliptical orbits are caused by gravity acting between the sun and each planet. The main cause of the deviations from elliptical motion is gravity acting between the different planets. Interplanetary dust and many other things also cause small deviations. Nevertheless, scientists have retained Kepler’s theory because it’s a good and useful approximation. Perhaps we should describe theories as good or useful rather than true. The fact that theories are never absolutely certain is a strength, not a weakness, of science. Absolute certainty can foster dogmatism and a rigid inability to change what needs changing. Theories can be good, useful, fruitful, or compelling, but they are never certain. If a theory cannot be tested against observations, then it tells us nothing about the observable universe and is not a scientific theory at all. Scientific theories must be testable by observations that could conceivably contradict the theory. For example, a notion such as “undetectable alien creatures are living among us” is not a scientific statement, not because this notion seems odd (most scientific theories are odd), but because the creatures are said to be undetectable. Scientifically, this statement is not true and it’s not even false. Being untestable, it is outside science. Nonscientific ideas can, of course, have their own validity. “Beethoven’s music is sublime,” or “May God bless this home,” can be meaningful statements, but they lie outside science. The elegant tools of Brahe and the inspired theories of Pythagoras and Kepler show that science thrives on creativity. It is one of nature’s mysteries that these beautiful inventions actually turn out to produce a consistent picture of the universe. Scientists generally believe in the Pythagorean ideal of a universe based on simple and elegant principles. Copernicus adopted a sun-centered theory over the hallowed Earth-centered theory because it was “pleasing to the mind.” Scientists such as Kepler strove passionately to perceive such an elegant framework. When creating his theories, Einstein used to ask himself how he would have constructed the universe if he were God. The scientific process of observing and theorizing is not very different from our ways of coping with daily life. In science, as in life, we learn from experience and by thinking carefully. It’s a very human activity. To summarize: The Scientific Process Science is a process, a way of learning, rather than a set of conclusions. It is the process of using evidence (experiments and observations) and reason (hypotheses and theories that correlate the evidence) to develop testable knowledge about the natural world. This basis in evidence and reason distinguishes science from other forms of knowledge based on belief, intuition, personal authority, or authoritative books.

I will return frequently to the theme of the scientific process. CONCEPT CHECK 8 The idea that proved fruitful (or useful) for Kepler as he developed his own ideas was (a) Copernicus’s theory; (b) Aristarchus’s hypothesis; (c) Ptolemy’s theory; (d) Plato’s hypothesis; (e) Newton’s theory. Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house. Jules Henri Poincaré, Scientist and Mathematician, 1854–1912

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CONCEPT CHECK 9 William is absolutely certain of a particular scientific principle. You can conclude from this that (a) this principle is correct; (b) this principle is wrong; (c) this principle is irrelevant; (d) William is being scientific; (e) William is being unscientific; (f) William is a blithering idiot.

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7 THE COPERNICAN REVOLUTION: DAWN OF THE MODERN AGE The scientific age has its roots in two historical developments. One is the Pythagorean belief in natural harmonies, an idea that captivated Greek philosophers and then spread to Europe and the world. Its central premise is that the universe is organized in a framework of principles that can be uncovered by observation. The second is the rejection of the “geocentric illusion” that Earth is at the center of, and therefore fundamentally different from, the rest of the universe. Copernicus started this development, which is justly called the Copernican revolution. Its thrust is that, just as Earth is a planet similar to the other planets, the natural world is fundamentally the same everywhere, differing in details at different places and times but always following the same general principles. When stated in this form, you can see a kind of symmetry in the Copernican viewpoint. Symmetry is an important theme throughout science. An object is commonly said to have symmetry when it can be viewed from several perspectives and still look the same. For instance, a square can be viewed from four directions and still look the same. The thrust of the Copernican viewpoint is that, no matter where you are in the universe, the fundamental operating principles are the same there as they are here on Earth. Others began thinking along these lines. It became apparent that the sun—considered by Copernicus and Kepler to be central to the universe—was a star like the other stars. We now know that the visible stars belong to a vast revolving aggregation of some 400 billion stars spread out in the shape of a giant pizza. Our sun, one of the stars in the outreaches of this aggregation, circles the center every 200 million years. But the center of this aggregation is not at the center of the universe, either. Instead, there are hundreds of billions of other similar aggregations of stars throughout the observable universe. And according to current theories, none of these aggregations is at the center, because the universe has no center—an odd idea that is the ultimate extension of Copernican astronomy. Each of these aggregations is called a galaxy. Ours is the Milky Way galaxy. Figure 23 shows a typical galaxy, one much like our own. The cloudlike glow in the night sky that is called the Milky Way is our galaxy seen from our position

NASA Headquarters

Figure 23

The Andromeda galaxy, photographed through a telescope. This is the nearest large galaxy outside of our own Milky Way galaxy. Like our galaxy, Andromeda is made of billions of stars, each one somewhat similar to our sun, whirling around a bright star-filled center. Andromeda is nearly invisible to the unaided eye and lies far beyond the visible stars of our own galaxy. Since light takes about 2.5 million years to reach here from there, you are looking at 2.5-million-year-old history.

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Humanity has perhaps never faced a greater challenge; for by [Copernicus’s] admission [that humanity is not the center of the universe], how much else did not collapse in dust and smoke: a second paradise, a world of innocence, poetry and piety, the witness of the senses, the conviction of a religious and poetic faith...; no wonder that men had no stomach for all this, that they ranged themselves in every way against such a doctrine. Johann Wolfgang Von Goethe, Nineteenth-Century German Poet and Dramatist

within it—it’s like standing in the middle of a giant pizza and looking into the dough. The glow comes from the stars in only a small, local portion of our entire galaxy. The center of our galaxy lies far beyond the visible Milky Way, in the direction of the constellation (group of stars) known as Sagittarius. There are lots of galaxies out there. Figure 24 is a photograph of the very distant galaxies in a typical narrow speck of sky containing hundreds of galaxies. In the entire observable universe—that part of the universe from which we can receive light—there are something like 100 billion galaxies. There are about as many galaxies in the observable universe as there are stars in our Milky Way galaxy! It’s a big place. Copernicus sowed the seeds of many revolutions. Once Copernicus announced that Earth is a planet, Isaac Newton could unify the heavens and Earth in a new physics based on principles that were uniform throughout the universe. And just as Copernicus unified Earth with the other planets, Charles Darwin conceived an evolutionary biology that unified all life and included humankind as one species among many. The Copernican/Newtonian conception of natural laws that apply democratically everywhere and to all people helped to propel the political transition from medieval authority to constitutional law and democracy. The U.S. Declaration of Independence, for example, refers to the “Laws of Nature” that entitle the people to assume a separate and equal station with their former rulers. This notion that natural law applies equally to all people stems partly from the universality of Newtonian physics. Copernican astronomy was correctly perceived as revolutionary by religious and philosophical authorities. Ptolemy’s system had been developed in parallel with Earth-centered Aristotelian physics, and Aristotle’s thinking was a foundation of Catholic theology. The perfection of heaven, the imperfection of Earth, and humankind’s centrality to God’s plan for the universe were threatened by the loss of the Ptolemaic system. Seventy years after Copernicus’s death, the Catholic church pronounced his theory “false and erroneous,” “altogether opposed to Holy Scripture,” and “heretical.” Science historians believe that during this period, science and religion fell out into two noncommunicating camps that still, today, feel they are at odds with each other.

This view of distant galaxies was taken with the Hubble Space Telescope. Nearly every object in the photograph is an entire galaxy, most of them so far away that the light we see from them started on its journey about 13 billion years ago. This is around the time we believe the galaxies were formed, and only about 1 billion years after the origin of the universe. The view covers only a narrow speck of sky one-thirtieth the diameter of the full moon, and reaches deeper into space than any previously visible image. Although this view shows a very small sample of sky, it is considered a typical representative of the distribution of galaxies in space because the universe, statistically, looks the same in all directions.

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NASA Headquarters

Figure 24

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This situation is a far cry from the Pythagoreans, who considered science and religion to be the same thing. Champions of the Copernican theory were denounced and persecuted by religious authorities.9 Leaders of the Protestant Reformation were even more extreme in denouncing the new astronomy. The main Protestant objection was that the new theory ran counter to a “literal” reading of the Bible. The Bible frequently mentions a moving sun and a fixed Earth, contrary to the Copernican theory. Even before publication of the new theory, Protestant leader Martin Luther heard about Copernicus’s ideas and condemned them for contradicting the Bible. In Luther’s opinion, “The fool [Copernicus] will turn the whole science of astronomy upside down. But, as the Holy Writ declares, it was the sun and not the Earth which Joshua commanded to stand still.” It is not surprising that Copernicus, prudent by nature, withheld publication of his planetary theory until his dying day. CONCEPT CHECK 10 Which of the following represent a continuation of the basic thrust of the Copernican revolution? (a) The universe was made for humans. (b) The moon is probably made of material that split off of Earth. (c) Our sun is just one star among billions of similar stars. (d) Our Milky Way galaxy is at the center of the universe. (e) The human species is not very different biologically from the other species. (f) Our galaxy is just one among billions of similar galaxies. CONCEPT CHECK 11 The most characteristic feature of science is (a) the use of precise mathematical relations; (b) precise quantitative observations; (c) the absolute truth of the scientific laws; (d) the mutually supporting relationship between theory and observation.

8 PSEUDOSCIENCE As part of understanding what science is, we need to understand what it’s not. Because science is so widely accepted today, it has become common for all manner of charlatans to hawk their wares by alleging some scientific basis for them. Thus we’ve been treated, during the past century or so, to a proliferation of pseudoscientific claims—claims presented so as to appear scientific even though they lack supporting evidence and plausibility and therefore aren’t scientific. Typically, pseudoscientists reverse the scientific process by assuming their desired conclusion at the outset and then searching for evidence that supports that conclusion while ignoring evidence and arguments to the contrary. Using such a biased and backwards approach, it’s possible to “prove” (in the eyes of the convinced believer) anything, including some of the most absurd nonsense imaginable. Pseudoscience comes in many guises (Table 1). Although some of the supposed phenomena in Table 1 are not entirely ruled out by the evidence, there is no real scientific evidence that actually supports any of them. Although they might have other, non-scientific virtues, these beliefs lie outside of science because they have not received support from the scientific process. It’s a significant issue. Pseudoscientific views are alarmingly popular: 52% of American adults believe astrological “predictions,” 46% believe in extrasensory perception, 42% believe that people can communicate with the dead, and 35% actually believe in ghosts. One-and-a-half centuries after Darwin’s The Origin of 9

Early in 1926 [the magician] Houdini made a pilgrimage to Washington to enlist the aid of President Coolidge in his campaign “to abolish the criminal practice of spirit mediums and other charlatans who rob and cheat grief-stricken people with alleged messages.” From Houdini, by B. R. Sugar

Every science that is a science has hundreds of hard results; but search fails to turn up a single one in “parapsychology.” John A. Wheeler, Physicist

In 1984, the Vatican stated that church officials had erred in condemning Galileo and called for increased dialogue between science and religion. Then in 1992, the pope announced that the church had wrongly accused Galileo, laying the blame on seventeenth-century church authorities who interpreted the Bible too literally.

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The Way of Science Table 1 A few of the better-known pseudosciences ancient astronauts

extrasensory perception

orgone boxes

astrological birth control

Falun Gong

parapsychology

astrology

flying saucers

perpetual motion machines

Bermuda Triangle

fortune-telling

phrenology

Big Foot

ghosts

psychic surgery

channeling

holocaust denial

psychokinesis

creationism

homeopathy

pyramid power

crop circles

intelligent design

quantum mysticism

crystal healing

Kirlian aura

remote viewing

crystal power

levitation

séances

dianetics

lost continent of Atlantis

spoon bending

dowsing

Noah’s flood

Velikovsky’s colliding worlds

emotions in plants

occult chemistry

witches

extraterrestrial visitations

To the best of my knowledge there are no instances out of the hundreds of thousands of UFO reports filed since 1947—not a single one—in which many people independently and reliably report a close encounter with what is clearly an alien spacecraft. Carl Sagan, Astronomer, Physicist, Educator, and Author

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Species, 46% believe that human beings did not develop from earlier animals. And 43% believe it likely that some of the reported unidentified flying objects are really space vehicles from other civilizations. More fundamentally, pseudoscience is a kind of mind pollution. By pretending to be what it’s not (namely science), pseudoscience weakens one’s ability to think honestly and rationally. Let’s look at three typical pseudoscientific beliefs: extraterrestrial visitations, astrology, and creationism. UFOs are unidentified objects in the sky, or “unidentified flying objects.” Two UFO beliefs have gained a following in the popular media. The first is that some UFOs are visitations by contemporary aliens; the second is that aliens visited Earth in the past. The problem with these ideas is not that UFO beliefs themselves are inherently antiscientific. The problem, instead, is in the nonscientific way these beliefs are supported. Let’s examine the evidence. There have been thousands of reports of sightings of strange lights, strange aircrafts, and people being captured by aliens. Upon investigation, these reports fall into three categories. Most have normal explanations: automobile headlights reflected off high-altitude clouds, a flight of luminescent insects, unconventional atmospheric effects, unconventional aircrafts, aircrafts using searchlights for meteorological observations, aerial refueling operations, orbiting satellites, sunlight reflecting from objects that are dropped from aircrafts, or the setting planet Venus distorted by the atmosphere. Although these are honest reports, “seeing what you want to believe” is common. For example, the U.S. Air Force collected 30 UFO reports in 1968 when a satellite reentered the atmosphere and broke into burning pieces in the night sky. Of these, 57% reported that the objects were flying in formation, implying intelligent control, and 17% claimed that the glowing objects were attached to a black “cigarshaped” or “rocket-shaped” object, sometimes with glowing windows. Other reports are hoaxes, often for profit. For example, a 1968 University of Colorado study, headed by physicist Edward Condon, established that many of the classic UFO photos are either fakes or photos of known natural phenomena. Nevertheless, these photos continue to reappear in new UFO publications. Great Britain’s widely publicized “crop circle” phenomenon reported around 1990 was caused by pranksters.

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Finally, a few UFO reports cannot be explained. In any investigation of unusual phenomena, there will always be cases that remain unexplained because of lack of data, false reporting, self-deception, and so forth. The unexplained UFO reports offer no positive evidence, such as unambiguous photographs or unambiguous sightings by many observers or an artifact (a tool or piece of material) left behind by aliens. Such a residue of unexplained cases, with no positive evidence, is not surprising and offers no support for UFO beliefs. In fact, the evidence points the other way. The only real evidence we have is negative: Extraterrestrials have not come right out and revealed themselves to us. So if they exist, they prefer to conceal themselves. Any extraterrestrial civilization able to mount a journey to Earth would surely be able to conceal themselves from us if they wanted to and would not make simple mistakes like flying around in visually observable vehicles. So reports of UFO sightings are inherently implausible. Furthermore, it’s surprising that such beings would want to conceal themselves. It seems more reasonable that they would want to contact and investigate us. At least, this is what human explorers have done when they discovered new cultures. A common fallacy of many UFO reports is that far from being overly fantastic, they are not nearly fantastic enough to be believable. The reported technologies are always just a little in advance of, or even behind, the current technology on Earth. The aliens are reported to have curiously humanlike features. But there is little reason to expect that alien technologies, or alien body features, would resemble ours. These are some of the reasons that scientists who have thought about this matter overwhelmingly reject the hypothesis that we are being visited. The second UFO belief, that we have been visited in the past, has even less supporting evidence. Ancient legends of superior beings mean little: Most cultures have had such legends, based on either real humans or stories promoted by the priesthood. An ancient legend containing “futuristic” information, such as instructions for an electronic circuit, might be convincing. Also convincing would be an ancient artifact that could not have been made by the ancient civilization, like an electronic microchip or an advanced metallic alloy. But such evidence hasn’t been found. Popular UFO mythology illustrates several common features of pseudoscience: mistaken observations attributed to exotic causes when simpler explanations suffice, deliberate fraud, using a small number of unexplained cases as proof of an exotic hypothesis, and self-deception caused by a desire to believe. It is for precisely such reasons that scientists ask, How do we know? What is the evidence? Astrology, the belief that events on Earth are influenced by the positions of the planets, began in ancient Babylonia. It seemed reasonable in an era that believed the planets existed for human purposes. Its central belief is that the configuration of the sun, moon, and planets at the moment of a person’s birth affects his or her personality or fortune. A simplified form of astrology, based only on the position of the sun, is the mainstay of newspaper astrology columns. Today, astrology is scientifically implausible, to say the least. The only known physical influences exerted on Earth by the planets are gravitational effects and electromagnetic radiation. It is hard to imagine how these effects at birth could influence our lives. For example, the gravitational effects10 exerted on a baby by the doctor and nurse and furniture in the delivery room far exceed the effects of the planets, the walls of the delivery room shield us from many radiations, and the variations in the sun’s radiation output (variations that are unrelated to a person’s astrological sign) are far larger than the total radiation received from the moon and all the planets added together. 10

In the prehistoric period, the human civilization sometimes lasted long, sometimes short. Some human civilization lasted very long. Mankind in every cycle takes a different way in the development of science. In fact, the moon was made by the prehistoric human beings. It is hollow inside. From Zhuan Falun, Volume II, Literature of the Chinese Cult Known as Falun Gong, by Falun Gong Leader Li Hongzhi, 1998.

The effects referred to here are “tidal effects,” by which the moon and sun cause tides in bodies of water on Earth. Tidal effects cause similar, but smaller, distortions in all objects on Earth.

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The Way of Science

Virtually every major move... the Reagans made was cleared in advance with a woman in San Francisco who drew up horoscopes to make certain that the planets were in a favorable alignment for the enterprise. Nancy Reagan seemed to have absolute faith in... this woman.... At one point, I kept a color-coded calendar—highlighted in green for “good” days, red for “bad” days, yellow for “iffy” days—as an aid to remembering when it was propitious to move the President from one place to another, or schedule him to speak in public, or commence [foreign] negotiations. Donald Regan, Chief of Staff to Former President Ronald Reagan

Science is the great antidote to the poison of ... superstition. Adam Smith, in The Wealth of Nations

For us, not to believe in inerrancy is not to believe in God.... [T]he Bible is literally without error in all respects—in history and science as well as religion.... Adam and Eve were real people. The historical narratives of the Bible are accurate. Miracles of the Bible were supernatural events. The authors stated by all the books were the authors of the book. Rev. M. H. Chapman, President of the Nation’s 14.9 Million Southern Baptists, 1990

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How do we know astrology is not credible? Despite its scientific implausibility, can we find any evidence that astrological predictions are correct? A number of researchers have studied this question, some of them using astrological predictions based on horoscopes (charts showing the orientation of the planets at the moment of birth) for thousands of people, and found no evidence that astrology has any power to predict personalities or lives. If astrology were valid, such evidence should be easy to find. For example, University of California educator and physicist Shawn Carlson conducted a rigorously controlled investigation of astrology that was published in the 5 December 1985 issue of the journal Nature. He studied 30 astrologers considered by their peers to be among the best practitioners of their art. Carlson asked the astrologers to interpret birth charts for 116 real-life but unseen “clients.” With each client’s chart, astrologers were provided three personality profiles, one from the client and two others chosen at random, and asked to choose the one that best matched the birth chart. Contrary to the astrologers own predictions that they would spot the correct chart significantly more often than the “guessing” frequency of one-third, Carlson found that they could correctly match only one of every three charts—the proportion predicted by chance. Even when astrologers expressed strong confidence in a particular match, they were no more likely to be correct. Carlson comments that astrologers may be successful because they draw clues about their clients from body language and verbal responses, but not because astrology itself has any scientific validity.

Even though astrology is incredible theoretically and disproved observationally, half of American adults say they believe in it; newspapers continue their daily astrological predictions; there are many times more professional astrologers than astronomers; and former president Ronald Reagan’s scheduled activities were determined partly by astrological horoscopes (see marginal quotation). Will humankind outgrow its most harmful instincts and develop a mature culture able to control its own technology? Facts such as these give little cause for optimism. Creationism is the belief that the Bible’s Old Testament can be read literally, as scientific and historical truth, and that Earth and the main biological organisms, including humans, all were created separately and at roughly the same time, just a few thousand years ago. Creationism, including its variations such as “intelligent design” (see below), is perhaps the most harmful pseudoscience in the United States because it is believed by so many people, it is fervently championed by many powerful religious organizations, and it tries to cast doubt on science education, especially biology education. For example, in 1999, creationists in Kansas removed from the state science standards all mention of the big bang, radioactive dating, continental drift, the age of Earth, global warming, and biological evolution. Like astrology, creationism was credible until a few centuries ago, and many scientists believed it. But today it conflicts with the observations and principles of astronomy, physics, chemistry, geology, biology, paleontology, and archaeology. There is a broad scientific consensus, supported by a consistent network of evidence from many sciences, that Earth is billions of years old, that humankind is millions of years old, and that humans are related through biological evolution to all other living creatures. Although scientific ideas are never certain, and honest doubts about established theories should never be arbitrarily dismissed, creationist arguments have found essentially no scientific support. Creationist beliefs are in direct conflict with physics on several counts. According to one creationist argument, biological evolution conflicts with the second law of thermodynamics, because evolution describes an increasingly organized biological realm while the second law says that things become more disorganized. But the same question arises in the growth of a leaf, which creates organization out

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of disorganized water and carbon dioxide. The answer to this apparent dilemma is that the leaf is not an isolated system but instead gets crucial assistance from the sun’s radiation, whereas the second law applies only to isolated systems. The same is true of the evolution of all life. The biological realm could not exist without the sun’s radiation to both energize and organize all life on Earth. Evolution does not violate the second law. This creationist argument, based on the second law, is an excellent example of pseudoscience because it sounds scientific but is in fact a misleading distortion of science. Other points of conflict between creationism and physics include the big bang and the conclusions of radioactive dating and other methods of determining the ages of objects. Four independent lines of evidence point to the big bang creation of the universe about 14 billion years ago. There is an enormous and consistent body of radioactive and nonradioactive evidence showing that Earth is billions of years old and dating the geological ages in a manner that confirms evolution but disproves creationism. One creationist view, known as intelligent design, put forth recently is that life is too complex in certain regards, such as complex cellular structures, to have evolved by Darwinian processes. Intelligent design’s key idea, that complex structures could not evolve through intermediate nonfunctional steps, is a new version of the old “argument from design” first proposed 200 years ago and discredited long ago by biologists. The intelligent design view argues (unconvincingly, in the view of the vast majority of biologists) against evolutionary explanations of complexity, but without putting anything in its place. If complex structures did not originate through the evolutionary process, then how did they originate? Arguing that an “intelligent designer” (in other words, God) did it explains nothing, tends to stifle further scientific research, and is beside the point because it doesn’t tell us how a complex structure came to be complex. It’s important to note that science is compatible with a belief in God and with many other religious beliefs. Many scientists are Christians, and many more believe in God. Science has nothing to say about the existence or nonexistence of God, because science studies only natural processes, not supernatural processes. Many scientists harbor a deep conviction that both science and religion are part of a single larger truth. In 2005, the American Association of Physics Teachers adopted a statement on the teaching of evolution and cosmology that demonstrates an admirable understanding of the scientific process. It says, in part:

It is our conclusion that creationism ... is not science. It subordinates evidence to statements based on authority and revelation.... Its central hypothesis is not subject to change in light of new data or demonstration of error. Moreover, when the evidence for creationism has been subjected to the tests of the scientific method, it has been found invalid. National Academy of Sciences, Committee on Science and Creationism, 1984

Nothing in biology makes sense except in the light of evolution. Theodosius Dobzhansky, Geneticist

No scientific theory, no matter how strongly supported by available evidence, is final and unchallengeable; any good theory is always exposed to the possibility of being modified or even overthrown by new evidence. That is at the very heart of the process of science. However, biological and cosmological evolution are theories as strongly supported and interwoven into the fabric of science as any other essential underpinnings of modern science and technology. To deny children exposure to the evidence in support of biological and cosmological evolution is akin to allowing them to believe that atoms do not exist or that the Sun goes around the Earth. We believe in teaching that science is a process that examines all of the evidence relevant to an issue and tests alternative hypotheses. For this reason, we do not endorse teaching the “evidence against evolution,” because currently no such scientific evidence exists. Nor can we condone teaching “scientific creationism,” “intelligent design,” or other non-scientific viewpoints as valid scientific theories. These beliefs ignore the important connections among empirical data and fail to provide testable hypotheses. They should not be a part of the science curriculum.

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School boards, teachers, parents, and lawmakers have a responsibility to ensure that all children receive a good education in science. The American Association of Physics Teachers opposes all efforts to require or promote teaching creationism or any other non-scientific viewpoint in a science course.

© Sidney Harris, used with permission.

CONCEPT CHECK 12 Creationists sometimes argue that all the evidence that Earth is billions of years old was actually created just a few thousand years ago in order to make Earth appear old without really being old. Is this argument scientific? (a) Yes, even though it is not especially credible. (b) Yes, even though it is impossible to prove. (c) Yes, even though it is impossible to disprove. (d) No, because it is impossible to prove. (e) No, because it is impossible to disprove.

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The Way of Science Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions

18. According to Kepler’s theory, what geometric shape fits the planetary orbits?

OBSERVING THE NIGHT SKY

THE SCIENTIFIC AND COPERNICAN REVOLUTIONS

1. What two reasons does this chapter give for studying science? 2. What is physics? 3. Distinguish astronomy from astrology. 4. What astronomical objects can you normally see in the night sky? Describe their motion as seen from Earth.

ANCIENT GREEK THEORIES 5. What did the Pythagoreans believe, and how did these beliefs influence the development of science? 6. According to the earliest Greek hypothesis, the planets orbit Earth in uniform circular motion. In what way does this hypothesis disagree with simple observations made without telescopes? 7. Give two observational reasons for believing that Earth is curved rather than flat. 8. How does Ptolemy’s theory explain the retrograde motion of the planets and the fact that planets are brighter during retrograde motion? 9. Did Ptolemy’s theory agree with the quantitative observations known in Ptolemy’s time? How were these observations made?

COPERNICUS’S THEORY 10. “Copernicus rejected Ptolemy’s theory because it disagreed with the data, and he proposed a new sun-centered theory that did agree with the data.” True or false? Explain. 11. Use Copernicus’s theory to explain the retrograde motion of the planets and the fact that they are brighter during retrograde motion. 12. Why did Copernicus propose his theory? 13. State at least one plausible argument against the notion that Earth moves around the sun. 14. How did new telescopic evidence decisively disprove Ptolemy’s theory?

KEPLER’S THEORY 15. “Kepler was attracted to Copernicus’s theory because the known data supported that theory.” True or false? Explain. 16. Describe Brahe’s work and its effect on the theories of Copernicus and Ptolemy. 17. What aspect of Kepler’s theory would have horrified all previous astronomers?

19. What is the most characteristic and significant feature of science? 20. Describe several characteristics of a good scientific theory. 21. Can a scientific theory be proved (can we show that the theory is certainly true)? Can it be disproved? Explain. 22. Strictly speaking, Kepler’s theory has been disproved. What has been found wrong with it? Why, then, do we still use it? 23. How does a hypothesis differ from a theory? 24. Distinguish between the Copernican theory and the Copernican revolution. 25. In what sense can evolutionary biology be said to be “Copernican”?

PSEUDOSCIENCE 26. What is pseudoscience? List several examples. 27. What are the two popular UFO beliefs? What is the scientific consensus about them, and why? 28. What is astrology? What is creationism? 29. Why do scientists consider UFO beliefs to be pseudoscientific? Answer the same question for astrology and for creationism.

Conceptual Exercises OBSERVING THE NIGHT SKY 1. How can you tell, from naked-eye observation alone, whether a particular object in the sky is a planet? 2. Draw a diagram showing the positions of Earth, the moon, and the sun at new moon, crescent moon, nearly full moon, and full moon. 3. Are the stars in Figure 4 circling clockwise or counterclockwise? A time-lapse photograph made in the Southern Hemisphere, looking toward the South Pole, would also show the stars moving in a circle around a fixed point in the southern sky. Would the stars in the southern view be circling clockwise or counterclockwise?

From Chapter 1 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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The Way of Science: Problem Set

ANCIENT GREEK THEORIES 4. Describe a naked-eye observation you could make to disprove the theory that the planets orbit Earth in a simple, uniform, circular motion. 5. Describe a naked-eye observation you could make to disprove the theory that the planets orbit Earth attached to transparent spheres that rotate in a complicated fashion but that are always centered on Earth. 6. In seeking an explanation of retrograde motion, why didn’t the Greeks just allow the planets to change their speed and direction of motion as the planets moved along circular paths around Earth, instead of resorting to circles within circles?

10. Use Copernicus’s theory to explain why Venus often appears as the morning star or the evening star.

KEPLER’S THEORY 11. Which aspects of Kepler’s theory would Copernicus have liked? Disliked? 12. Would Kepler’s theory have agreed with the data available in Ptolemy’s time? In Copernicus’s time? 13. Did Brahe’s data prove that planets move in ellipses? Explain. 14. Is there anything in Kepler’s theory that resembles the displaced centers of Ptolemy and Copernicus? 15. Who is the “observer” mentioned by Kepler? 16. Kepler says that God has waited 6000 years. Why 6000? 17. Explain how to get a highly elliptical (elongated) orbit from the tack-and-string construction of Figure 20.

COPERNICUS’S THEORY 7. Is it possible that on some evenings the planet Mars is the evening star? Is this very likely? (see figure below) 8. Use Copernicus’s theory to predict whether Mars goes through moonlike phases. Do we ever see a “full Mars”? A “new Mars”? 9. It is possible, but difficult, to see the planet Mercury with the unaided eye. How, then, would you go about finding it?

Sphere of the stars

Copernicus’s sun-centered theory of the layout of the universe, 1543 CE. The diagram is simplified; the planets all move on epicycles, similar to those in Ptolemy’s theory, but here there are far fewer epicycles, and no equants.

Jupiter Mars

Mercury

Venus Sun

Moon Earth

Saturn

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Figure 14

The Way of Science: Problem Set

THE SCIENTIFIC REVOLUTION 18. Because Darwinian evolution is only a theory, we need not take it seriously. Comment on this statement. 19. What is the most important and characteristic feature of science? 20. Can two different theories both be true in the sense that at some particular time in history, they correctly predicted the known data? Defend your answer with a historical example. 21. “If Earth is curved, it must have a spherical shape, because a sphere is the most perfect curved solid form.” Does an aesthetic argument like this have any place in science? 22. A sensationalist tabloid “news”paper carries this headline “SCIENTISTS PREDICT THAT THE UNIVERSE AND EVERYTHING IN IT WILL DOUBLE IN SIZE AT THE BEGINNING OF THE NEXT NEW YEAR!” Is this a testable hypothesis? If so, how could you test it, and if not, why not? Is this good science, bad science, or neither? 23. What is the scientific attitude toward beliefs such as astrology, dianetics, extrasensory perception (ESP), visitations by extraterrestrials, a 6000-year-old Earth, the Bermuda Triangle, and pyramid power? 24. Aristotle, a careful observer of living organisms, wondered where the material that contributes to the growth of a plant comes from. He hypothesized that all of it comes from the soil. Based on your knowledge of biology, do you consider this hypothesis to be correct? Propose an experiment to test this hypothesis. 25. Some people believe that plants will grow better if they are talked to. Is this a testable hypothesis? If so, propose an experiment to test it. 26. “Certain people are gifted with extrasensory perception (ESP), such as the ability to move material objects with their own minds. However, ESP is so delicate that every attempt to verify it always destroys it.” Is this a scientific hypothesis? 27. Isaac Newton predicted that because of its spinning motion, Earth would bulge out near the equator and be flattened near the poles. In 1735 the French Academy of Sciences sent an expedition to the Arctic to measure the exact shape of Earth. When they returned, reporting the predicted results, the philosopher Voltaire mocked them with the following couplet: To distant and dangerous places you roam To discover what Newton knew staying at home. Was Voltaire’s sarcasm justified? Why or why not? 28. Consider the flat Earth hypothesis. Give evidence for this hypothesis. Give evidence against it.

THE COPERNICAN REVOLUTION 29. Since there are some 100 billion stars in a typical galaxy, and since there are at least 100 billion galaxies in the known parts of the universe, how many stars are there in the known universe? Write this number out. 30. An astronomical unit (AU) is the distance from Earth to the sun. The radius of the approximately circular orbit of Mars is about 1.5 AU. As Earth and Mars orbit the sun, what is their greatest and least distances apart, measured in AU? 31. A light-year (LY) is the distance light travels in one year. Our nearest neighboring star is 4 LY away. Using the fact that light gets here from the sun in 8 minutes, how many AU (preceding exercise) is it to our nearest neighboring star?

32. The astronomical object known as the Crab Nebula is the remnant of an exploded star. The explosion was seen, by the Chinese, in 1054 CE. However, the Crab Nebula is about 3500 LY (preceding exercise) distant from Earth. In what Earth year did the star actually explode?

PSEUDOSCIENCE 33. Jeane Dixon, who also claims to have forecast John F. Kennedy’s assassination, once claimed that an incredible vision informed her that aliens from another planet in our solar system would visit Earth the following August and announce their arrival to the entire Earth. She claimed that this other planet lies directly on the other side of the sun, which is why we have never seen it. Give one good scientific argument against the existence of any such planet. 34. Continuing Exercise 33: One answer is that any such planet should have a gravitational effect on the other planets and that this effect has not been observed. Suppose that Jeane Dixon then replied, “But these aliens are so advanced that they have been able to completely mask the effects of their planet’s gravity, as well as all other observable effects of their planet.” What is your response to this explanation? Does this supposed planet fall within the realm of science? 35. Continuing Exercise 34: Jeane Dixon’s forecast was published on the front page of the National Enquirer on September 14, 1976. Have you heard of any reports, the following August, that her forecast was correct? Do you suppose that the National Enquirer then printed a front-page story reporting that her forecast was wrong? Can you recall any instance when such forecasts were later reported as false when they turned out to be false? 36. Some supporters of ESP (extrasensory perception—for example, mind reading, causing objects such as spoons to move by means of mental concentration, and the like) claim that ESP really exists but that it cannot be checked scientifically because scientific experiments always cause the ESP effect to vanish. What is your response to this argument?

Answers to Concept Checks 1. No looking until you’ve formed your own answer! The

answer is (d). 2. The moon’s phases (new, crescent, quarter, and so on) show

that it shines by means of reflected light from the sun, (b). 3. Ptolemy’s theory, Figure 11, places the centers of the orbits 4. 5. 6. 7. 8. 9. 10. 11. 12.

of Mercury’s and Venus’s epicycles on the line joining Earth with the sun, (b). (d) (a) (e) (c) Note that this shows that a circle is a particular kind of ellipse. (a) A scientific idea is never absolutely certain, because the next observation could disprove it, (e). Answers (c), (e), and (f) are correct, because each one says that our particular place in the universe is not unique. (d) (e)

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The Way of Science: Problem Set

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. Follow its position in the sky for a few weeks. If its position relative to the surrounding stars changes, it is a planet. 3. We are looking northward, and stars rise in the east (righthand side of photo) and set in the west (left-hand side), so the stars in the photo are circling counterclockwise. If we were looking southward, toward the South Pole, the stars would be circling clockwise around a point in the southern sky. 5. Follow a planet every night until it noticeably brightens or dims—an indication that it is closer to or further from Earth. 7. This is possible, if Mars happens to lie a little to the east of the sun in the sky (in other words, close to a line joining Earth to sun in Figure 14) and if Venus is below the horizon. However, this combination of events isn’t very likely. 9. Look near the rising or setting sun, just before it rises and just after it sets. If the sun is visible above the horizon, the dim light from Mercury will be obliterated by the light from the sun. 11. Aspects Copernicus would have liked: Earth is a planet; Earth goes around the sun; Kepler’s theory is fairly simple and straightforward (compared to Ptolemy’s theory). What Copernicus would not have liked: In Kepler’s theory the planets do not move in circles or in combinations of circles. 13. Specific data can never prove a general theory, so Brahe’s data could not prove that planets move in ellipses. 15. The observer is Brahe. 17. Move the thumbtacks far apart.

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19. The interplay between theory and observation. 21. Yes, in fact the theories of Ptolemy, Copernicus, and Kepler

were partly based on aesthetic considerations. 23. Scientifically, the best attitude is “let’s look at the evidence.”

25.

27. 29. 31.

33. 35.

Does the “theory” make clear-cut observational predictions? Can they be checked? What are the results? From the purely theoretical standpoint, one should also ask whether the “theory” is clear and logically consistent. This is testable. To test it, ask a neutral person (one who could care less about talking to plants—so as not to bias the experiment) to raise two identical plants in identical surroundings, with the sole significant difference that one plant is talked to and the other is not. To draw any firm conclusion, this experiment should be repeated several times. Do the talked-to plants actually grow significantly better, on the average? Voltaire’s sarcasm was not justified. New scientific predictions must be verified by observation, even though they are sometimes first predicted by theory. 100 billion times 100 billion = 10,000,000,000,000,000,000,000 (a one followed by 22 zeros). Divide 4 years by 8 minutes. First, we must express 4 years in minutes: 4 yrs * 365 days>year * 24 hrs>day * 60 min>hr = 374,400 min. Then 374,400>8 = 46,800. Thus the distance is 46,800 AU. Any such planet should have a gravitational effect on the other planets, and this effect has not been observed. There were no such reports. There were no stories reporting this fact. Generally, when far-fetched predictions such as this are made, there is little or no attempt to follow up on them.

Atoms The Nature of Things

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is Á that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling each other upon being squeezed into one another. In that one sentence Á there is an enormous amount of information about the world. Richard Feynman, Physicist

L

et’s turn now from stars to atoms.

One of science’s key principles is that everything is made of imperceptibly small particles. As Richard Feynman points out above, this explains an extraordinary range of observations. We’ll learn that science profoundly changed its view of atoms during the twentieth century, a development crucial to one of this text’s four themes:1 modern physics and its significance. Section 1 presents the 2500-year-old idea that everything is made of small particles. Section 2 discusses the atom as chemists have understood it for 200 years and distinguishes atoms from molecules. Section 3 explores the wide-ranging explanatory power of the atomic idea. In order to discuss atoms quantitatively, Section 4 takes a brief excursion into “powers of 10” and metric units. Section 5 ponders the incredible tininess of atoms. Section 6 looks at the philosophical implications of the atomic idea. Section 7 compares this chapter’s model of the atom with two other models. Finally, Section 8 explores some significant examples of “chemical reactions”—combining and recombining atoms.

1 THE GREEK ATOM: THE SMALLEST PIECES The ancient Greeks produced an astonishing number of original thinkers. They wanted to think their way to the bottom of everything. Because of their prointellectual attitude, this small group of people, during just two centuries centering on the

1

The three other themes are the scientific process, the social context of physics, and energy.

From Chapter 2 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Atoms

fifth century BCE, laid the foundations for most of the Western world’s great ideas. One thing they thought about was the nature of matter: material substances such as wood, cotton, sausage, ice, water, soil, and gold. They speculated about the underlying unity that they believed lay behind the different substances. What do sausage and gold have in common? What is matter? One Greek, Democritus, sharpened his focus on this question with a “thought experiment,” an imagined experiment that seemed possible in principle but difficult in practice. Suppose, he argued, you cut a piece of gold in half, and then cut one of the halves in half, and so forth. How far could you continue making such divisions? Either the divisions could go on forever, or there would be a limit at which no further divisions would be possible. That is, matter is either continuous—divisible without limit—or it is made of particles that cannot be divided. The first alternative seemed absurd to him. Matter, he concluded, is made of imperceptibly small, “a-tomic” (Greek for “not divisible”) particles. He called these smallest particles “atoms.”2 CONCEPT CHECK 1 Today, this idea should be classified as (a) a scientific fact; (b) an experimental observation; (c) a hypothesis; (d) a scientific theory; (e) scientifically false; (f) gibberish.

I’ll call this idea

The Atomic Theory of Matter All matter is made of tiny particles, too small to be seen.

This is a good example of a scientific principle or law or theory: a well-confirmed idea that explains a broad range of observations. In Democritus’s time, the atomic idea had not yet been confirmed by observations; rather, it was an educated guess or hypothesis. When observations confirmed it during the nineteenth and twentieth centuries, it then became an established theory. A general idea like this cannot, however, be called a fact, no matter how often observations may have confirmed it, because we cannot observe every possible material object to prove for certain that everything really is made of atoms. Theories are never certain. What do your thumbnail, beer, an acorn, Saturn, the Amazon River, and the period at the end of this paragraph have in common? Each is made of atoms. This is the kind of underlying unity that the Greeks loved and that scientists seek.

2

50

“Atom” is used in a slightly different sense today. Today, the “atom” or “chemical atom” is the smallest particle of a chemical element. This chemical atom is actually made of smaller, subatomic parts: electrons, protons, and neutrons. In fact, the protons and neutrons are themselves made of quarks. As far as we know, electrons and quarks are the smallest particles, of which the others are made. These smallest parts are what the Greeks meant by an atom.

Atoms

How do we know that things are made of atoms? Science’s power comes from its “show-me” attitude, its insistence on evidence. So I will frequently ask, “How do we know?” The ancient Greeks had no direct microscopic evidence for atoms, but Democritus had some ingenious indirect evidence. He argued that since we can smell a loaf of bread from a distance, small bread particles must break off and drift into our noses. This is still an acceptable explanation of odors today (Section 4). John Dalton discovered the first specific evidence for atoms around 1800. He found that whenever certain substances combine chemically to form other substances, they combine in simple ratios by weight. For example, when hydrogen and oxygen combine to form water, the ratio of the weights of the two substances is always 1 to 8. Such ratios are difficult to understand if matter is infinitely divisible, but there is a simple explanation if matter is made of atoms. If, for example, one atom of hydrogen and one atom of oxygen have a simple weight ratio, and if these atoms always combine in simple ratios to create water, the weight ratios of hydrogen and oxygen in water will be simple numbers also. Today we know that individual atoms of hydrogen and oxygen have a weight ratio of 1 to 16 and that it always takes two hydrogen atoms for every one oxygen atom to form a water molecule. So you can see, today, why the weight ratio should be 1 to 8. So the atomic theory explains Dalton’s simple ratios. But does this prove the theory? The answer is no! It’s possible that atoms don’t exist and that there is some other explanation for the simple ratios. Observations cannot prove a general theory, but they can make it more plausible. A few decades after Dalton, botanist Robert Brown, using a microscope, observed that tiny pollen grains suspended in liquid move around erratically (Figure 1), even though the liquid itself had no observable motion. His first hypothesis was that the grains were alive. But lifeless dust grains suspended in liquid executed the same erratic dance, disproving the hypothesis. Hypotheses and theories cannot be proved but they can be disproved. It was suggested that submicroscopic motions of atoms (or “molecules,” as we’ll see later) caused this Brownian motion. The idea was that atoms moved around constantly and Brownian motion resulted from numerous atoms colliding with each pollen or dust grain every second. This hypothesis got strong support in 1905 from an unknown young physicist, Albert Einstein. He used an already established theory to calculate how particles such as dust grains are jostled when bombarded randomly by moving atoms. He made several quantitative (numerical) predictions, such as the rate at which a collection of grains should spread out in a liquid. Such predictions could be checked by measurements, and the measurements agreed with Einstein’s predictions. It was difficult to dispute this evidence. Either unseen atoms really did cause Brownian motion, or Einstein’s calculations were fabulously lucky in giving all the right numbers. Since Einstein’s work, scientists have not questioned the atomic theory.

CONCEPT CHECK 2 A sulfur atom has twice the weight of an oxygen atom. When sulfur and oxygen combine to form sulfur dioxide, one sulfur atom is required for every two oxygen atoms. In the formation of sulfur dioxide, the weight ratio of sulfur to oxygen is (a) 4 to 1; (b) 2 to 1; (c) 1 to 1; (d) 1 to 2; (e) 1 to 4. CONCEPT CHECK 3 Following up on the preceding question: In the formation of sulfur trioxide, the weight ratio of sulfur to oxygen is (a) 6 to 1; (b) 3 to 1; (c) 3 to 2; (d) 2 to 3; (e) 1 to 3; (f) 1 to 6.

Figure 1

Brownian motion. This erratic path is typical of a small particle such as a single dust grain, suspended in water, observed under a microscope. The atomic theory explains this behavior by the ceaseless, rapid, random motion of water molecules. Although the molecules are far too small to be seen even under a microscope, the effect of numerous molecules impacting a dust grain every second can be seen in the erratic motion of the grain.

2 ATOMS AND MOLECULES Think of all the different material substances around you: this paper, your shirt, your hair, and so forth. You could list hundreds. Nineteenth-century chemists found that they could transform most substances into a much smaller number of simpler

51

Atoms

substances but that they could not further transform this small number. Any process3 that changes a single substance into other simpler substances is called a chemical decomposition of the original substance. An example: By passing electricity through it, water can be decomposed into two distinct substances, called hydrogen and oxygen, neither of which are anything like water. But hydrogen and oxygen turn out to be among that small (fewer than 100) group of substances that nineteenth-century chemists could not decompose. No matter how they tried to decompose hydrogen, it remained hydrogen, and the same was true for oxygen. Apparently, these roughly 100 substances that cannot be chemically decomposed are particularly fundamental. They are called chemical elements, or simply elements. By studying the weight ratios just discussed, Dalton and others soon recognized that each element was made of only one kind of atom and that elements differed because their atoms differed. So an atom is the smallest particle of a chemical element. But water is compounded of two kinds of atoms, hydrogen and oxygen, which is why you can decompose water but not oxygen or hydrogen. Today, 117 elements are known—117 different kinds of atoms. Eighty-eight of these occur naturally on Earth, while the remaining 29 are created in laboratories. The most recently discovered, but yet unnamed, element is number 118, created in 2006 when a total of just three atoms of it were produced in the lab during a 1080-hour high-energy physics experiment. Each atom existed for only about 0.89 milliseconds (0.00089 seconds). Element number 117 is predicted but not yet observed. Each number, known as the element’s atomic number, represents a particular kind of atom. Higher atomic numbers correspond to heavier atoms. Scientists found that certain groups of elements have similar properties. For example, elements 2, 10, and 18 (helium, neon, and argon) are “inert gases,” meaning that they are gases that will not combine chemically with other elements. For another example, elements 3, 11, and 19 (lithium, sodium, and potassium) are soft, silver-white metals melting at moderate temperatures. If we list these groups having similar properties vertically and also list the elements in order of increasing atomic number, the result is the periodic table. This table is a nice example of the regularities that scientists find in natural phenomena. Scientists use it as a predictive device, by noting the table’s unfilled gaps and searching for elements with properties that just fit those gaps. What about all the other substances, those made of more than one element? A pure substance, such as pure water (with no impurities like salt or dirt), that is made of more than one element is called a chemical compound. Imagine dividing a cup of water into smaller and smaller amounts. If the water is pure, you will always get just water—not something else such as salt or dirt. Working downward in size, you will eventually arrive at the smallest particle of water. In pure water, every one of these smallest particles must be a particle of water, and so they should be identical. And every particle must contain atoms of both hydrogen and oxygen, because we know that water can be chemically decomposed into these elements. This reasoning shows that every pure chemical compound must be made of tiny particles that are identical and that are themselves made of two or more atoms

3

52

More precisely, any low-energy (lower than nuclear energies) physical process.

Atoms

attached together into a single identifiable unit. Such a particle, the smallest particle of a compound that still has the characteristics of that compound, is called a molecule. Chemists can deduce a compound’s molecular structure by decomposition and by combining it with other compounds. Such experiments show, for example, that the water molecule is made of two hydrogen atoms and one oxygen atom (Figure 2). Some elements are made of two-atom molecules. For example, a molecule of hydrogen gas is made of two hydrogen atoms (Figure 3).4 Helium gas, on the other hand, is made of individual helium atoms (Figure 4). The common forms of oxygen gas and nitrogen gas are made of two-atom molecules. Air (Figure 5) is made primarily of these two kinds of molecules. We represent compounds and elements by abbreviated formulas. For example, water is represented by H 2O, where the subscript 2 belongs to the symbol preceding it and indicates the number of atoms of that type in each molecule. Hydrogen gas is represented by H 2, oxygen gas by O2, and helium by He. Molecules can get pretty complicated, especially the molecules of life, such as your protein and DNA. Biological molecules are among the most varied and complicated known. Hemoglobin, the protein responsible for the red color of blood, has the formula C3023H 4816O872N780S8Fe 4. DNA molecules contain millions of atoms and vary from one individual to the next. They carry the instructions that make you you.

Figure 2

A simplified drawing of liquid water, magnified 50 million times. In a liquid the molecules are in close contact and slide past one another. Each water molecule is made of one oxygen atom (blue) and two hydrogen atoms (black). This and other microscopic drawings in this chapter view only a tiny region within a much larger container.

CONCEPT CHECK 4 Which of the following elements are chemically similar to chlorine? (a) Iodine. (b) Sulfur. (c) Xenon. (d) Bromine. (e) Krypton. (f) Fluorine.

4

Figure 3

Figure 4

Figure 5

A simplified drawing of hydrogen gas. Each molecule of hydrogen is made of two hydrogen atoms. Each molecule moves rapidly in a nearly straight line, changing direction only when it collides with another molecule or with the container wall.

A simplified drawing of helium gas. Each molecule of helium is simply an unattached atom of helium.

A simplified drawing of air, magnified 50 million times. Air is a mixture mostly of nitrogen (gray) and oxygen (green) molecules, both two-atom molecules.

However, outside Earth’s atmosphere nearly all the universe’s hydrogen is in the “atomic” (single-atom) form rather than the “molecular” (two-atom) form, because the universe began with atomic hydrogen and these atoms are separated so widely in space that they never combined into molecules. These “primordial” hydrogen atoms have not been altered in 14 billion years (the age of the universe)!

53

Atoms

CONCEPT CHECK 5 What elements, and how many atoms of each, does the simple sugar C6H 12O6 (“glucose”) contain? (a) 6 chlorine, 12 helium, 6 ozone. (b) 6 carbon, 12 hydrogen, 6 oxygen. (c) 1 chlorine, 1 hydrogen, 1 oxygen. (d) 1 carbon, 1 hydrogen, 1 oxygen. (e) 1 carbon, 2 hydrogen, 1 oxygen. CONCEPT CHECK 6 The chemical formula for carbon dioxide is (a) CaO; (b) Ca2O; (c) CO; (d) CD; (e) C2O; (f) CO2.

3 THE ATOM’S EXPLANATORY POWER: THE ODOR OF VIOLETS5

The power of the atomic theory, and the virtues of careful observation, are both illustrated by the question “What are smells?” Have you ever thought about this? Please stop reading and think about it for a moment.... Observation shows that you are surrounded by an invisible substance, air. You know it’s there because you can feel the wind. The atomic theory tells us every material substance, anything you can pick up or touch, is made of atoms. It’s reasonable to suppose that air is a material substance, too, because you can feel it blow on you. A careful measurement would show that air has weight, further confirming our hypothesis. We conclude, from the atomic theory, that air is made of atoms. As you know from the Brownian motion experiment, the atoms in a liquid move all the time, even when the liquid appears to be motionless. So it’s reasonable to suppose that air molecules are in constant motion, too, even in still air. Consider the odor of violets. Among a violet’s various molecules, there must be some that make it smell the way it does. Chemists have learned how the violet’s odor molecule is strung together (Figure 6). Scientifically, at least, that’s what the smell of violets is—those molecules. In order for a violet’s smell to spread out, odor molecules must break loose from the violet. Once in the air, moving air molecules knock odor molecules around, causing odor molecules to spread out in all directions. Eventually, they reach your nose. Think about that the next time you smell something. The atomic theory links human-scale or macroscopic phenomena that you can see to phenomena at the unseen microscopic level. The microscopic perspective is especially helpful in understanding the states of matter. Water, for example, comes in three states: ice, liquid, and vapor (or steam). We call these the solid state, liquid state, and gas state of water. Nearly every substance can exist in any one of these three states.6 At the macroscopic level, the three states of matter are distinguished by the shapes they assume when placed in a container. In a closed container, a solid maintains its shape; a liquid spreads out over the bottom; and a gas fills the volume. How do they differ at the microscopic level? A little thought, guided by the atomic theory and by some simple macroscopic observations, can answer this. As I hope Figure 6

The odor of violets, in air. The funny-looking thing is the odorof-violets molecule, made of carbon (horizontal stripes), hydrogen (black), and a single oxygen atom (green).

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5

6

This section is partly based on Richard Feynman’s Lectures on Physics (Addison-Wesley, Reading, MA, 1963), Vol. I, Chapter 1, where the odor of violets example was first presented. Although these three are the normal states of matter on Earth, many other states are common throughout most of the universe. Such “exotic” states of matter include plasma (electrically charged gas, common in stars), three kinds of superdense matter (found in white dwarf stars, neutron stars, and black holes), and supercold states in which substances display large-scale “quantum” behavior (superfluids, superconductors, and Bose-Einstein condensates).

Atoms

you’ll discover throughout this text, it’s amazing what careful thought guided by simple observations can accomplish! Because solids maintain a fixed shape, their molecules must be locked into a fixed arrangement. Because solids are difficult to compress into a smaller volume, their molecules must be crowded against one another. The precise arrangement is determined by the ways in which the substance’s molecules push and pull on one another when they get close together. If you have ever seen a large number of balls tossed one by one into a big box, or gunshot (BBs) filling a small container, you can guess that molecules tend to lock into an orderly pattern that repeats itself throughout the solid.7 Orderly molecular patterns are responsible for the regular surfaces and beautiful symmetries seen in macroscopic crystals such as diamonds. Figure 7 shows the microscopic six-sided crystal pattern of ice, and Figure 8 shows the macroscopic snowflake crystals that are formed from it (note the hexagonal, or six-sided, symmetry). Because liquids have no fixed shape, their molecules must not be rigidly attached to one another. But liquids are about as difficult to compress into a smaller volume as are solids, so we expect that a liquid’s molecules are crowded together about as closely as possible. This reasoning leads us to a microscopic picture of a liquid that is similar to a jumbled bowlful of marbles that assume different shapes depending on the shape of the bowl. The molecules in a liquid are free to migrate throughout the liquid by sliding past one another. Most liquids take up slightly more volume than do the solids of the same substance, but water is an exception to this rule; because of its open crystal structure (Figure 7), ice takes up slightly more volume than does liquid water. Because gases can be compressed into a much smaller volume, their molecules must be widely separated. Gas molecules dart back and forth, bouncing off the container’s walls and colliding and rebounding from one another. From this microscopic picture we would expect that, because of the continual torrent of gas molecules hitting the surrounding surfaces, a gas should press outward against its container. This outward press is called gas pressure. It’s as though hundreds of baseballs were thrown at a wall, pressing the wall backward. You can see the effect of gas pressure when you blow up a balloon; the elastic material is pressed outward by trillions of gas molecules hitting the inner surface every second. Figure 9 shows the differences between solids, liquids, and gases. A complete absence of air and all other forms of matter is called a vacuum. A perfect vacuum is impossible to achieve in any ordinary macroscopic volume on Earth, but it is not difficult to achieve a partial vacuum in which the container holds far less air than it would if filled with air at its normal density. A good vacuum in a laboratory still contains a trillion molecules in every cubic centimeter! But in space the large regions between galaxies are nearly perfect vacuums—neighboring molecules are some 2 meters apart! When you warm a substance, what happens to its molecules? With our understanding of solids, liquids, and gases, we’re in a position to answer this important question. Consider an air-filled balloon tied shut so that no air enters or leaves. What happens if you heat or cool it? Try it! First put the balloon in the freezer for a few minutes. Then hold it over boiling water. What happens? The balloon expands as the air warms inside it. Returning to our microscopic picture of a gas, you can 7

Figure 7

Ice. Compare this diagram of solid water with that of liquid water in Figure 2. In the solid state, atoms vibrate around their average position in the crystal pattern but they do not migrate throughout the material as they do in the liquid state.

National Oceanic and Atmospheric Administration/Seattle Figure 8

The hexagonal symmetry we see in snowflakes mirrors their underlying microscopic symmetry (compare Figure 7).

However, some solid materials, including plastics and glasses, have an irregular arrangement at the microscopic level.

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Atoms

(a)

(b)

see that as the air in the balloon warms, the molecules inside must bounce harder off the inner walls in order to cause the expansion. This means that, as the air warms, the molecules move faster. Experiments amply confirm this hypothesis. It’s true not only in gases such as air but also in liquids and solids. That is, molecules are always in random (or disorganized) motion, whether in a solid, liquid, or gas, and those motions get faster as the solid, liquid, or gas gets warmer. Warmth is measured quantitatively by devices called thermometers. In a simple household thermometer the liquid inside responds to warmth by changing its volume in a measurable way when placed into a solid, liquid, or gas. The resulting reading is called the temperature of the solid, liquid, or gas. This connection between warmth and molecular motion is so close that scientists consider warmth (or temperature) and molecular motion to be, respectively, the macroscopic and microscopic aspects of the same phenomenon. Because of this connection, this molecular motion is called thermal motion. Summarizing this important idea: The Microscopic Interpretation of Warmth At the microscopic level, warmth (temperature) is the random, or disorganized, motion of a substance’s molecules. This thermal motion cannot be directly observed macroscopically but is observed instead as temperature or warmth.

The atomic theory explains and unifies the odor of violets, the three states of matter, chemical compounds, gas pressure, warmth, and much more. It’s a good theory.

(c)

Figure 9

Microscopic views of the (a) solid, (b) liquid, and (c) gas states of matter.

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CONCEPT CHECK 7 Which of the following observations confirm that you are surrounded by air? (a) Trees bending in the wind. (b) Your ability to observe light. (c) Air that you can feel entering your nose as you breathe. (d) A breeze brushing against your cheek. (e) An air-filled balloon. (f) A rock falling when you drop it.

4 METRIC DISTANCES AND POWERS OF 10 Before discussing atoms quantitatively in Section 5, we need to take a brief excursion into metric units and powers of 10. When you measure a quantity such as the length of a table or the weight of a rock, that measurement is made relative to a particular standard or unit. For example, you might measure length in inches, feet, or miles, or in the metric system in centimeters, meters, or kilometers. If you wanted to tell someone the distance to the next town, you wouldn’t say “53.” You must give the unit, for example 53 kilometers or 53 miles. One curious feature of American life is its use of the “English” system of measurements, based on feet and pounds and so forth. Since only Liberia, Burma, and the United States use it officially anymore (it’s still used unofficially in the United Kingdom and Ireland), I’ll henceforth call it the U.S. system of units. We’ll shun this confusing traditional system (inches, feet, yards, miles, horsepower, pounds, ounces, pints, quarts, gallons, degrees Fahrenheit, etc.), a modern patchwork codification of medieval trade units. This quaint system can be costly: During NASA’s

Atoms

Mars Climate Orbiter space probe in 1999, one engineering group used U.S. units for navigation while another assumed the numbers were metric. This caused problems. The $125-million spacecraft veered off course while approaching Mars, lost contact with Earth, and crashed. Not good. For further excellent reasons why the United States should convert to all-metric units, see Concept Check 10 and the marginal quotation by Valerie Antoine. The basic metric distance unit is the meter (abbreviated m). It’s about 39 inches, a little over a yard. Table 1 lists other metric distances and relates them to the meter. The most important are the kilometer (km), centimeter (cm), and millimeter (mm). Table 2 lists six common prefixes that can be attached to any metric unit. For example, the kilowatt is 1000 watts, and the megawatt is 1 million watts (later, we’ll see what’s a watt). For handling large and small numbers, a technique known as powers of 10 is invaluable. A power of 10 means 10 raised to some power. So 102 means 10 * 10, which equals 100, and 105 means 10 * 10 * 10 * 10 * 10 = 100,000. For example, the solar system’s diameter (distance across) is 12,000,000,000,000 m. You can write this as 1.2 * 10,000,000,000,000 m or 1.2 * 1013 m. Each multiplication by 10 moves the decimal point one place to the right, so to write out 1.2 * 1013, you begin with 1.2 and move the decimal point 13 places to the right. The number in front (the 1.2) is usually written as a number between 1 and 10. If there is no number in front, you can think of a 1 in front; for instance, 105 is the same as 1 * 105. Negative powers are used for small numbers. For instance, 10-2 means 1/102, which equals 1/100, or 0.01, and 10-5 means 1/105 = 0.000 01. The minus sign indicates that the power of 10 is to be divided into 1. For example, the diameter of an atom is about 0.000 000 000 11 m, which can be written as 1.1 * 10-10 m. Since each division by 10 moves the decimal point one place to the left, to write 1.1 * 10-10, you begin with 1.1 and move the decimal point 10 places to the left. Thousand 11032, million 11062, billion 11092, and trillion 110122 all represent various powers of 10. Similarly, thousandth 110-32, millionth 110-62, and so forth represent negative powers of 10. To multiply two powers of 10, just add their powers. For instance, 102 * 105 = 2+5 10 = 107, and 102 * 10-5 = 102 + 1-52 = 10-3. The numbers in front of the power of 10 can be grouped together first, before multiplying. For example,

Benefits to U.S. industry if it converts to metric usage Á can be summed up in one word: survival. Overseas countries already are refusing entry to some U.S. inch–pound goods, Á our industries will lose the buying power of 320 million people in the European Community (EC) if we don’t wake up and begin producing to the EC metric standards Á U.S. industry must convert to metric production if it wants to survive. Valerie Antoine, Executive Director of the U.S. Metric Association

11.5 * 1022 * 13 * 1052 = 11.5 * 32 * 1102 * 1052 = 4.5 * 107 To divide two powers of 10, subtract the denominator’s power from the numerator’s power. For instance, 102/105 = 102 - 5 = 10-3, and 102/10-5 = 102 - 1-52 = 107. Table 2

Table 1

Metric prefixes

Metric distances Name of unit

Kilometer (km)

Distance 3

1000 m = 10 m

Meter (m) Centimeter (cm)

0.01 m = 10-2 m

Millimeter (mm)

0.001 m = 10-3 m

Micrometer 1mm2 Nanometer (nm)

0.000 001 m = 10

Conversion to U.S. units

Giga (G)

one billion, 109

1 km = 0.62 mi, 1 mi = 1.6 km

Mega (M)

one million, 106

1 m = 3.3 ft = 39 in., 1 ft = 0.30 m

Kilo (k)

one thousand, 103

1 cm = 0.39 in., 1 in. = 2.5 cm

Milli (m)

one-thousandth, 10-3

Micro 1m2 -6

m

0.000 000 001 m = 10

Nano (n) -9

one-millionth, 10-6 one-billionth, 10-9

m

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Atoms

Numbers in front of the powers of 10 can be grouped together first, before dividing them. For example, 11.5 * 1022/13 * 1052 = 11.5/32 * 1102/1052 = 0.5 * 10-3 = 5 * 10-4 Using powers of 10, you can calculate all sorts of fabulous things. For example, the solar system’s diameter divided by an atom’s diameter (the ratio of the two diameters) is 1.2 * 1013 m/1.1 * 10-10 m = 11.2/1.12 * 11013/10-102 = 1.1 * 1013 - 1-102 = 1.1 * 1023

This number, 110,000,000,000,000,000,000,000, is the number of atoms you would have to line up side by side in order for them to stretch across the solar system. Is that fabulous or what? CONCEPT CHECK 8 The universe is only seconds old, a million trillion seconds old, in fact. In powers of 10, this is (a) 1014 s; (b) 1015 s; (c) 1016 s; (d) 1017 s; (e) 1018 s. CONCEPT CHECK 9 The diameter of an atomic nucleus is about a hundredth of a trillionth of a meter. In powers of 10, this is (a) 10-10 m; (b) 10-12 m; (c) 10-14 m; (d) 10-15 m; (e) 10-16 m; (f) 10-18 m. CONCEPT CHECK 10 Answer either one of the following two questions, without using a calculator. For those of you who prefer U.S. units: How many inches are there in 6 miles? For those who prefer metric units: How many centimeters are there in 6 kilometers? Answers (U.S. system): (a) 463,173 in. (b) 380,160 in. (c) 263,150 in. Answers (metric system): (a) 6,000,000 cm; (b) 600,000 cm; (c) 60,000 cm.

Did you choose to answer Concept Check 10 in U.S. units? Do I make my point clear?

5 THE INCREDIBLE SMALLNESS OF ATOMS The most convincing evidence for atoms is to see one. But it’s impossible to see atoms with ordinary light, even with the best optical microscopes. The reason lies in the nature of light. Light is a wave, similar in some ways to water waves on the surface of a pond. The wavelength of light, the distance from one crest to the next, is very small—10 to 100 times smaller than the smallest visible dust particle. That’s small, but a single atom is 5000 times smaller still. To visualize this, imagine that light has a wavelength of 5 meters. On this scale, an atom would be a speck just 1 millimeter across! So light waves are too big to respond to tiny individual atoms. How do we know that atoms exist? Before 1970, Brownian motion was probably the closest we had come to seeing atoms. In 1970, scientists developed a more direct way: the scanning electron microscope. It shoots a steady stream, or beam, of tiny material particles called electrons at the object to be detected. Similarly, your television set sprays an electron beam across the inside of the screen’s face to make each picture. An electron beam is fundamentally different from a light beam because electrons are made of matter—material substance having weight—whereas light is not made of matter. During

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Atoms the 1920s, physicists discovered that every particle of matter, such as an electron, has a certain kind of wave, called a “matter wave,” associated with it. Matter waves are terrific for detecting individual atoms, because they have wavelengths thousands of times smaller than the wavelength of light. As an electron microscope’s electron beam sweeps over, or “scans,” atoms, the beam’s matter wave is disturbed. The disturbed matter wave is monitored by devices that collect and record the patterns made by the electrons. An even more precise device called the scanning tunneling microscope was developed in 1983. It employs a tiny probe, shaped like a sharp pencil tip a few atoms wide, that scans a surface from just above the surface. In order to sense the microscopic structure of the surface, electrons move across the narrow gap between the surface and the probe in a uniquely quantum process called “tunneling.” Figure 10 shows a typical image. Because the tip of the probe can pick up individual atoms and drag them from place to place, scanning tunneling microscopes can perform the thought experiment that Democritus could only imagine 2500 years ago. In 1990, scientists picked up 35 individual atoms of xenon gas and rearranged them to spell out the name of their laboratory (Figure 11).8 They had divided xenon gas into its individual atoms, just as Democritus had imagined.

Although atoms are small, the nucleus at the atom’s center (see Section 7) is smaller still (Figure 12). The electron, also found within the atom, is known to be smaller than the smallest distance yet measured, which makes it at least 100,000 times smaller than the nucleus. Since it might in fact have zero size, the electron does not appear in Figure 12. At the other end of the scale, galaxies are among the largest objects known. Larger still are clusters of galaxies, forming thin “sheets” of galaxies that can individually stretch across as much as 1% of the known universe. The largest structures ever detected are the ripples in the faint afterglow of the big bang origin of the universe, stretching across two-thirds of the known universe. Humankind stands roughly in the middle, somewhere between the atoms and the stars (Figure 12). Atoms and molecules are pretty small. If you put a million atoms side by side, the lineup would be no longer than the period at the end of this sentence. The head of a pin contains more than 1018 atoms. One breath of air, about 1 liter (1000 cm3, about a quart), contains more than 1022 molecules. Now, 1022 also happens to be about the number of liters of air in Earth’s atmosphere, which leads to an interesting conclusion. Any particular parcel of air, such as the liter of air you will exhale in your next breath, mixes throughout Earth’s atmosphere within a few years. This means that of the air you exhaled a few years ago in any particular breath, about one atom is now in every liter of air on Earth and inside the lungs of every person on Earth! And about one atom that was breathed out by every person on Earth, in any particular breath, is in your lungs now. One from George Washington’s first breath, one from his dying breath, and one from every other breath that dear George ever took are in your lungs right now, along with atoms from each of the breaths of all the other people who have ever lived. You are a walking museum of history. Atoms are forever. Earth’s atoms have been here since Earth formed 5 billion years ago, and very few have changed during that time.9 It is only the connections

8 9

National Institute for Materials Science, Japan Figure 10

Scanning tunneling microscope (STM) image of a horizontal layer of silicon “dimers” (pairs of silicon atoms) with a single tungsten atom that the STM has deposited onto the surface. Note the nanometer (nm) distance scale.

IBM Corporation Figure 11

Thirty-five individual xenon atoms have been manipulated into position by the tip of a scanning tunneling microscope. The distance between atoms in the pattern is about 10-9 m—one-billionth of a meter, or 10 times the width of a single atom.

Xenon atoms do not combine easily with other atoms, making them easy to manipulate. Only those relatively few atoms that have been involved in radioactive decay, fission, or fusion have changed into different types of atoms.

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Atoms Meters 1025

1020

Size of known universe Ripples in cosmic background radiation Clusters and sheets of galaxies Milky Way galaxy

between atoms that have changed. A particular oxygen atom might be part of a nerve cell in your brain today, part of an atmospheric water molecule a century from now, and part of a tree a century after that. “Your” atoms, the ones you are carrying around right now, have just been borrowed from the air, from Earth, to be given back perhaps soon, perhaps later, to be given back entirely when you die. MAKI NG ESTI MATES

1015

Distances to nearby stars Solar system

1010

Sun Earth

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1

Tall mountain Tall building Human

10⫺5

Fine dust particle Wavelength of light

10⫺10

Typical atom

10⫺15

Smallest nucleus

10⫺20

Of all the molecules you have ever exhaled, about how many will your class instructor inhale during his or her next breath? This sounds hard. However, it’s surprisingly easy to make a rough estimate. In estimating very large numbers, an estimate to the nearest power of 10 is usually good enough: Is the answer closer to 10, or 100, or 1000, and so on? Such estimates are called “order-of-magnitude estimates.” We will make such estimates of all sorts of things throughout this text, but don’t expect a single correct answer. Different people will make different estimates, but all should be in the same ballpark. Suppose you exhale 12 times per minute (measure it!). Note how per is used—it always means “in each.” Multiplying by 60 minutes per hour, 24 hours per day, and 365 days per year, you’ll get your number of exhales per year. Since we want only a rough estimate, round off these numbers for easy multiplication: 10 * 60 * 25 * 400 = 6 * 106 exhales per year. If you are 20 years old, you have exhaled 20 * 6 * 106 or about 100 million times. How many of these exhaled molecules will your instructor inhale in one breath? As discussed above, he or she will inhale about one molecule from every one of your exhaled breaths. So the answer is 100 million! And in that same breath, he or she will also inhale some 100 million molecules from the exhaled breaths of each person living on Earth and from each person who has ever lived on Earth. And all of this will only be a tiny fraction of the total number of molecules your instructor will inhale in that breath. It’s something to consider when you take a breath.

Smallest distance yet measured

10⫺25

Figure 12

The range of sizes in the universe.

MAKI NG ESTI MATES About how many millimeters thick is one sheet of paper? (Hint: Roughly how thick is a 500-sheet package of typing paper?) My solution is at the bottom of the page. MAKI NG ESTI MATES

The U.S. national debt is about $12 trillion. If you stacked this up in new $100 bills, about how many kilometers high would the stack be? (Hint: Assume that they stack like typing paper and see the preceding question.)

6 ATOMIC MATERIALISM: ATOMS AND EMPTY SPACE Every time you drink a glass of water, you are probably imbibing at least one atom that passed through the bladder of Aristotle. A tantalizingly surprising result, but it follows [from the simple] observation that there are many more molecules in a glass of water than there are glasses of water in the sea. Richard Dawkins, Zoologist, Oxford University

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From 1550 to 1700, the revolutionary ideas of Copernicus and others became widespread, and educated people no longer viewed Earth as the central focus of the universe. Humans became passengers on one planet among many, inhabitants of a less SO LUTION TO MAKI NG ESTI MATES A 500-sheet package of typing paper is about 4 to 6 cm (a few inches) thick, say 5 cm. So the thickness of one sheet is about 5 cm/500 = 0.01 cm = 0.1 mm. SO LUTION TO MAKI NG ESTI MATES The number of $100 bills needed is 12 * 1012/100 = 1.2 * 1011. We saw in the preceding question that the thickness of one bill is 0.01 cm = 10-2 cm = 10-4 m. The height of the stack is 1.2 * 1011 * 10-4 m = 1.2 * 107 m = 12,000 km, which is more than twice the distance across the United States.

Atoms

personal universe (Figure 13). The new view stimulated an advancing scientific tide whose high point was Newtonian physics, the remarkably effective ideas about motion, force, and gravity developed by Isaac Newton (1642–1727) and others. During 1700 to 1900, its cultural influence spread far beyond science, affecting the way people thought about themselves, their society, and their place in the universe. Today, the Newtonian worldview still dominates our culture. In summing up his scientific career, Isaac Newton once stated, “If I have seen farther than others, it is by standing on the shoulders of giants.” Two such giants, René Descartes (1596–1650) and Galileo Galilei (1564–1642), helped establish the philosophy behind Newton’s physics. Descartes, Galileo, and Newton were the leading founders of science as we know it today. Although in Newton’s time there was very little evidence for atoms, the atomic idea underlies much of Newtonian physics. It was an idea that went pretty deep, a philosophical idea. As Democritus put it: By convention sweet is sweet, bitter is bitter, hot is hot, cold is cold, and color is color. But in reality there are only atoms and empty space. That is, the objects of sense are supposed to be real, and it is customary to regard them as such, but in truth they are not. Only the atoms and empty space are real.

All these things being considered, it seems probable to me that God in the Beginning formed Matter in solid, massy, hard, impenetrable, movable Particles, of such Sizes and figures, and with such other Properties, and in such Proportion to space, as most conduced to the end for which he formed them. Isaac Newton, 1704

Corbis/Bettmann

This goes far beyond the atomic theory. Democritus is saying that not only matter but everything is made of atoms and that atoms are all there is. So when you say

Figure 13

When medieval beliefs gave way to the new science of Copernicus and Newton, the cozy pre-Newtonian universe was replaced by a vast impersonal mechanical universe. This woodcut was made during the nineteenth century, long after the transition had taken place.

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“the water is hot” or “the shirt is red,” you really mean that the atoms in the water and shirt are moving in a certain way. There really is no such thing as hot or red— there are only atoms. According to Descartes, sense impressions such as hot and red are merely “secondary qualities” that exist only in our minds. The real universe, the universe outside our minds, contains only the atoms and their physical properties, such as weight and size. Descartes calls these the “primary qualities.” For Descartes, science’s task is to study the primary realm and explain it by means of atoms. Because such views about what is real go beyond what can be observed, they lie outside of science. They are philosophical, rather than scientific, views. These views are one version of materialism, the philosophy that matter is the only reality and that everything is determined by its mechanical motions. Not that Descartes, Galileo, or Newton were materialists themselves. They all subscribed to nonmaterialistic religious beliefs such as a belief in God. But the new scientific philosophy provided little room for the God of the Middle Ages, a God who is continually and intimately involved in the world. Instead, the founders of modern science believed in a God who created the universe and established the physical laws all at once, and who then merely maintained those laws. Newtonian physics is a remarkable achievement that is incredibly effective in explaining countless observed phenomena. Are its materialistic philosophical underpinnings then necessarily correct? Is it true that atoms are all there is? One reassuring thing about philosophy is that, for most good philosophical ideas, there are also good arguments to refute those ideas. As physicist Niels Bohr put it, “The hallmark of a profound idea is that its converse is also profound.” And so it is with atomic materialism. The respectable arguments on the other side include the following: 1. Science always starts from evidence, which comes ultimately from sense impressions. Materialism, which says that atoms are primary and sense impressions only secondary, has the cart before the horse. Theoretical ideas such as atoms are secondary to sensory evidence, rather than the other way ’round. 2. Although materialism is rooted in science, science is only one way of viewing reality. Other views—religious, aesthetic, intuitive—have equal claim to being “real.” Scientists themselves cover the gamut of religious views, from devout to atheistic. 3. All scientific ideas are only tentative, including atomic materialism. 4. Since 1900 scientists have found that Newtonian physics is only approximately correct over only a limited range of phenomena. Outside that limited range it is not even approximately correct. For example, we’ll find that some things (such as light) are nonmaterial and not made of atoms and that matter itself is made of nonmaterial “fields.” Whereas Newtonian physics is congenial to materialism, recent theories are more neutral or even uncongenial to materialism.

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Atoms

7 THREE ATOMIC MODELS: GREEK, PLANETARY, AND QUANTUM Science’s way of working back and forth between observations and theories comes with a kind of guarantee of success. If an observation agrees with a theory, that’s fine for that theory. And if an observation disagrees with a theory, that’s too bad for that theory, but it’s fine for science because science makes its greatest progress by repairing, or replacing, disproved theories. The atomic theory of matter is a good example. Science’s notion of the nature of atoms has changed several times. In the original Greek model of the atom, an atom was an unchangeable, single object like a small rigid pea. The Greek atom was also Newton’s way of looking at the atom, and it got support from the nineteenthcentury discoveries of elements, compounds, and Brownian motion. Around 1900, experiments with electricity showed that an “atom,” as that word was understood in 1900, was more complex than the Greek model of the atom. Electricity had been studied throughout the nineteenth century. Nobody suspected that an entirely new model of the atom would be needed to explain these experiments until, in 1897, scientists discovered a new, very lightweight, “electrified” particle. It was the first discovery of a particle that weighed less than an atom. It was, apparently, one part of the so-called atom (remember, the word means “indivisible”). This particle was the electron. A few years later, in 1911, physicists discovered that an atom is itself nearly entirely empty space and that nearly all of an atom’s material substance resides in a tiny central core, or nucleus. Experiments indicated that each atom also contained electrons moving through the large empty regions outside the nucleus. Scientists developed the theory that electrons orbit the nucleus much as planets orbit the sun. Later, scientists learned that the nucleus itself is made of two kinds of subnuclear particles, protons and neutrons. Figure 14 shows an atom of the element helium and an atom of the element carbon according to this planetary model of the atom. During the 1920s, new experiments involving electrons contradicted the planetary model and even contradicted the principles of Newtonian physics itself. An entirely new theory of matter called ”quantum theory” was needed to explain the new results. To date, no disagreements have been found between experiments and the new quantum theory of the atom. So our understanding of the atom has evolved through at least three different theories. At each stage, new experiments disproved the old theory, and scientists invented a broader theory that explained both the old and the new observations. But scientists did not discard the Greek and planetary models, despite their shortcomings, because these models are useful within their proper range. For instance, we can use the Greek atom to explain many common observations like air pressure. We need not resort to the planetary atom or the quantum atom to explain these things, because the atom’s internal structure and its quantum nature are irrelevant to these phenomena. Restricted to its proper range, the Greek atom is just fine. Once again, theories are best described as useful rather than true.

Helium

Carbon

Figure 14

An atom of helium and an atom of carbon, according to the planetary model of the atom. The small black dots are electrons in orbit around the nucleus, and the blue circles and white circles represent protons and neutrons in the atom’s nucleus. This is not drawn to scale! The nucleus should be 100,000 times smaller than the electron orbits, and electrons might have no size at all.

CONCEPT CHECK 11 The quantum theory of the atom agrees with every experiment to date. (a) Thus it can now be called an accepted fact rather than merely a theory. (b) Thus it can now be called a scientific hypothesis. (c) Thus it is now known to be certainly true, although we still refer to it as a “theory.” (d) Nevertheless, it remains

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Atoms

only a theory and thus is basically just a guess. (e) Thus it is properly called a scientific theory although, like all theories, it is somewhat tentative. (f) Nevertheless, it could still be disproved by future experiments.

8 CHEMISTRY AND LIFE: WHAT DID ATOMS EVER DO FOR YOU? You can get atoms to do fantastic things by connecting them in sufficiently subtle ways. It’s possible, for example, for a pile of atoms to acquire additional atoms from its environment, to move itself from one place to another, to respond to external events such as the presence of particular molecules, and to create copies of itself. We would call such a pile of atoms alive. Indeed, you are just such a pile of atoms. The pile of atoms that is you has an especially surprising property: It is aware of itself, and in this scientific age it is even aware that it is a pile of atoms. It’s something to think about. The chemical element that gives life its powerful abilities is carbon, which is plentiful on Earth and connects readily to a variety of other plentiful elements. Other elements that are abundant in biological molecules are oxygen, hydrogen, and nitrogen (remember “COHN”). Any rearrangement of molecules into new molecular forms is called a chemical reaction. As examples, this section looks at three chemical reactions that are important in your life: burning, respiration, and photosynthesis. It was once believed that fire was one of the substances of which things are made and that a burning object was releasing the fire that it already contained. The prevailing theory said that this intangible and nonmaterial substance—fire—had weight and carried its weight away when any object burned. Around 1780, Antoine Lavoisier studied burning more closely. He accurately weighed all the materials involved when an object burned, including the gases consumed and given off. Although the prevailing theory predicted that the weight should decrease, Lavoisier found that there was no net change in weight. This disproved the prevailing theory and initiated the modern science of chemistry—the study of the properties and transformations of substances (chemical compounds). The key to this new science was the principle that chemical reactions are rearrangements of atoms that are themselves changeless and indestructible. It followed that the total amount of matter involved in any chemical reaction is the same before and after the reaction. This idea is known as conservation of matter. We now know that, although it’s a useful theory that is very nearly correct in chemical reactions, experiments have proved it entirely wrong in other situations. Most burnable substances are derived from biological materials that contain carbon or hydrogen. As you can demonstrate by placing an inverted jar over a burning candle, burning requires air. Air is not a pure substance but is a mixture of many different substances. Nitrogen and oxygen dominate: Nearly 80% of air’s molecules are nitrogen 1N22, about 20% are oxygen 1O22, and 1% are single argon atoms (Ar). All the other gases added together total far less than 1%. These “trace gases” include all sorts of compounds. Some, such as water vapor 1H 2O2 and helium (He), are natural. Others, such as carbon dioxide 1CO22 and ozone 1O32, come from both

64

Atoms

natural and industrial sources. And still others, such as carbon monoxide (CO), come almost entirely from industry. For burning,10 the crucial component needed from air is oxygen. Carbon from the burning substance combines with oxygen from the air to form carbon dioxide. We abbreviate the preceding sentence symbolically as C + O2 ¡ CO2 The plus sign means “combined with,” and the arrow means “changes into.” If the fuel contains hydrogen, it too combines with oxygen to form water vapor. For example, methane gas, CH 4, is the simplest of the hydrocarbon (hydrogen and carbon) fuels. It is the main component of natural gas. It burns in air to form carbon dioxide and water vapor:11 CH 4 + O2 ¡ CO2 + H 2O An important feature of any chemical reaction is its energy balance. For now, I’ll use the important word energy to mean either of two things: the ability to move things around, and heat or, as I’ll call it, “thermal energy.” Thermal energy is related to, but not the same thing as, warmth. As you know, warmth—perhaps from friction or a burning match—is needed to start a substance burning. Once it starts, the burning reaction itself creates more than enough thermal energy to maintain itself, so excess thermal energy is given off. Including thermal energy, the reaction formula for burning a typical fuel such as methane is CH4 + O2 ¡ CO2 + H2O + excess thermal energy Turning to biology, animals get their bodily material and their energy from the food they eat and the air they breathe. Your blood absorbs carbon-based molecules from food and oxygen from air and ferries them all over your body. When they arrive at, say, your thumb, they enter a biological cell there. In a reaction known as respiration, the cell uses these substances to create biologically useful energy. In a typical case, a simple sugar called glucose reacts with oxygen to create carbon dioxide and water: C6H12O6 + O2 ¡ CO2 + H2O + useful energy As you can see, this is similar to burning. Animal life is a slow burn. In respiration, part of the useful energy goes into making a high-energy molecule known as ATP, and the rest appears as thermal energy. ATP is the energy carrier in animals; it can remain in storage, or it can move from place to place within the cell. It can be used for all sorts of things, like bending your thumb. As you can see from the reaction formula, respiration generates water and carbon dioxide as wastes that are excreted in your sweat, urine, and exhaled breath. Plants and animals have different energy strategies. Whereas animals gather energy by eating plants and other animals, plants use energy directly from the sun. Plants gather carbon dioxide and water from their surroundings and put them 10 Burning

is one form of combustion, which means any chemical reaction that generates warmth and light. formula is not quantitatively “balanced.” For example, there are four Hs on the left, but only two on the right. The balanced formula is CH 4 + 2O2 ¡ CO2 + 2H 2O. Since we are not interested here in how much of each compound enters into a reaction, we omit these numbers.

11 This

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Atoms

together to form high-energy carbohydrates (carbon compounded with water) such as glucose. From the formula for respiration, you can see that this process in plants is exactly the reverse of respiration in animals! Since respiration generates useful energy, the reverse reaction in plants must require an input of energy. Plants have worked out a complicated process that gets this needed energy from the sun. Since animals consume oxygen, we also expect that plants must generate oxygen. So a typical reaction is CO2 + H2O + solar energy ¡ C6H12O6 + O2 It’s called photosynthesis (putting together by light). Animals depend on plants not only for food and fuel but also for oxygen, the crucial component of the air we breathe. Nearly all of Earth’s oxygen comes from photosynthesis and did not exist in large, breathable amounts until after the rise of photosynthesizing bacteria some 2.7 billion years ago. Without plants, animals would soon be out of food, fuel, breath, and luck.

© Sidney Harris, used with permission.

CONCEPT CHECK 12 Some people have suggested that another substance, other than carbon, might conceivably be the key element in forming living organisms elsewhere in the universe. Which of these is the most plausible choice? (a) silicon. (b) oxygen. (c) chlorine. (d) argon. (e) neon. (f) peanut butter. (Hint: See the periodic table. Note: Scientists have rejected this suggestion as unrealistic.)

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Atoms Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions THE GREEK ATOM 1. What macroscopic evidence is there for atoms? 2. What light-based microscope evidence is there for atoms? 3. What experiment did the ancient Greek atomists imagine doing, and what did they believe the result would be? 4. An experiment such as the Greeks (previous question) imagined was actually carried out recently. Describe it. 5. Which is bigger, an atom or the wavelength of light? A little bigger or a lot?

ATOMS AND MOLECULES 6. From the microscopic point of view, what is the difference between an element and a compound? 7. From a macroscopic point of view, what is the difference between an element and a compound? 8. What is the difference between an atom and a molecule? 9. Why is the periodic table arranged in the way that it is? 10. If you chemically decompose water, will you get anything like water? What will you get?

THE ATOM’S EXPLANATORY POWER 11. Describe the microscopic process by which a flower gives off an odor that you can smell some distance away. 12. How do solids, liquids, and gases differ macroscopically? Microscopically? 13. Which is easiest to compress: solids, liquids, or gases? Why? 14. Is a perfect vacuum ever attained on Earth, over a volume as large as 1 cubic centimeter? Elsewhere? 15. What is the microscopic difference between hot water and cold water?

ATOMIC MATERIALISM AND ATOMIC MODELS 16. Name several things that people ordinarily regard as real but that, according to atomic materialism, are not real. 17. Describe the philosophy of materialism. 18. Give arguments for the materialist philosophy. 19. Give arguments against the materialist philosophy. 20. Name three different models of the atom. Describe two of them.

CHEMISTRY AND LIFE 21. What is meant by a chemical reaction? 22. Name three different chemical reactions. 23. Is air a single substance (a single compound)? Describe its chemical composition. 24. Describe an experiment involving burning that supports the notion of conservation of matter. 25. In what types of experiments is the conservation of matter correct to a very good approximation? 26. Describe the following reactions: burning, respiration, and photosynthesis.

Conceptual Exercises THE GREEK ATOM 1. Is the atomic theory known, for certain, to be true? 2. Carbon atoms are about 25% lighter than oxygen atoms (the ratio of their weights is 3 to 4). What is the weight ratio of the carbon and oxygen that go into the formation of carbon monoxide? Answer the same question for carbon dioxide. 3. A carbon atom is 12 times heavier than a hydrogen atom. If methane 1CH 42 is chemically decomposed, what will be the ratio of the weights of the resulting carbon and hydrogen? 4. A carbon atom is 12 times heavier than a hydrogen atom, and an oxygen atom is 16 times heavier than a hydrogen atom. If glucose 1C6H 12O62 is chemically decomposed, what will be the ratio of the weights of the resulting elements?

ATOMS AND MOLECULES 5. How many atoms are in a molecule of H 2SO4 (sulfuric acid)? 6. How many atoms are in the alcohol molecule C2H 5OH? 7. Which of these is a pure compound, which is an element, and which is neither: helium gas, carbon dioxide, polluted water, C6H 12O6, gold, steam? 8. Which of these is a pure compound, which is an element, and which is neither: pure water, oxygen gas, liquid mercury, H 2SO4, U, air? 9. Suppose you obtained the smallest single particle of each of the following substances. In which cases would this particle be a molecule made of more than one atom, and in which cases would it be a single unattached atom: pure water, oxy-

From Chapter 2 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Atoms: Problem Set

10. 11.

12.

13. 14. 15. 16.

17.

gen gas in the form found in Earth’s atmosphere, H 2SO4, U, He, carbon dioxide, H 2, H? Helium is an inert gas, meaning that it does not readily enter into chemical reactions with other substances. List five other substances that you would expect to also be inert gases. Chlorine has a strong tendency to combine with a single hydrogen atom to form HCl. Look in the periodic table and list at least three other elements that you would expect to combine with hydrogen the way that chlorine does. Consider a pure chemical substance A. Suppose that it can be chemically decomposed into two other pure substances B and C. Can we then conclude that B and C must be elements? That B and C must be chemical compounds? That A must be a chemical compound? What is the chemical formula for methane (carbon and four hydrogens)? What is the chemical formula for sulfur dioxide? What is the chemical formula for carbon tetrachloride (tetra means “four”)? On the simplifying approximation that oxygen and carbon atoms have the same weight, how many tons of carbon dioxide gas are formed when 1 ton of coal burns (coal is nearly pure carbon)? In a typical large coal-fed electrical generating plant, a ton of coal is burned every 10 seconds. About how many tons of carbon dioxide enter the atmosphere every hour from such a plant?

THE ATOM’S EXPLANATORY POWER 18. Why can’t you observe Brownian motion in easily visible objects such bits of paper floating in water? 19. What is the chemical formula for the odor of violets (see Figure 6)? 20. A dog follows an escaped convict’s trail by putting its nose to the ground. Explain this from a microscopic point of view. 21. If air is put into a sealed container that is then compressed (reduced in volume), what do you predict will happen to the air pressure on the container walls? Explain this from a microscopic point of view. 22. If air is put into a sealed container and warmed, what do you predict will happen to the air pressure on the container walls? Explain this from a microscopic point of view. 23. If a balloon is partially filled with air (so that it isn’t fully expanded), sealed, and then warmed, what do you predict will happen to the balloon? Explain this from a microscopic point of view. What if the balloon is cooled instead? 24. Why is it so difficult to remove the lid from a vacuumsealed jar? 25. Suppose that you observe the Brownian motion of tiny pollen grains floating in still air enclosed in a glass bottle. What would happen if you increased the amount of air? What would happen if you warmed the air? 26. Why does the air pressure in a tire increase as you add air? Why does the air pressure in a tire increase as you warm the tire? 27. Suggest an experiment that would show that air has weight. 28. What if, in addition to its random molecular motion, all the air molecules in some volume of air had an overall collective motion, all of them moving, say, eastward. Could this collec-

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Figure 6

The odor of violets, in air. The funny-looking thing is the odorof-violets molecule, made of carbon (horizontal stripes), hydrogen (black), and a single oxygen atom (green).

tive motion be observed macroscopically? What would we call it?

METRIC DISTANCES AND POWERS OF 10 29. Write as ordinary numbers: 109; 1026; 3.6 * 1013; 5.9 * 10-8. 30. Write in powers of 10 notation: 3 trillion; five-thousandths; 730,000,000,000,000; 0.000 000 000 082. 31. The distance to the moon is 384,000 km. Express this using powers of 10. How far is this in meters? In millimeters? 32. Our Milky Way galaxy contains perhaps 400 billion stars. Suppose that 0.05% (i.e., 0.000 5) of these stars have planetary systems containing at least one Earthlike planet (one that could conceivably support Earthlike life). Express these two numbers using powers of 10, and then multiply them together to find how many Earthlike planets there are in our galaxy. Express this number in words (thousands or millions, etc.). 33. The universe is a million trillion seconds old. Write this number in ordinary (not powers of 10) notation.

Atoms: Problem Set

THE SMALLNESS OF ATOMS 34. Put these in order from lightest to heaviest: water molecule, oxygen atom, raindrop, hydrogen atom, glucose molecule, electron, DNA molecule. 35. Put these in order from lightest to heaviest: H 2 molecule, methane molecule, fine dust particle, hemoglobin molecule, proton, glucose molecule. 36. How old are a baby’s atoms? Are they older than an old person’s atoms? What about a baby’s DNA molecules? 37. MAKING ESTIMATES One sheet of paper is about 0.1 mm thick. An atom is about 10-10 m across. About how many atoms thick is one sheet of paper? 38. MAKING ESTIMATES The average weight, per atom, of the atoms in your body is about 10-26 kg 12 * 10-26 pounds2. About how many atoms are there in your body? 39. MAKING ESTIMATES The smallest dust particle visible to the unaided eye measures about 0.05 mm across. About how many atoms across is this? In other words, if we line up atoms side by side, about how many would it take to make a line of atoms 0.05 mm long? 40. MAKING ESTIMATES Referring to the preceding exercise: Assume the small dust particle is shaped like a cube. About how many atoms does it contain?

CHEMISTRY AND LIFE 41. What is the chemical reaction formula for burning hydrogen gas in air? What substance is created by this reaction? 42. For safety, gas-filled balloons are filled with helium instead of hydrogen. What does this tell you about the behavior of helium in the atmosphere? 43. Gasoline is a hydrocarbon fuel. What are the two main compounds created when gasoline burns in a car engine? 44. NOX (nitrogen oxide and nitrogen dioxide) is one pollutant from automobiles. What elements must combine to form NOX? 45. Gasoline contains neither oxygen nor nitrogen. So where must these elements come from when NOX is formed in car engines? 46. Are there any molecules in your body that you could claim are “your” molecules, unique to your body and probably unlike any other molecules in the universe?

Answers to Concept Checks 1. [Are you reading this before forming your own answer? If

2. 3. 4. 5. 6.

so, do you exercise by watching somebody else jog? Exercise your mind by providing your own input to the Concept Checks!] In Democritus’s time, this was a hypothesis, but today it is an established scientific theory, (d). It is incorrect to call a general idea, such as this one, a fact or observation. The weight of one sulfur atom is the same as the weight of two oxygen atoms, so the ratio is 1 to 1, (c). Now the ratio is 1 to 1.5, which is the same as 2 to 3, (d). (a), (d), and (f) (b) (f)

7. (a), (c), (d), and (e). Note that we can observe the light

8. 9. 10. 11. 12.

from stars [answer (b)], despite the absence of air in outer space. And a rock would still fall [answer (f )] even if there were no air. 106 * 1012 = 1018, (e) 10 - 2 * 10 - 12 = 10 - 14, (c) (b) (e) and (f). Note that answer (d) is wrong: A hypothesis is an educated guess, but a theory is far more than that. Silicon, (a), in the same column with carbon in the periodic table.

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. No. General scientific principles are never certain. 3. 12 to 4, in other words 3 to 1. 5. 2 + 1 + 4 = 7. 7. Helium is an element, carbon dioxide is a pure compound, polluted water is neither (it is a mixture), C6H12O6 is a pure compound, gold is an element, and pure unpolluted steam is a pure compound (H2O). 9. Molecule made of two or more atoms: pure water (H2O), atmospheric oxygen (O2), H2SO4, carbon dioxide (CO2), H2. Single unattached atom: U, He, H. 11. The elements lying in the same column with chlorine in the periodic table are fluorine, bromine, iodine, astatine. 13. CH4. 15. CCl4. 17. Since 1 ton of coal is burned every 10 seconds, about 3 tons of CO2 enters the atmosphere every 10 seconds. In 1 hour there are 3600 seconds, or 3600>10 = 360 of these 10-second intervals. So the number of tons of CO2 entering the atmosphere in 1 hour is roughly 3 * 360 = 1080 tons. 19. Figure 6 shows that the molecule is made of 14 carbon atoms (C), 22 hydrogen atoms (H), and 1 oxygen atom (O), so the chemical formula is C14H22O. 21. When the container’s volume is reduced, an individual air molecule hits the inner walls of the container more often because it has less space in which to move around. So the walls will be struck more often by moving air molecules. In other words, the pressure will increase. 23. When the air is heated, the balloon will expand a little because the molecules are moving faster and hit the walls harder, pushing the walls further apart (since the balloon is not fully expanded to begin with). When the air is cooled, the balloon will shrink a little. 25. Since there are now more air molecules, the pollen grains will be hit more often, and they will not travel as far between hits (changes in velocity). If we heat the air, the air molecules will be moving faster, the pollen grains will be hit harder, and the pollen grains will gain greater speeds. 27. Weigh two identical rigid containers, one containing air and one that has had some of its air removed. If air has weight, the container with more air should weigh a little more.

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Atoms: Problem Set 29. 109 = 1,000,000,000

31. 33. 35. 37. 39.

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10 - 6 = 0.000 001 3.6 * 1013 = 36,000,000,000,000 5.9 * 10 - 8 = 0.000 000 059 3.84 * 105 km, 3.84 * 108 m, 3.84 * 1011 mm. 1,000,000,000,000,000,000 seconds. Proton, H2, methane (CH4), glucose molecule (C6H12O6), hemoglobin molecule, dust particle. How many times does 10 - 10 m go into 0.1 mm? Since -4 - 10 = 0.1 mm = 10 - 4 m, the answer is 10 >10 10 - 4 + 10 = 106 atoms, or one million atoms thick. An atom is about 10 - 10 m across (Section 3). Since 0.05 mm = 5 * 10 - 5 m, the number of atoms needed to -5 - 10 = stretch across a dust particle is 5 * 10 >10

5 * 10 - 5 + 10 = 5 * 105 atoms, or 500,000 atoms (half a million). 41. Hydrogen and oxygen come in the two-atom form, H2 and O2. They combine to give water: H2 + O2 : H2O. 43. Hydrocarbons are made of hydrogen (H) and carbon (C). When these burn in air containing O2, the H should combine with O2 to create H2O, and the C should combine with O2 to create CO2. 45. From the atmosphere, which contains an abundance of N2 and O2. These elements don’t combine at normal atmospheric temperatures, but at the high temperatures prevailing in automobile engines they do combine.

How Things Move

From Chapter 3 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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How Things Move Galileo Asks the Right Questions

I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use. Galileo

W

hen you look around, your eye falls on a book, a flower, your foot. What are these things made of? The ancient Greeks answered that things are made of atoms. You notice that the book falls, the flower sways, your foot taps. Why, and how, do things move? Again, the ancient Greeks asked such questions (and so did others, such as Chinese naturalists of that same time). The Greek philosopher and scientist Aristotle (Figure 1) developed the earliest theory of motion. His theory was intuitively plausible and had some observational support, but later scientists such as Galileo and Newton discarded Aristotelian physics in favor of powerful new ideas that then dominated science for three centuries. Beginning in 1900, science again changed its view of motion, when the relativity and quantum theories altered most of Newtonian physics. Although we now know that these ideas are inaccurate outside of the situations encountered in everyday life on Earth, Newtonian physics continues to be useful for understanding the way the macroscopic world around us works and forms the basis for many of the technologies we rely on every day. Perhaps more importantly, Newtonian views have retained their powerful cultural influence. After a look at Aristotelian physics (Section 1), this chapter discusses Galileo’s ideas about motion. Section 2 presents Galileo’s objections to Aristotelian physics and the experimental background for the law of inertia, the foundation of Newtonian physics. Section 3 examines this law. Sections 4 and 5 explore speed, velocity, and acceleration, ideas needed to describe motion. Section 6 applies all of this to a familiar phenomenon, falling.

1 ARISTOTELIAN PHYSICS: A COMMONSENSE VIEW Aristotle’s physics agrees with most people’s common sense. But these plausible notions are precisely the ones that Newtonian physics discarded. Because Aristotelian physics is so ingrained in our intuitions, we had best see where it went wrong.

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Aristotle noticed that some motions maintain themselves without assistance, and he called these natural motions. For example, a rock pushed off a ledge falls toward the ground with no obvious assistance, so he considered the fall of a solid object to be a form of natural motion. In his view, solid objects fall because they are made of the Aristotelian element “earth” and thus seek to get as close as possible to their natural resting place, which is the center of the solid Earth. In addition to solid objects falling, Aristotle perceived three other sorts of natural motion on Earth: water falling or running downhill, air rising, and flames leaping upward. All these natural motions are downward or upward. Horizontal motions seem different. When you pull a cart along a road, throw a stone horizontally, or push a box along the floor, your activity maintains the motion: Pushes and pulls are needed to keep the cart, the ball, and the box moving. Aristotle believed these pushes and pulls were needed because objects must be forced to behave contrary to their own natural motion. Aristotle classified such motion as “violent motion,” meaning that an external push or pull was needed to maintain it. He believed that all motion on Earth was either natural or violent, but he perceived in the heavens an entirely different kind of motion. He believed that the moon, sun, planets, and stars were made of a substance called ether (from the Greek word for “blaze”), which was not found on Earth. Ether had no weight and was incorruptible (unchangeable, eternal). Perfect in every way, ether’s natural place was in the heavens, and it naturally moved in perfect circles around Earth. This third kind of motion was called “celestial motion.” Aristotle’s theory explained lots of observations. It gained wide acceptance, partly because of its plausibility. It does seem to us that a rock falls all by itself, that a push is needed to maintain horizontal motion, and that motion in the heavens really is different from motion on Earth. But as you will see, Newtonian physics contradicts all three of these notions.

Erich Lessing/Art Resource, N.Y.

How Things Move

Figure 1

The Greek philosopher and scientist Aristotle, 384–322 BCE. He developed the earliest theory of motion. His theory, which agreed with our common intuitive notions, was replaced by the far less intuitive theory of Newton.

2 HOW DO WE KNOW? DIFFICULTIES WITH ARISTOTELIAN PHYSICS Aristotelian physics had its weaknesses. You can demonstrate some of these for yourself. Drop a piece of notebook paper to the ground. Now crumple it into a tight ball and drop it again. Does it fall faster?1 Aristotelian physics has a hard time explaining this result. After all, the crumpled paper is still the same paper, so it should “seek” Earth’s center equally as strongly as the flat sheet, and should fall no faster than the flat sheet. Now try dropping two objects that have the same shape but very different weights, such as a rock and a tightly crumpled piece of newspaper of about the same size (Figure 2). What do you find? According to Aristotelian physics, the heavier object, containing more of the element “earth,” should seek out Earth’s center more strongly, and so should fall noticeably faster. But it does not. If you do this experiment carefully and from a high place such as a second-story window, you might detect that the heavier object actually does fall a little faster, but not a lot faster as is predicted by Aristotle’s physics. Today’s theories predict that for two objects of the same shape, the lighter one will fall a little slower because of 1

Figure 2

Hold a rock and a wadded-up piece of paper above the ground and drop them simultaneously. What does Aristotelian physics predict? What do you observe?

Please do these simple experiments when they are suggested, to reinforce your learning. There’s nothing like observing the real thing!

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How Things Move

air resistance—the resistance to the motion of an object through the air due to the object’s collisions with numerous air molecules. According to today’s theories, air resistance also explains why a flat sheet of paper falls slowly. One can test the hypothesis that these small differences in falling are due to air resistance by letting two objects fall in a vacuum, perhaps inside a container from which the air has been removed. One then finds that the light and heavy objects fall at precisely the same speed. A feather dropped in a vacuum falls as fast as a rock, in clear contradiction of Aristotelian predictions (Figure 3)! Since Galileo Galilei was one of the first scientists to challenge Aristotle on this point, I’ll summarize these conclusions as:

Galileo’s Law of Falling If air resistance is negligible, then any two objects that are dropped together will fall together, regardless of their weights and their shapes, and regardless of the substances of which they are made.

Figure 3

Editorial Photocolor/Art Resource, N.Y.

A feather dropped in vacuum falls as fast as a rock.

Figure 4

Galileo Galilei, 1564–1642. He helped overthrow Aristotelian physics, helped formulate the law of inertia, made astronomical discoveries that supported the Copernican view of the universe, and much more. But his most important contribution might have been his development of the scientific process: the notion that we learn not from authority but rather from experience and rational thought.

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This is an amazingly general statement. It applies to any two objects. Each object could be anything: cannon ball, frog, feather, helium-filled balloon, even an individual atom,2 just as long as air resistance is negligible. Aristotle’s concept of violent motion also has problems. If you shoot an arrow, it can travel a great distance horizontally while hardly slowing down. A brief strong push from the bowstring starts it, but what external assistance keeps it moving once it is released from the bow? Aristotle himself had difficulty reconciling this sort of example with his own theory, and later scientists had similar difficulties. Galileo (Figure 4) was a brilliant, cocky Italian who supported Copernican astronomy, issued sarcastic opinions about Aristotelian physics, and generally annoyed the authorities. His writings eventually earned him a visit by the Catholic Inquisition, which “persuaded” the now elderly man to “confess” and then confined him to house arrest for his remaining 10 years of life. Even under house arrest, the irrepressible Galileo pursued his experiments and wrote a large physics book. To focus his thinking, Galileo imagined the following experiment: Let a ball roll down an incline. Its speed will increase. Now give the ball a starting push and let it roll up an incline. It slows down (then stops and rolls back down). Suppose we make the inclines nearly horizontal (Figure 5). If you have ever let a ball roll down a very slight incline, you know that it’s likely to slow down and come to rest, even though it’s going downhill. Galileo understood that this slowing is due to the roughness of the incline and ball. Today it’s called friction. Galileo’s crucial step was to idealize the experiment by neglecting, at least in his mind, the effect of friction. He saw that if there were no friction, the ball would speed up on any downward incline, no matter how slight, and would slow down on any upward incline. Then he took another brilliant step: He imagined the “limiting case” of slight inclines, namely, a perfectly horizontal surface. On a frictionless horizontal surface, the ball could neither speed

2

In 1999, researchers were for the first time able to compare the motion of freely falling individual atoms with the fall of a macroscopic object such as a rock. It was not an easy experiment because, in order to observe only their falling, the atoms’ thermal motion had to be removed by cooling them to within two-millionths of a degree of “absolute zero” (the lowest temperature allowed by the laws of physics). The atoms fell just like rocks.

How Things Move

(a)

(b)

(c)

Figure 5

A smooth ball on a smooth incline always (a) speeds up going down and (b) slows down going up, even for a very slight incline. In the limiting case (c) of a perfectly smooth and level surface, the ball should keep going forever.

up nor slow down because the surface was intermediate between downhill and uphill. Galileo concluded that in absence of friction, a ball that once started rolling on a horizontal surface would roll forever. This radically contradicted Aristotle’s theory, which stated that continued pushing or pulling was needed to maintain violent (that is, horizontal) motion, and led directly to the law of inertia (next section), the foundation of post-Aristotelian physics. Galileo’s methods have been crucial to science ever since. They included the following: • Experiments, designed to test specific hypotheses. • Idealizations of real-world conditions, to eliminate (at least in one’s mind) any side effects that might obscure the main effects. • Limiting the scope of the inquiry by considering only one question at a time. For example, Galileo separated horizontal from vertical motion, studying only one of them at a time. • Quantitative methods. Galileo went to great lengths to measure the motion of bodies. He understood that a theory capable of making quantitative predictions was more powerful than one that could make only descriptive predictions, because quantitative predictions were more specific and could be experimentally tested in greater detail. Galileo was one of the first people to practice what we recognize today as the scientific process: the dynamic interplay between experience (in the form of experiments and observations) and thought (in the form of creatively constructed theories and hypotheses). This notion that scientists learn not from authority or from inherited beliefs but rather from experience and rational thought is what makes Galileo’s work, and science itself, powerful and enduring.

The Holy Spirit intended to teach us in the Bible how to go to heaven, not how the heavens go. Galileo

A grain of sand falls as rapidly as a grindstone. Galileo

3 THE LAW OF INERTIA: THE FOUNDATION OF NEWTONIAN PHYSICS Galileo and other scientists eventually arrived at a profound non-Aristotelian insight. It involved an extreme idealization: Suppose you could get away from the effects of friction and air resistance and also gravity. It isn’t easy to imagine such a

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How Things Move

Oh, my dear Kepler, how I wish that we could have one hearty laugh together! Here at Padua is the principal professor of philosophy, whom I have repeatedly and urgently requested to look at the moon and planets through my glass, which he perniciously refuses to do. Why are you not here? What shouts of laughter we should have at this glorious folly! And to hear the professor of philosophy at Pisa labouring before the Grand Duke with logical arguments, as if with magical incantations, to charm the new planets out of the sky. Galileo, in a Letter to Kepler, Commenting on Libri, Teacher of Philosophy at Padua, Who Refused Even to Look into Galileo’s Telescope in Order to View the Newly Discovered Moons of Jupiter

Libri did not choose to see my celestial trifles while he was on Earth; perhaps he will do so now he has gone to Heaven. Galileo’s Further Comment on Libri, Who Died Soon After the Telescope Incident

thing, because gravity is so omnipresent that you scarcely notice it. French philosopher and scientist René Descartes was the first to imagine the absence of gravity and understand its consequences. What if you could turn off gravity? This question would have been meaningless to Aristotle, because for him there was no such thing as gravity. Objects just fell, by themselves, because that was their nature. But Descartes realized that, if you released a stone in midair and there were no gravity or friction or air resistance, the stone would not fall. It would hang, motionless, in midair. And if you flicked that motionless stone with your finger, it would coast in a straight line with no change in speed, forever! Descartes is saying that without gravity or friction or air resistance, an object that was moving to begin with would keep moving without external assistance. And an object that was at rest to begin with would stay at rest; it would hang in midair, for instance. This is strange, counterintuitive. Because gravity, friction, and air resistance are all around us, our intuition tells us that objects can keep moving only if they are pushed or pulled, and objects can hang in midair only if something holds them up. As a way of holding on to our intuitive notion that something must assist an object if it is to keep moving, scientists give a word to an object’s tendency to keep moving or to remain at rest: “inertia.” In other words, an object’s inertia is its tendency to maintain its state of motion, whether moving or remaining at rest. This word inertia doesn’t really explain anything; it is simply a word that stands for the unexplainable fact that unassisted objects do keep moving. I’ll summarize all of this as: Law of Inertia3 A body that is subject to no external influences (also called external forces) will stay at rest if it was at rest to begin with and will keep moving if it was moving to begin with; in the latter case, its motion will be in a straight line at an unchanging speed. In other words, all bodies have inertia.

You see the law of inertia in action in any motion that is horizontal (to eliminate the effect of gravity) and nearly frictionless, such as a bowling ball rolling slowly down a bowling alley, or an object coasting on a cushion of air. Figure 6a is a multiple-flash photograph of such an air-coaster (see Figure 6b), made in a completely dark room with a camera whose shutter remained open, using a rapidly flashing light to illuminate the coaster only briefly at several equally spaced times. As you can see by checking the meter stick next to the coaster, the coaster is moving in a straight line at an unchanging speed: It moves the same distance (so far as it is possible to measure this in the photo) in each time interval. Although careful measurement would show that air resistance slows the coaster slightly, this example is close enough to Galileo’s ideal case that if you view it in a lab, it will give you an intuitive feel for the law of inertia. Outer space furnishes lots of nice examples. Outer space refers to those regions of the universe outside Earth and outside other astronomical objects, where “Earth” 3

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This is often called Newton’s first law, even though Descartes invented it, because Newton listed it first among his three basic principles of motion. I will refer to these three principles as the law of inertia, Newton’s law of motion, and the law of force pairs, rather than by their more common but less accurate, less descriptive, and more boring titles: Newton’s first law, Newton’s second law, and Newton’s third law.

Uri Haber-Schaim

How Things Move

Uri Haber-Schaim

(a)

(b)

Figure 6

(a) Multiple-flash photo of the motion of an air coaster on a smooth horizontal surface, viewed from above. (b) The coaster at rest.

includes the atmosphere. The atmosphere thins out at higher altitudes, becoming so thin above 100 km that the drag (air resistance) on satellites is nearly negligible. Beyond about this altitude lies outer space. Figure 7 puts this altitude into perspective. Let me clarify some common misconceptions about the word space. Space is all around you. There is space between you and objects on the far side of the room you are in. The space within a few miles of Earth’s surface is filled with air—but not completely filled because the empty spaces between the air’s molecules are far larger than the molecules themselves. There’s nothing dramatically different about this near-Earth space and the outer space that lies above Earth’s atmosphere. The major difference is simply that there’s far less air up there than down here; that is, outer space is closer to being empty space than is the space in your room. But the same laws of physics, such as the law of inertia and the law of gravity, that operate down here also operate up there. When astronauts traveled to the moon, their spacecraft’s rocket engines first boosted (pushed) them up and into orbit around Earth. Then they fired their rocket engines for a few minutes to leave Earth’s orbit and start toward the moon. Then they shut down their engines and coasted, for three days, to the moon. The spacecraft became a long-distance coaster and a great example of the law of inertia. But this coasting was not entirely free of external influences. Although there is no significant air resistance in outer space, there is still significant gravity unless the spacecraft is extremely far from all large bodies such as Earth, the sun, and the moon. Gravity has a very long-range effect. For example, at one-sixth of the distance to the moon, gravity is still 1% as strong as it is on Earth’s surface—a muchreduced effect but still not negligible. The spacecraft to the moon slowed during the first part of its journey because of the pull of Earth’s gravity and then sped up during the last part because of the moon’s pull.

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How Things Move Layer of very thin atmosphere, too thin for breathing, about 90 km high Black line represents the thin layer of breathable air, 10 km high. This is the height of Mount Everest.

A typical low-orbit satellite, greatly enlarged.

Radius of solid Earth, 6000 km Outer space begins here, at about 100 km above Earth’s solid surface.

Figure 7

The solid Earth, Earth’s atmosphere, and outer space. The drawing is roughly to scale, except that the artificial satellite is far too large. A real orbiting satellite, several meters in size, would be a microscopic dot on this diagram.

A big difficulty for the sun-centered astronomy proposed by Copernicus around 1550 was the problem of how Earth could keep moving with nothing to push it. This problem perplexed Copernicus, and it gave his opponents powerful ammunition. The answer came a century too late to help Copernicus. Earth is a coaster in space! Like astronauts coasting to the moon, Earth coasts around the sun. It keeps going because there’s nothing to stop it. But why, you might ask, does it move in a circle rather than in a straight line at unchanging speed? And what started it moving in the first place? The answers are that the sun’s gravity bends Earth’s path into a circle, and Earth started moving because the gas and dust from which it was formed were already moving. Another difficulty for a moving-Earth theory of astronomy is that it seems as though birds and other objects not attached to the ground should be left behind as Earth moves through outer space. The answer is that birds have inertia too. A bird standing on the ground is participating in Earth’s 30-kilometer-per-second coaster

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ride around the sun, so when the bird takes to the air, it is already moving at this speed around the sun, and the law of inertia says that there’s no reason for it to stop. CONCEPT CHECK 1 According to Aristotelian physics, which outside influences act on a stone while it falls? (a) Air resistance (b) Inertia (c) Gravity (d) The tooth fairy (e) There are no outside influences. CONCEPT CHECK 2 According to Newtonian physics, which outside influences act on a stone while it falls to the ground? (a) Air resistance (b) Inertia (c) Gravity (d) There are no outside influences. CONCEPT CHECK 3 Suppose that, because of unknown causes, the sun suddenly appeared to “stand still” in the sky. From the viewpoint of Copernican astronomy, this would mean that (a) Earth stopped spinning around its own center; (b) Earth stopped moving in its orbit around the sun; (c) the sun stopped moving in its orbit around the center of the Milky Way Galaxy; (d) the sun stopped moving from east to west; (e) the sun stopped moving from west to east.

4 MEASURING MOTION: SPEED AND VELOCITY Sometimes quantitative methods are needed to get at nature’s deeper secrets. At other times, quantitative details are superfluous and qualitative descriptions are preferable. For a clear understanding of motion and related topics like force and energy, you need to think both qualitatively and quantitatively. And quantities (numbers) are needed to understand important practical matters such as world energy problems. Scientists like to specify their measurements in terms of just a few basics. To describe motion, only two are needed: distance and time. For example, suppose you want to describe quantitatively the motion flash-photographed in Figure 6 using only the meter stick shown and a clock. Suppose the flasher flashes steadily at 0.40-second intervals and that you start the clock at the first flash, when the coaster is at the 10-centimeter (cm) mark. Then successive flashes occur at 0.40 seconds (s), 0.80 s, 1.20 s, 1.60 s, 2.00 s, and 2.40 s. Table 1 tabulates these data, with distances estimated to the nearest millimeter (0.1 cm) from Figure 6. How can you use these data to describe how fast the coaster is moving? In other words, how can these data tell you what a speedometer attached to the coaster would read? A speedometer tells you the distance traveled per (in each) unit of time, such as the number of kilometers per hour or the number of centimeters per second. So if you were to walk 6 kilometers (km) in 2 hours, the number of kilometers in each hour would be 6 divided by 2, or 3 km per hour. From this, you can see that speed is distance divided by time. Returning to Table 1, during each 0.40-s time interval the coaster travels 14.1 cm. So a speedometer attached to the coaster should register 14.1 cm divided by 0.40 s, or 35.2 centimeters per second. I’ll call this the coaster’s speed. I’ll abbreviate it as 35.2 cm/s, where the divide sign (/) is to be read “per” or “in each.” But there’s a catch. Suppose you ride your bicycle 24 km in 3 hours. According to this definition, your speed would be 8 km/hr. But surely you did not maintain exactly this speed during every minute of the 3-hour trip. Rather, the 8 km/hr is an

Table 1 Positions and times for air coaster of Figure 6 Clock time (s)

Position (cm)

0.00

10.0

0.40

24.1

0.80

38.2

1.20

52.3

1.60

66.4

2.00

80.5

2.40

94.6

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overall speed, actually the speed you would have had to maintain in order to make the 3-hour trip at an unchanging speed. I’ll call this the average speed for the trip. But a car’s speedometer gives a value at every instant of time. What does a speedometer read? A speedometer is based on wheel rotation rates. If you look closely at the way it operates, you will find that it actually reads only an average speed during some time interval, but that this time interval is very short—so short that the speed is practically unchanging during this time interval. This average speed during a time interval so short that the speed hardly changes is called the instantaneous speed. It’s defined just like the average speed, but with the understanding that the time interval is short. It’s what a speedometer reads. I will use the unmodified word speed to mean “instantaneous speed,” and I’ll use average speed when that's what I mean. Quantitative statements, such as our definition of average speed, are often easier to grasp if written as an abbreviated formula: average speed =

distance traveled traveling time

This can be further abbreviated using symbols. We can choose the symbols to suit ourselves. I will use s to represent instantaneous speed, s for average speed, d for distance traveled, and t for traveling time. Then the formula is s =

d t

Please don’t be intimidated by formulas like this. A formula is just an abbreviation for words. It’s the idea, not the formula, that’s essential. And many of the most important principles, such as the law of inertia, are best stated without formulas. In fact, if you do use a formula, be sure you can first state it carefully in words, because otherwise you could fool yourself into thinking you understand the idea when all you’ve done is memorize some symbols. For example, the t in the speed formula’s denominator is not just any arbitrary time—it means something very specific: the duration of the time interval during which the object traveled the distance indicated in the numerator. When you travel, it makes a difference which direction you move. Jogging at 10 km/hr northward will get you to a different place than will jogging at 10 km/hr westward. Speed and direction of motion occur together so frequently in physics that it’s useful to have a separate word for the combination. I will use the word velocity to mean speed and direction. The words speed and velocity are interchangeable in everyday language, but in physics they are not. Test your understanding of speed and velocity by trying these questions: CONCEPT CHECK 4 A car travels 12 km in half an hour, while a bicyclist “sprints” for 1 minute at a steady 30 km/hr. The one with the higher average speed is (a) the car; (b) the bicyclist. CONCEPT CHECK 5 In which of the following cases is the car’s speed increasing? (a) A car covers longer and longer distances in equal time intervals. (b) A car takes longer and longer time intervals to cover equal distances. (c) A car covers equal distances in equal time intervals. (d) A car covers equal distances in shorter and shorter time intervals. (e) A car takes equal time intervals to cover equal distances. (f) In equal time intervals, a car covers shorter and shorter distances.

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CONCEPT CHECK 6 Two bicyclists, both moving at 10 km/hr, pass each other on a straight road, one moving north and the other moving south. These bicyclists have (a) the same speeds and the same velocities; (b) different speeds and different velocities; (c) different speeds but the same velocities; (d) the same speeds but different velocities.

5 MEASURING MOTION: ACCELERATION The law of inertia says that a body that feels no external forces must either move at unchanging speed in a straight line or remain at rest if it were at rest to begin with. But remaining at rest is just a special case of an unchanging speed (the speed remains zero), so we can state the law of inertia as simply “a body that feels no external forces must move at unchanging speed in a straight line.” But motion in a straight line at an unchanging speed is motion at an unchanging velocity, and the condition of remaining at rest is also a condition of unchanging velocity. So we can state the law of inertia more concisely: Law of Inertia (more concise form) A body that is subject to no external forces must maintain an unchanging velocity.

Now suppose that there are external forces (external influences). How will they affect an object’s motion? It’s not too difficult to guess the answer once you understand the law of inertia: External forces must cause changes in velocity. Any object whose velocity is changing is said to be accelerated. Concept Check 7 will exercise your thinking about this idea. Remember: An object is accelerated only if its velocity is changing, and velocity refers to the combined instantaneous speed and direction. CONCEPT CHECK 7 During a trip, a car executes several kinds of motion. In which of the following cases is the car accelerated? (a) Moving along a straight, level road at a steady 70 km/hr. (b) Moving along a straight, level road while slowing down from 70 km/hr to 50 km/hr. (c) Rounding a curve at a steady 50 km/hr. (d) Moving uphill along a straight incline at a steady 50 km/hr (Figure 8). (e) Rounding the top of a hill at a steady 50 km/hr (Figure 9). (f) Starting up from rest along a straight, level road.

You have seen how to describe velocity in terms of measured quantities. What about acceleration? To answer this, imagine a car moving north along a straight, level highway. Suppose it speeds up, say from 60 km/hr to 72 km/hr. Its change in speed is then 12 km/hr. Imagine how this would feel to you if you were in the car. It would make a difference to you how fast this change took place. If it took place over an entire hour, you would hardly notice it, but if it occurred during one-tenth of a second, you could wind up with a whip-lashed neck! So the rate at which the speed changes—the amount of speed change per second—is important. Suppose that the time interval is 8 s. Then the amount of speed change per second is 12 km>hr

Figure 8

Illustration for Concept Check 7.

Figure 9

Illustration for Concept Check 7.

8s

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or 1.5 “kilometers per hour per second.” These units tell us that in every second, the speed changes by 1.5 km/hr. We’ll write this as 1.5 (km/hr)/s. This useful quantity, which measures the rate of speeding up, is called the “acceleration” of the car. But remember that an object is said to be accelerated whenever its velocity changes and that the velocity changes not only when the object speeds up but also when it slows down or changes direction. This means that an object’s acceleration is its change in velocity (and not simply the change in speed) divided by the time to make the change. Written as a formula, acceleration =

change in velocity time to make the change

Please note that this physicists’ definition is a little different from the popular definition. The scientific meaning includes not just speeding up (the common meaning of acceleration) but also slowing down (commonly called deceleration) and changing direction. Definitions of words like acceleration are arbitrary, in the sense that nature does not tell us that we must define these words in any particular way. Physicists define their words for maximum convenience. The distinction between velocity and acceleration is important and often misunderstood. “Velocity” refers to motion itself—an object has a velocity whenever it is moving. But “acceleration” refers only to changes in velocity. CONCEPT CHECK 8 Which of these have a high velocity and low acceleration? (a) A speeding bullet moving through air. (b) A race car just as it begins to “dig out” from rest. (c) A fast train as it moves around a long and gentle curve. (d) A fast car as it collides with a brick wall. (e) A golf ball at the instant it is struck by a fastmoving golf club. CONCEPT CHECK 9 In the preceding question, which ones have a low velocity and high acceleration?

6 FALLING Galileo’s law of falling tells us that, because all objects fall in the same way, we can learn about the fall of any object by studying the fall of just one particular object— a book, for example. How do we know objects speed up as they fall? Hold a book, flat, above the floor, and let it go. ——— This is a pause, for dropping your book. What is the numerical value of the book’s speed at the instant you release it? ——— Another pause, to think about that. While you are holding it, the book’s speed is zero, so its speed at the instant of release must also be zero. But does its speed remain zero? Obviously not. At the beginning of the motion, the speed increases from zero to something bigger than zero. So the book must accelerate, at least at the beginning. But does your book accelerate all the way down—does its speed keep on changing? To answer this, drop your book from about half a meter above the floor. Pick it up and drop it again from about 2 m above the floor. Listen as it hits the floor. Did it hit noticeably harder

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How Things Move the second time? If so, the book must have been moving faster at the end of the second drop. Apparently, objects keep on moving faster and faster as they fall farther. So falling objects accelerate all the way down. The preceding paragraph demonstrates the power of careful observation. As the Yankee catcher and famous sage Yogi Berra put it, “You can learn a lot just by looking.”

The multiple-flash photo in Figure 10 shows how to measure falling. A billiard ball falls past a 2-m stick while being photographed at several equally spaced times. The time interval between photos is 1/30 s. You can see the ball’s acceleration: The images get farther and farther apart. Figure 11 is an idealized drawing of a ball falling through a larger distance, neglecting air resistance. A real object falling farther than about 20 m is strongly affected by air resistance because air resistance becomes stronger as an object moves faster. We will, however, imagine that there is no air resistance, in order to focus on the effects due to gravity alone. An object like this whose falling is influenced by gravity alone is said to be in free fall. Suppose you measure distances downward from the release point and that a speedometer attached to the ball measures the ball’s speed as it falls. At the instant of release, you start a clock that tells you the time elapsed since the ball was dropped. Figure 11 gives some of the data you would get in this idealized experiment. In order to highlight the pattern, the actual experimental data are rounded.4 You can see, in three different ways, that the ball accelerates: First, you can see directly from the drawing that the distances fallen during each successive second (0 to 1 s, 1 to 2 s, and so on) get larger and larger. Second, the speed is greater at the end of each successive second, so the ball is moving faster and faster. Third, if you look closely at the distance data, you’ll see that the distances covered during each successive time interval grow larger and larger. For instance, the distance covered during 0 to 1 s is 5 m, and the distance covered during 1 to 2 s is 15 m. How big is this acceleration? Recall that the acceleration is the change in velocity divided by the duration of the time interval. Now look at each successive 1-s time interval. All the changes in speed are precisely the same! During 0 to 1 s, the speed change is 10 m/s. During 1 to 2 s, the speed changes by 10 m/s. And so forth. So, calculating the acceleration for any one of these time intervals, acceleration =

10 m>s change in speed = time interval 1s

which is 10 (m/s)/s, abbreviated as 10 m/s2. This means that the speed changes by 10 m/s in every second. So the acceleration is the same, 10 m/s2, all the way down. And Galileo’s principle of falling tells us that it must be the same for every freely falling object. The numerical value of this acceleration due to gravity is, more accurately, 9.8 m/s2. There is a pattern in the speed data. At times of 0 s, 1 s, 2 s, 3 s, 4 s, and 5 s, the speed (in m/s) is 0, 10, 20, 30, 40, 50, and so on. Just by looking at this string of Figure 10 4

The rounding process introduces a 2% error. More precise distances are 4.9 m, 19.6 m, 44.1 m, 78.4 m, and 122.5 m. More precise speeds are 9.8 m/s, 19.6 m/s, 29.4 m/s, 39.2 m/s, and 49 m/s. Using these more accurate numbers, the calculated acceleration is (9.8 m>s)>(1 s) = 9.8 m>s2.

Multiple-flash photo of a falling billiard ball. The position scale is in centimeters, and the bulb flashed every 1/30 s.

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0s 1s

Approximate distance 0m 5m

2s

20 m

20 m/s

3s

45 m

30 m/s

Time

Approximate speed 0 m/s 10 m/s

numbers, you can guess what should come next: 60. The reason this can be guessed is that there is a recognizable pattern here. Patterns are what scientists look for in nature! Your expectation that the sun will rise tomorrow is also based on a pattern in nature. One way to describe this pattern is with proportionalities. One quantity is proportional to a second quantity if doubling the first means you must double the second, tripling the first means you must triple the second, and so forth. In general, whatever you multiply the first quantity by, you must also multiply the second quantity by. Looking at the speed data, doubling the time (from 1 to 2 s, say) results in doubling the speed (from 10 to 20 m/s), and similar proportionalities hold throughout the data. So, for free fall, speed is proportional to time; s r t

4s

80 m

40 m/s

5s

125 m

50 m/s

Figure 11

A freely falling ball dropped from the top of a tall building. The effect of air resistance is neglected in this illustration.

The symbol r means “is proportional to.” Note that a proportionality such as s r t is not the same thing as an equation. Distance cannot equal time, for the same reason that apples cannot equal oranges. Similarly, speed is proportional to the time for any object that starts from rest and moves in a straight line with unchanging acceleration. For example, an object dropped onto the surface of the moon falls freely (there’s no air resistance on the moon, because there’s no air) with an unchanging acceleration of 1.6 m/s2. So a falling object reaches a speed of 1.6 m/s at the end of 1 s of falling, 3.2 m/s at the end of 2 s, 4.8 m/s at the end of 3 s, and so forth. As on Earth, speed is proportional to time. The pattern in the position data of Figure 11 is not as easy to recognize. To make it easier to find, let’s express the position data in multiples of the position at 1 s, in other words, in 5-m units. The data tell us that at 2 s, the position (the total distance fallen) is 20 m, or 4 of these units; it is 9 units at 3 s, 16 units at 4 s, 25 units at 5 s. So the positions, measured in 5-m units, are 0, 1, 4, 9, 16, 25. What’s the next number in this sequence? Have you got it? Each of the numbers is a perfect square: 02, 12, 22, 32, 42, 52. Next comes 62, or 36 of our 5-m units! To get the distance in meters, multiply this by 5, getting 180 m. This pattern can also be described quantitatively with proportionalities. The distances, in meters, are proportional to the square of the time. This means that doubling the time multiplies the distance by 4, tripling the time multiplies the distance by 9, and so forth. In general, whatever number you multiply the time by, you must square this number and then multiply it by the distance. For instance, if you multiply the time by 3 (from 1 to 3 s, say), then you must multiply the distance by 32, or 9. So for free fall: distance fallen is proportional to the square of the time; d r t2 Again, this is a proportionality that is valid for any case of unchanging acceleration. For instance, distance is proportional to the square of the time for an object falling freely onto the moon.5 5

With further analysis, we could arrive at two equations for the speed and position of any object that starts from rest and moves in a straight line with unchanging acceleration s = at,

1 d = a b at2 2

where a represents the acceleration. For freely falling objects on Earth, a = 9.8 m>s2.

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CONCEPT CHECK 10 In four times as much time, a freely falling object dropped from rest falls (a) twice as far; (b) 4 times farther; (c) 8 times farther; (d) 12 times farther; (e) 16 times farther; (f) into your soup. CONCEPT CHECK 11 Would the answer to the preceding question be altered if the object were falling freely onto the surface of Mars? (a) Yes. (b) No. CONCEPT CHECK 12 In four times as much time, a freely falling object gets going (a) twice as fast; (b) 4 times faster; (c) 8 times faster; (d) 12 times faster; (e) 16 times faster.

© Sidney Harris, used with permission.

CONCEPT CHECK 13 For a freely falling object, (a) the total distance covered (distance fallen) keeps increasing; (b) the distance covered during each second keeps increasing; (c) the speed keeps increasing; (d) the change in speed during each second keeps increasing; (e) the acceleration keeps increasing.

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How Things Move Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions ARISTOTELIAN PHYSICS 1. Describe the four kinds of motion that Aristotle considered to be natural on Earth. 2. Give two examples that Aristotle considered to be violent motions. 3. According to ancient Greek thought, in what fundamental way does Earth differ from the heavens? 4. Give an example of a motion that contradicts Aristotelian physics. 5. According to Aristotelian physics, why does a stone fall when it is released above the ground? According to Newtonian physics? 6. Describe at least one principle of Aristotelian physics that seems intuitively plausible but that Newtonian physics rejects.

THE LAW OF INERTIA 7. What does it mean to say that an object has inertia? 8. What does the law of inertia say about the velocity of a body that is subject to no external influences? What does this law say about such a body s acceleration? 9. Which of the following describes how high you must go before you will first reach outer space : anywhere above the ground, about 100 m, about 1 km, about 100 km, beyond the moon, beyond the solar system? 10. Is there air in outer space? 11. Give at least one example that demonstrates, at least approximately, the law of inertia in others words, an example of unassisted motion at constant, or nearly constant, velocity.

SPEED AND VELOCITY 12. Describe how you could use a clock and a meter stick to measure a moving object s speed. 13. When we say 5 centimeters per second, what does the per mean? 14. What is the difference between speed and average speed? In what circumstances are they the same? 15. Can you give an example in which the speed is unchanging but the velocity changes? If so, give one. 16. Can you give an example in which the velocity is unchanging but the speed changes? If so, give one.

ACCELERATION 17. A car speeds up along a straight line. Describe how you could use clocks and meter sticks to measure its acceleration. 18. How is acceleration related to velocity? 19. If an object s position is changing, can we be certain that it has a nonzero velocity? Can we be certain that it has a nonzero acceleration? 20. If an object is moving in a circle at an unchanging speed, is it accelerated? 21. If an object is slowing down, is it accelerated?

FALLING 22. An object is released above the ground and falls freely. At which of the following places during the fall is its velocity greatest: the top, the midpoint, or a point near the bottom? At which position is its acceleration greatest? 23. What is the meaning of the phrase acceleration due to gravity ? What is its approximate value on Earth? 24. What does speed is proportional to time mean? 25. In twice the time, does a freely falling object fall (from rest) twice as far? Does it gain twice as much speed?

Conceptual Exercises ARISTOTELIAN PHYSICS 1. You roll a ball. It soon rolls to a stop. How would Aristotle interpret this? How would Galileo interpret it?

THE LAW OF INERTIA 2. Most meteoroids pebble-sized to boulder-sized rocks in outer space have been moving for billions of years. What, if anything, keeps them moving? 3. If you ride on a smooth, fast train at an unchanging speed and throw a baseball upward inside the train, will the baseball then get left behind and come down toward the rear of the car? Explain. 4. If a ball is moving at 20 m/s and no forces ever act on it, what will its speed be after 5 s? After 5 years? 5. Do you suppose that the photo of Earth shown in Figure 7a was taken from a low-orbit satellite (see Figure 7b on next page)?

From Chapter 3 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright ' 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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How Things Move: Problem Set Figure 7a

NASA Headquarters

A whole-world view showing Africa and Saudi Arabia taken 7 December 1972 as Apollo 17 left Earth s orbit for the moon. The cultural impact of photos like this, showing Earth as a single, freely moving ball in space, may be among the space program s most important benefits.

Layer of very thin atmosphere, too thin for breathing, about 90 km high Black line represents the thin layer of breathable air, 10 km high. This is the height of Mount Everest.

A typical low-orbit satellite, greatly enlarged.

Radius of solid Earth, 6000 km Outer space begins here, at about 100 km above Earth’s solid surface.

Figure 7b

The solid Earth, Earth s atmosphere, and outer space. The drawing is roughly to scale, except that the artificial satellite is far too large. A real orbiting satellite, several meters in size, would be a microscopic dot on this diagram.

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How Things Move: Problem Set 6. In order to experimentally verify the law of inertia, would you need to be able to measure time? Weight? Distance? 7. When a moving bus comes rapidly to a stop, why do the riders who are standing lurch toward the front of the bus?

SPEED AND VELOCITY 8. Can you drive your car around the block at a constant velocity? 9. Mary passes Mike from behind while bicycling. As she passes him, do the two have the same velocity? The same speed? 10. Mary is bicycling straight north at 15 km/hr, and Mike is bicycling straight south at 15 km/hr. As they pass each other, do they have the same velocity? The same speed? 11. Figure 12 represents a multiple-flash photo of two balls moving to the right. The figure shows both balls at several numbered times. The flash times are equally spaced. Which ball has the greater acceleration? The greater speed? The greater velocity? Does either ball pass the other, and if so, when? 1

1

2

3

2

5

4

3

4

6

6

Figure 12

12. The automobile beltway around many cities is approximately circular. Suppose that you start driving at point A on the east side of the beltway and drive counterclockwise at an unchanging 90 km/hr (Figure 13). As you pass point B on the north side, what is your speed? Your velocity? B

C

City

1

1

2

2

3

4

3

5

4

5

6

6

7

7

8

Figure 14

7

5

17. Is the motion sickness that some people get in a car actually due to motion per se or to something else? Describe one form of motion that would not make people sick. 18. When you drive a car, might you depress the accelerator pedal without actually accelerating? Could you accelerate without having your foot on the accelerator? Explain. 19. One car goes from 0 to 30 km/hr. Later another car goes from 0 to 60 km/hr. Can you say which car had the greater acceleration? Explain. 20. Figure 14 represents a multiple-flash photo of two balls. Describe each ball s motion. Does either ball pass the other? When? Do they ever have the same speed? When?

A

Figure 13

13. Referring to the preceding exercise, as you pass point C on the west side, what is your speed? Your velocity? 14. In Figure 12, suppose the large divisions on the measuring rod are centimeters and that the time intervals each have a duration of 0.20 s. Find the speed of each ball. 15. Find the average speed of a jogger who jogs 3 km in 15 min. Give your answer in km/hr.

ACCELERATION 16. A French TGV train cruises on straight tracks at a steady 290 km/hr (180 mi/hr). What is its acceleration?

21. In each of the following cases, is the motion accelerated or not accelerated? (a) A rock falling freely for 2 m. (b) A meteoroid (a rock in outer space) that is so far from all planets and stars that gravity is negligible. (c) An artificial satellite orbiting Earth at a steady 30,000 km/hr. (d) The moon. (e) An ice-skater coasting on smooth ice, neglecting friction and air resistance. 22. Can a slow-moving object have a large acceleration? Can a fast-moving object have a small acceleration? 23. Which devices in a car are designed to cause acceleration? 24. For an unassisted (unforced, or isolated) moving object, which of the following quantities change, and how do they change: distance (from the starting point), speed, velocity, acceleration? 25. An automobile moves along a straight highway at an unchanging 80 km/hr. During the motion, which of the following quantities change, and how do they change: distance (from the starting point), speed, velocity, acceleration? 26. A ball rolls down a straight ramp. During the motion, which of the following quantities change, and how do they change: distance (from the starting point), speed, velocity? 27. A bicyclist increases her speed along a straight road from 3 m/s to 4.5 m/s, in 5 s. Find her acceleration. 28. A car accelerates from 0 to 100 km/hr in 10 s. Find its acceleration. Drag racers can get to 400 km/hr from rest in 5 s. How big is this acceleration? 29. Find the acceleration of a car as it speeds up from 70 to 82 km/hr in 4 s. From 70 to 82 km/hr in 16 s. From 70 to 94 km/hr in 8 s. From 70 to 76 km/hr in 8 s.

FALLING 30. Multiple choice: Two metal balls are dropped from a thirdstory window at the same time. They are the same size, but one weighs twice as much as the other. The time to reach the ground will be (a) about twice as long for the heavy ball; (b) about twice as long for the light ball; (c) about the same for both; (d) considerably longer for the heavy ball, but not necessarily twice as long; (e) considerably longer for the light ball, but not necessarily twice as long.

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How Things Move: Problem Set 31. A falling object has a speed of 10 m/s at t = 1 s. So it seems that it should move 10 m in 1 s. Yet the data say that the object moves only 5 m in 1 s. What is wrong here? 32. Figure 15 represents a multiple-flash photo of a falling ball. Neglect air resistance. At which point, A or B, is the ball s acceleration larger? At which point is its velocity larger?

A

B

Figure 15

33. By how much does a freely falling object s speed increase during its third second of fall (from t = 2 s to t = 3 s after release)? During its fourth second of fall? 34. For an object that is freely falling to Earth, which of the following quantities increase during the fall: distance (from the starting point), speed, velocity, acceleration? 35. As an object falls freely, what will be its speed at the end of the first second? Second second? Third second? 36. As an object falls freely, what will be its acceleration at the end of the first second? Second second? Third second? 37. An astronaut on another planet, one that has no atmosphere, drops a rock off a cliff. How much faster is the rock moving at the end of 3 s as compared with 1 s? How much farther (measured from the release point) does the rock fall in 3 s as compared with the distance it falls in 1 s? 38. Neglecting air resistance, would the answers to the preceding exercise be different if the rock is dropped on Earth? Neglecting air resistance, would the distance fallen in 3 s on Earth be likely to be the same as the distance fallen in 3 s on the other planet?

1

Problems VELOCITY 1. Worldwide sea levels are predicted to rise at a rate of at least 5 mm per year during the next century, due to global warming. At this rate, how long will it be before sea levels have risen by 0.5 m? 2. It takes light about 8 minutes to travel here from the sun. Given that the speed of light is 300,000 km/s, how far is it to the sun? 3. It is 3.8 * 108 m to the moon. How long does it take a radar beam, traveling at the speed of light (300,000 km/s), to get from Earth to the moon and back? 4. You wish to travel from downtown New York City (NYC) to downtown Washington DC (DC), a distance of 330 km. You consider two options: train and plane. The high-speed train takes 1.5 hr, plus 30 min in stations. The airplane flies the 330 km (airport to airport) in just 30 min, but the drive to the NYC airport takes 30 min, you must arrive 2 hr before departure time, the plane waits 15 min for takeoff, and it takes 45 min to get your luggage and drive into DC. Find the train s track speed, the plane s flying speed, the total travel time for each option, and the overall average speed for each option. 5. You drive from New York City to Washington DC by car. You drive in traffic for the first hour at an average of 50 km/hr. You cover the next 250 km in 3.0 hr, and then drive the remaining 30 km into Washington in 30 min. Find your total time and average speed.

ACCELERATION1 6. A car starts from rest and maintains an acceleration of 4.5 (km/hr)/s for 5 s. How fast is it going at the end of the 5 s? 7. A car is moving at 30 km/hr. The driver then presses harder on the accelerator, causing an acceleration of 2.25 (km/hr)/s, which she maintains for 4 s. How fast is the car going at the end of the 4 s?

FALLING 8. You drop a rock off a cliff and note that it hits the ground below in 6 s. How high is the cliff, assuming that the air is so thin that air resistance can be ignored during the entire fall? Unless the air is extremely thin, air resistance will not be negligible. What does this tell you about your calculation of the cliff s height: Is the cliff actually higher, or is it lower, than your calculated answer? 9. In the preceding problem, how fast is the rock moving when it hits the ground, still assuming negligible air resistance?

With further analysis, we could arrive at two equations for the speed and position of any object that starts from rest and moves in a straight line with unchanging acceleration s = at,

1 d = a bat2 2

where a represents the acceleration. For freely falling objects on Earth, a = 9.8 m>s2.

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How Things Move: Problem Set 10. You drop a rock down a well and hear a splash 3 s later. As Charlie Brown would say, the well is 3 seconds deep. But how many meters deep is it, assuming that air resistance is negligible and that the time for sound to travel back up the well is also negligible? 11. In the preceding problem, how fast is the rock moving when it hits the water? 12. You drop an apple out of a third-story window. When does it pass the second-story window 4 m below? 13. When does the apple in the preceding question hit the ground 8 m below? 14. A car speeds up from rest, along a straight highway. Its acceleration is unchanging. How much farther (measured from the starting point) does it get in 10 s, as compared with 1 s? 15. In the preceding problem, how much faster is it moving in 10 s, as compared with 1 s? 16. On the planet Mars, a free-falling object released from rest falls 4 m in 1 s and is moving at 8 m/s at that time. How fast would such an object be moving after 2 s? 3 s? 17. In the preceding problem, how far would such an object fall in 2 s? 3 s?

Answers to Concept Checks 1. Artistotle believed that it was part of the nature of stones and 2. 3.

4. 5. 6. 7.

8. 9. 10. 11. 12. 13.

other heavy objects to fall, so no outside influence was required, (e). (a) (if the stone is falling through air) and (c). (a) It s interesting to note that, because of Earth s daily spin, a typical point on Earth s surface moves around Earth s center at about 1600 km/hr (1000 mi/hr). So the law of inertia tells us that, if Earth suddenly stopped spinning, people and houses would find themselves sliding across Earth s surface at a speed of 1600 km/hr. Eventually, friction would bring them to rest. Most objects would slide for a few minutes across a distance of about 25 km (15 miles) and their surfaces would be heated by friction to nearly the boiling point of water. The car s average speed is 24 km/hr, which is less than the bicycle s 30 km/hr, (b). (a) and (d) (d) The car is accelerated when its velocity is changing that is, whenever either the speed or the direction of motion is changing. Answers: (b) (speed is decreasing), (c) (direction of motion is changing), (e) (direction of motion is changing), (f) (speed is increasing). Note that there is no acceleration in (d), because neither the speed nor the direction is changing at the instant shown. (a) and (c) (b) and (e) (e) (b) (b) (a), (b), and (c)

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. Aristotle: This is the ball s natural motion. Galileo: Friction slowed it to a stop. 3. The baseball keeps up with the train and comes back down in your hand, just as though you were standing still on Earth s surface. Explanation: Because of the law of inertia, the ball keeps moving in the forward direction with no change in its forward speed, even though you have released the ball. Your throw simply gives the ball an upward (and then downward) motion, on top of the forward motion that the ball had before you threw it. 5. Judging from Figure 7b, an observer in low orbit would see only a small portion of Earth. Thus Figure 7a must have been taken from a much greater distance. 7. They lurch forward because of their own inertia their bodies have a tendency to keep on moving. 9. They must have different speeds, because she is passing him. Since their speeds are different, their velocities are different, because velocity means speed and (not or) direction; that is, both the speeds and the directions of motion must be the same before we can say that the velocities are the same. 11. Neither ball has any acceleration. The lower ball has the larger speed and the larger velocity. It starts out behind the upper ball, catches up at time 2, and then passes the upper ball. 13. Speed 90 km/hr, velocity 90 km/hr toward the south. 15. Speed = 3 km>0.25 hr = 12 km>hr. 17. It is due to bouncing, or shaking back and forth. These are accelerations. Nonaccelerated motion would not make people sick. 19. No, you cannot say, because you don t know how long each acceleration took. 21. (a) Accelerated. (b) Not accelerated. (c) Accelerated, since it is moving in a circle. (d) Accelerated, since it is moving in a circle. (e) Not accelerated. 23. Accelerator pedal, brake, steering wheel. 25. The object moves with an unchanging velocity. Thus, distance changes; it gets larger and larger, at a steady rate. Speed does not change. Velocity does not change. Acceleration remains zero, so it does not change. 27. accel = change in speed>time

(4.5 m>s - 3 m>s) 5s 1.5 m>s = 0.3 (m>s)>s = 5s 12 km>hr = 3(km>hr)>s 29. 4s 12 km>hr = 0.75(km>hr)>s 16 s 24 km>hr = 3(km>hr)>s 8s 6 km>hr = 0.75(km>hr)>s 8s =

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How Things Move: Problem Set 31. The object begins with a speed of 0 m/s and speeds up to a

speed of 10 m/s by the end of the first second, so its average speed during the entire first second (from 0 to 1 s) is only 5 m/s. 33. By 10 m/s. By 10 m/s. 35. 1st second: about 10 m/s. 2nd second: about 20 m/s. 3rd second: about 30 m/s. 37. Three times as fast, because speed is proportional to time; nine times as far, because distance is proportional to the square of the time. Problems 1. 0.5 m = 500 mm. Solving v = d>t for t gives t = d>v. So the number of years until the level has risen 500 mm is 500 mm>(5 mm>y) = 100 y. 3. The distance to the moon and back is 7.6 * 108 m. The speed is 300,000 km>s = 3 * 105 km>s = 3 * 108 m>s. Solving d = st for t, we get t = d>s = (7.6 * 108)> (3 * 108 m) = 2.5 s.

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5. total time = 1.0 hr + 3.0 hr + 0.5 hr = 4.5 hr.

7.

9. 11. 13. 15. 17.

total distance = 50 km (during first hr) + 250 km + 30 km = 330 km. average speed = 330 km>4.5 hr = 73 km>hr. The change in speed is at = 2.25(km>hr)>s * 4 s = 9 km>hr. This must be added to the initial speed of 30 km/hr. Thus the final speed is 30 km>hr + 9 km>hr = 39 km>hr. s = at = (9.8 m>s2) * (6 s) = 59 m>s. s = at = (9.8 m>s2) * (3 s) = 29 m>s. d = (1>2)at2. Solving for t, t = 2(2d>a) = 2(2 * 8>9.8) = 1.28 s. Speed is proportional to time, so the car is moving 10 times faster after 10 s than after 1 s. Distance is proportional to the square of the time, so it has moved four times as far, or 16 m, after 2 s. It has moved nine times as far, or 36 m, after 3 s.

Why Things Move as They Do Nature and nature’s laws lay hid in night; God said, “Let Newton be,” and all was light. Alexander Pope, Eighteenth-Century British Poet

T

he world changed in 1687. In that year, Isaac Newton (Figure 1) published his Mathematical Principles of Natural Philosophy. To take just one example, by making Descartes’s and Galileo’s law of inertia the foundation of his work, Newton undermined our intuitive view of how things move, a view accepted by all educated people for 2000 years. The Newtonian world is surprisingly simple. Using only a few key principles, Newton was able to give quantitative explanations for all manner of things: planets, moons, comets, falling objects, weight, ocean tides, Earth’s equatorial bulge, stresses on a bridge, and more. It was an unparalleled expansion and unification of our understanding of nature. Newton’s influence ranged far beyond physics and astronomy. Not only the sciences but also history, the arts, economics, government, theology, and philosophy were shaped by the general patterns of Newtonian physics. For example, the ideals of inalienable human rights that inspired the American and French revolutions stemmed largely from a populace steeped in a Newtonian culture of universal natural law that applied equally to all people, to commoners and kings alike. Newtonian physics worked almost too well. Unchallenged for over two centuries, it was eventually regarded as absolute truth. The very word understand came to mean “to explain in terms of Newtonian physics.” Most importantly, people eventually took for granted many subtle Newtonian habits of mind that had profound but unstated and unexamined implications having to do with determinism, cause and effect, the mechanical nature of the universe, and other philosophical conclusions.1 Eventually, everyone from the laborer to the scholar assumed that Newton had laid the framework for all human knowledge. During the twentieth century, relativity and quantum physics superseded Newtonian physics. But Newtonian cultural habits remain, partly because there is no agreed-upon philosophical framework for the new physics and partly due to the failure of science educators to teach the new physics to all people. Thus, our culture remains largely Newtonian while our science is post-Newtonian...not a healthy situation. In order that 1

Two classic historical studies of physical science from the early Greeks through Newton examine the transition to the new worldview. The very title of Arthur Koestler’s The Sleepwalkers (New York: Universal Library and The Macmillan Company, 1963) refers to the philosophically naive manner in which the Newtonian view developed. E. A. Burtt’s The Metaphysical Foundations of Modern Science (originally published in 1932; reissued by Humanities Press, Atlantic Highlands, NJ, 1980) is a close examination of the history and implications of these unstated philosophical assumptions.

From Chapter 4 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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you may develop the tools to help pull all of us into the post-Newtonian age, I’ve chosen modern post-Newtonian physics and its significance as one of the themes of this text. Newton’s physics starts from only a few concepts and principles. You have already learned two concepts, velocity and acceleration, and two principles, the law of inertia and the law of falling. Newton’s other key concepts are force and mass (Sections 1 and 2). His other key principles are the law of motion (Sections 3 and 4), the law of force pairs (Section 5), and the law of gravity. Section 6 applies these ideas to the motion of a device that has, for better or worse, reshaped our cities, our landscapes, and our lives: the automobile. Section 7 presents another way of looking at Newtonian physics based on the concept of “momentum.” American Institute of Physics/ Emilio Segre Visual Archives Figure 1

Isaac Newton, 1642–1727. His Mathematical Principles of Natural Philosophy, summarizing his, Descartes’s, and Galileo’s studies on the motion of material objects on Earth and in the heavens, may well be the single most important book in the history of science. Newton was not only the greatest genius that ever existed, but also the most fortunate, inasmuch as there is but one universe, and it can therefore happen to but one man in the world’s history to be the interpreter of its laws. Pierre-Simon De Laplace, Scientist

Figure 2

Both Sam and Sally are exerting a force on the ball, but the ball is not accelerating. When we say “Sam exerts a force on the ball” we mean that Sam would cause the ball to accelerate if no other forces were acting. In the case pictured, Sally’s force on the ball prevents Sam’s force from causing it to accelerate.

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1 FORCE: WHY THINGS ACCELERATE We have used the word force synonymously with “external influence.” Now I need to be more specific. Since the law of inertia says that bodies having no external influence (no force) on them are unaccelerated, it is natural to define a force as any external influence that causes a body to accelerate. So a body “exerts a force on” another body whenever the first body causes the second body to accelerate. Some examples: If a ball is at rest on a table and you push it with your hand, the ball will accelerate into motion. If the ball is already moving across a table and you push it briefly from behind, it will speed up. If you “pat” a moving ball lightly on the front side, toward the rear, it will slow down. If you pat it lightly sideways, it will change directions. In all four cases, your hand push accelerated the ball. In fact, a little experimentation shows that you cannot push the ball without accelerating it.2 So every hand push is a force. When you use the word force, it is useful to remember that a force is like a push. There are many misconceptions about the word force. Just like the word push, a force is an action rather than a thing. An object cannot be a force or possess force. Instead, force is something that one object does to another, just like “pushing.” A body can “exert a force on” another body. Pulling is another example. Starting with a ball at rest on the table, you can grasp it and pull it toward you, accelerating it into motion. So pulls are forces. Instead of pulling the ball with your hand, you could attach a string and pull the string, which pulls on the ball to accelerate the ball. So strings, when they pull, exert forces. Now suppose that Sam and Sally both push on a ball in opposite directions (Figure 2). If they adjust their pushes, they can get them to balance so that the ball remains at rest. Yet both are pushing on the ball. Even though the ball is unaccelerated, we will say that Sam exerts a force on the ball and that Sally does, too. In cases involving more than one force, a body exerts a force on another body if, in the absence of the other forces, the first body would cause the second body to accelerate. You could tap a ball with a hammer instead of pushing or pulling it with your hand. Since the hammer tap accelerates the ball, it exerts a force on the ball. Try

2

Assuming that there is only one push at a time. Two simultaneous pushes in opposite directions could cancel each other.

Why Things Move as They Do

tapping a motionless or moving ball from various directions yourself, and observe it carefully. Exactly when is it accelerating? ———This is a pause for finding a ball and a hammer and trying this. Observe carefully. The ball accelerates only during the fraction of a second when the hammer is touching it. So the hammer exerts a force on the ball only during this fraction of a second. After the tap, the ball moves at an unchanging speed in a straight line, so there is no force exerted on it. Note that the moving ball does not “have force” and it does not “carry force along with it.” A force is like a push. You wouldn’t say that the ball “has push.” Friction and air resistance are two more forces. If you briefly shove a book and let go so that it slides across a table, the book will slow down as it slides. Some force must cause this acceleration (recall that slowing down is one type of acceleration). This force results from the contact between the book and the tabletop, and is exerted by the tabletop on the bottom of the book. This force exists because both surfaces are rough or uneven at the microscopic level, as you can verify by sliding the book across a smoother surface and observing that the (de-)acceleration is reduced. Such a force, by one surface on another surface due to the roughness of the surfaces, is called friction. A fast bullet moving horizontally through air slows a little, so there must be a force on the bullet. This force is caused by the bullet hitting air molecules as it travels. It is called air resistance. Air resistance is similar to friction: The bullet slides through the air in somewhat the same way that a book slides across a table. You know that an apple falling freely to the ground accelerates all the way down. Since the apple accelerates, there must be a force on it, commonly called the force of gravity. But remember that forces are always actions by one object on another object. Gravity is a force on the apple, but what is this force exerted by? The answer is that it is exerted by planet Earth. The experimental evidence for this is that no matter where you go on Earth, a falling apple always accelerates downward, toward Earth’s center. You can think of a gravitational force as a pull, although not a human, muscular pull. It’s a pull by Earth on nearby bodies. There’s an interesting difference between gravitational forces and the other forces we’ve looked at. Hand pushes, hand pulls, hammer taps, string pulls, friction, and air resistance all are contact forces: forces exerted by an object that is touching another object. The gravitational force by Earth on a falling apple is different, because Earth is not actually touching the apple while it falls. Air is touching the apple, but you could imagine removing the air and the apple would still fall. The gravitational force acts at a distance, across empty space.

I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Newton

2 CONNECTING FORCE AND ACCELERATION The crux of Newton’s theory of motion is really simple: forces cause accelerations. It’s a surprising idea. Our Aristotelian intuitions tell us that outside influences are needed to keep something moving, that is, that forces cause (or maintain) velocities. But Newton says no force is needed to keep a thing moving, and that forces instead cause accelerations. Newton formulated the specific relation between force and acceleration. To follow his reasoning, suppose that you put a smooth ball on a smooth table and tap it once with a hammer. As you know, it accelerates during the tap. Experience

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shows that the more strongly you tap, the faster it will be moving after the tap. So stronger forces cause larger accelerations. When this kind of experiment is done quantitatively, it is found that as the force on an object increases, the object’s acceleration increases in exactly the same proportion: A doubled force causes a doubled acceleration, a tripled force causes a tripled acceleration, and so forth. So an object’s acceleration is proportional to the total force exerted on it. In symbols, a r F How do we know that acceleration is proportional to force? The proportionality between acceleration and force can be demonstrated by using a setup such as Figure 3: A coaster glides without friction, pulled by a spring. Measuring the coaster’s acceleration with clocks and rulers, one finds that an unchanging pulling force (Figure 3a) causes an unchanging acceleration and that a doubled pulling force (Figure 3b) causes a doubled acceleration.

Now imagine exerting forces on a light ball and a heavy ball. You’ll find that, if you give them equal taps, the light ball accelerates into faster motion than does the heavy ball. So the light ball has a larger acceleration during the tap. It’s useful to extend the concept of inertia to this situation. Recall that an object’s inertia is its ability to maintain its velocity. Since the heavier ball changes its velocity the least, we say that it has more inertia than the light ball, using the word inertia to mean a body’s resistance to acceleration. It might seem, offhand, that “inertia” means pretty much the same thing as “weight,” because the heavier or “weightier” ball has more inertia. And in fact an object’s inertia and its weight are pretty much the same thing so long as the object is near Earth’s surface. However, weight and inertia are actually different things. Convincing evidence of this key fact comes from the study of objects in outer space, objects such as the many isolated rocks moving through our solar system. If you were in distant space holding such a rock in your hand, and then released it, the rock wouldn’t “fall”; it would instead remain “floating” in front of you. But if you Doubled force Unchanging force

Unchanging acceleration

Doubled acceleration

Air coaster

(a)

(b)

Figure 3

Quantitative demonstration that acceleration is proportional to force. (a) An unchanging pulling force causes an unchanging acceleration of a frictionless coaster along a horizontal surface. (b) If the pulling force is doubled, the acceleration is also doubled.

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push it with your hand, you’ll find that it resists your push. A huge push is needed to get a large boulder moving at even a slow speed. A rock in space has inertia, even though it has no weight. Such an object has no weight because weight is the force of gravitational attraction and is too small to notice in deep space. It’s useful to define inertia quantitatively. When inertia is made quantitative, it’s called mass. That is, the mass of an object is its amount of inertia. To establish a measurement scale for mass, scientists choose one particular object, called the standard kilogram, and define its mass to be one kilogram, abbreviated kg (Figure 4). There are good duplicates of it in most physics laboratories. Any object having the same inertia as the standard kilogram is said to have a mass of 1 kilogram. And any object having the same inertia as 2 kilograms bundled together has a mass of 2 kilograms. The mass of any object is defined in this way, by comparing its inertia with that of 1 or more kilograms (or with half a kilogram or some other fraction). Now, suppose you conducted further “coaster” experiments such as those shown in Figure 3, but that this time you maintained an unchanging pulling force while varying the amount of material being pulled (Figure 5). Figure 5a shows a single coaster being pulled, and Figure 5b shows a double coaster (two identical coasters linked together) being pulled by the same force. The acceleration should be smaller in case (b), because the greater amount of material has greater inertia. But how much smaller? The experimental answer turns out to be that the doubled coaster has half as much acceleration as the single coaster. And three coasters would have onethird as much acceleration, and so forth. We express this by saying that an object’s acceleration is proportional to the inverse of its mass. Since the inverse of a number is 1 divided by that number, this is abbreviated as a r 1>m. You learn something else from this experiment: An object’s mass (its inertia) is a measure of the “quantity of matter” (amount of material, number of atoms) it contains. For example, there is twice as much matter in the doubled coaster as in the single coaster, and also twice as much mass.

Acceleration Force

National Institute of Standards and Technology Figure 4

The U.S. National Standard Kilogram no. 20, an accurate copy of the International Standard Kilogram kept at Sèvres, France. It is stored inside two bell jars from which air has been removed.

Half as much acceleration Force Two coasters

(a)

(b)

Figure 5

Quantitative demonstration that acceleration is inversely proportional to mass. (a) Pulling on an air coaster causes the coaster to accelerate, as in Figure 3. (b) Pulling on two air coasters (twice as much mass), with the same force as in (a), causes half as much acceleration.

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Our two proportionalities, a r F and a r 1>m, can be put together to read acceleration r force>mass;

a r F>m

In words: An object’s acceleration is proportional to the force exerted on it divided by its mass. We need a measurement scale for force. The unit of force, called the newton (abbreviated N), is defined as the amount of force that can give a 1 kg mass a 1 m/s2 acceleration. So the proportionality becomes an equality: acceleration = force>mass;

a = F>m

This formula gives the acceleration in m/s2, provided the force is in N and the mass in kg. In the U.S. system of units, force is measured in pounds. One newton is a little less than a quarter of a pound. Think of a quarter-pound (a single stick) of butter. CONCEPT CHECK 1 A giant rock several kilometers across is at rest in outer space, far from all outside influences. A small, slow-moving pebble lightly “taps” the rock and bounces off. What does the rock do? (a) It accelerates up to a slow speed during the tap, and then comes quickly back to rest. (b) It accelerates up to a slow speed during the tap, and then comes gradually back to rest. (c) It accelerates up to a slow speed during the tap, and then continues moving at that speed. (d) It doesn’t accelerate at all. (e) It accelerates up to a high speed during the tap, and then continues moving at that speed. (f ) It turns into a frog. CONCEPT CHECK 2 Imagine you are in space and so far from all astronomical bodies that gravity is negligible, with two blocks of metal in front of you. They look identical, but you have been informed that one is made of aluminum and the other of lead (which, on Earth, would be much heavier than aluminum). You could determine which one is which by (a) giving them equally strong hammer taps—the one that then moves more slowly is aluminum; (b) giving them equally strong hammer taps—the one that then moves more slowly is lead; (c) holding them in your two hands—the heavier one is aluminum; (d) holding them in your two hands—the heavier one is lead; (e) actually, none of these methods would work.

3 NEWTON’S LAW OF MOTION: CENTERPIECE OF NEWTONIAN PHYSICS I must discuss two other points about Newton’s theory of motion. First, what happens when more than one force acts on an object, perhaps pushing or pulling it in different directions? In this text, we’ll only be concerned with situations in which the individual forces all act along a single straight line. When the individual forces all act in the same direction, their effects simply add up, and the overall effect is the sum of the individual forces. What if two forces act in opposite directions on the same object? If the forces are of equal strength (Figure 2) you know from experience that the object will not accelerate, so the overall effect must be zero. This suggests subtracting the two forces. This suggestion is correct.

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So two or more forces acting in the same direction have the same overall effect as a single force equal to their sum, while two forces acting in opposite directions have an overall effect equal to their difference and acting in the direction of the larger force. We call this overall effect the net force. For instance, if you push your book along a tabletop with a force of 10 N and the tabletop exerts a 3 N frictional force on the book (Figure 6), the net force on the book is 7 N. These 7 newtons represent the net, overall effect of the external environment pushing and pulling on the book. It is this 7 N net force, and not just your 10 N pushing force, that accelerates the book. The second point is the direction of the acceleration. Since forces have directions, and since acceleration is proportional to force, it is not surprising that acceleration should have a direction, too. So far, the only accelerations I have discussed quantitatively were ones in which an object was speeding up along a straight line. In this case, the direction of the acceleration is forward, because the change in velocity is in the forward direction (Figure 7). What about an object slowing down along a straight line? Since the velocity gets smaller, the change in velocity is backward (Figure 8). This means that the acceleration is backward, opposite to the velocity. Since an object’s acceleration is determined by the net force on it, it seems plausible that the acceleration’s direction should be the same as the net force’s direction. Simple experiments verify this: If you give a motionless ball a brief hammer tap, it will accelerate into motion along the direction in which you tapped, so the acceleration is along the direction of the force. If the ball is already moving and you tap it from behind, it will speed up; the acceleration is forward, again along the direction of the force. And if you tap a moving ball lightly from in front, the ball will slow down; this is a backward acceleration, which again is along the direction of the force.

Initial velocity

10 N push

Figure 6

How strong and in what direction is the net force on the book?

2

1

Change in velocity

Initial velocity

Final velocity

1

3N friction

Change in velocity Final velocity

2

Figure 7

Figure 8

When an object speeds up along a straight line, its change in velocity is along the direction of the motion.

When an object slows down along a straight line, its change in velocity is opposite to the direction of the motion.

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In summary: Newton’s Law of Motion3 An object’s acceleration is determined by its mass and by the net force exerted on it by its environment. The direction of the acceleration is the same as the direction of the net force. Quantitatively, the acceleration is proportional to the net force divided by the mass: acceleration r

net force ; mass

a r

F m

If force is measured in newtons, mass in kilograms, and acceleration in m/s2, the proportionality becomes an equality: acceleration =

net force ; mass

a =

F m

CONCEPT CHECK 3 A slow car moves at a steady 10 km/hr down a straight highway while another car zooms past at a steady 120 km/hr. Which car has the greater net force on it? (a) The slower one. (b) The faster one. (c) The one having the greater air resistance and rolling resistance. (d) None of the above. CONCEPT CHECK 4 You push your 2 kg book along a tabletop, pushing it with a 10 N force. If the book is greased so that friction is negligible, the book’s acceleration (a) is 5 m/s2; (b) is 10 m/s2; (c) is 20 m/s2; (d) is 0.2 m/s2; (e) keeps getting larger and larger as long as you keep pushing; (f) keeps getting smaller and smaller as long as you keep pushing. CONCEPT CHECK 5 Follow-up on Concept Check 4: A nongreased book also has a mass of 2 kg and is pushed with a 10 N force, but now there is a 4 N frictional force. The book’s acceleration is (a) 12 m/s2; (b) 20 m/s2; (c) 28 m/s2; (d) 3 m/s2; (e) 5 m/s2; (f ) 2 m/s2.

4 WEIGHT: GRAVITY’S FORCE ON A BODY As you know, Earth exerts a gravitational force on objects that are falling to the ground. This force is called “weight.” An object still has weight even when it’s not falling, as when it’s at rest on the ground. We’ll discover that the sun, moon, planets, stars, and all other astronomical bodies exert gravitational forces, too. It’s useful to extend the meaning of the word weight to include all such possibilities. In other words, the weight of an object refers to the net gravitational force exerted on it by all other objects. Since weight is a force, it can be measured in newtons. In U.S. units, weight is measured in pounds. Weight and mass are related concepts, but they certainly are not the same thing. An object’s weight is the force on it due to gravity, whereas its mass is its quantity of inertia. Weight is measured in newtons (or pounds, in U.S. units) while mass is measured in kilograms. An object’s weight depends on its environment; for 3

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Often called, boringly, Newton’s second law.

Why Things Move as They Do

instance, an object’s weight is less when it is on the moon than when it is on Earth, because the force of gravity is smaller on the moon than on Earth. But an object’s mass is a property of the object alone and not of its environment, so its mass is the same on the moon as on Earth. For example, a kilogram has a mass of 1 kilogram regardless of whether it is on Earth or on the moon or in distant space, but its weight is about 10 N (or 2.2 pounds) on Earth, only 1.6 N on the moon, and essentially zero in distant space. If you drop a stone and a baseball, Galileo’s law of falling tells us that their accelerations will be the same (neglecting air resistance). If the stone and baseball happen to have the same mass, Newton’s law of motion tells us that the forces on them are the same. But this force is just the force of gravity, which means that their weights are equal. This is a plausible and important conclusion: Two objects of equal mass also have equal weight, so long as you measure both weights at the same place (you wouldn’t want to measure one on Earth and the other on the moon). So you can compare masses by comparing weights, for instance on a balance beam (Figure 9). Any object that balances a kilogram has a mass of 1 kilogram, for example. The metric ton, or tonne (it’s always spelled this way), equal to 1000 kilograms, is useful for measuring larger masses. The similar U.S unit—the ton—is 2000 pounds. On Earth, the mass of a ton is about 900 kilograms, so a ton is a little less massive than a tonne. As you can see, the U.S system gets needlessly confusing, so you’ll perhaps be glad to hear that I’ll henceforth dispense with it entirely. For example, consider a book resting on a table. Suppose it weighs 12 N, meaning that the gravitational force by Earth on the book is 12 N. This force has a downward direction. But the book is obviously not accelerating downward through the table. Since the book’s acceleration is zero, Newton’s law of motion tells us that the net force on it must also be zero. So there must be an upward force of 12 N acting on the book to balance the downward weight. The table must exert this force, because if the table vanished the book would fall. It may seem strange that an inanimate object could exert a force. Why should a table push on a book? The tabletop doesn’t seem to be doing anything. A microscopic view is enlightening. The upward force is exerted by the atoms in the tabletop on the atoms in the book’s bottom.4 When the book presses against the tabletop, the tabletop is squeezed down and slightly deformed. And like a squeezed spring, the atoms then push upward against the book (Figure 10). The direction of this force by the tabletop is directly away from the surface, perpendicular to it. A force similar to the upward force by the table on the book is exerted when any object touches a solid surface. Physicists call any such force a normal force, because “normal” means perpendicular. Figure 11 shows the forces exerted on the book. Each force is represented by an arrow. A force diagram like this can help in analyzing forces and motion. When you draw a force diagram, show every one of the individual forces acting on whatever object is of interest. Show each force as an arrow pointing in the direction in which that force pushes or pulls on the object, and name each force. As another example, suppose that a rocket at liftoff weighs 150,000 N and has a mass of 15,000 kg and that the rocket engines exert a 210,000 N “thrust” force on

Figure 9

You can compare masses by comparing weights, for instance on a balance beam. Since they balance, the stone and the baseball have equal masses.

Figure 10

As an explanation of the normal force by a table on a book, imagine that the tabletop is covered with small springs. When the book rests on the table, it squeezes the springs, causing the springs to push back against the book. Normal force on book

Weight of book

Figure 11 4

More precisely, this force is an electric force by the electrons in the table’s atoms on the electrons in the book’s atoms. Electrons repel one another strongly when they get very close together.

The forces exerted on a book at rest on a table.

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Thrust, 210,000 N

the rocket (Figure 12). (You’ll learn more about the thrust force in the next section.) How large is the net force on the rocket, and how big is the rocket’s acceleration? Solution: The net force is 210,000 N – 150,000 N = 60,000 N upward. It is only this 60,000 N that actually accelerates the rocket. To find the acceleration, Newton’s law of motion says to divide the net force by the mass: 60,000 N/l5,000 kg = 4 m/s2. CONCEPT CHECK 6 Suppose the rocket develops a thrust of only 165,000 N. The acceleration is then (a) 4 m/s2; (b) 3 m/s2; (c) 2 m/s2; (d) 1 m/s2; (e) 0 m/s2.

Weight, 150,000 N

CONCEPT CHECK 7 An astronaut on Earth has a mass of 70 kg and a weight of 700 N. On the moon, the astronaut’s mass and weight will be (a) 11 kg and 700 N; (b) 70 kg and 110 N; (c) 11 kg and 110 N; (d) 70 kg and 700 N. CONCEPT CHECK 8 Would it be easier to lift a book on Earth or on the moon? (a) On the moon, because the book’s weight would be smaller. (b) On the moon, because the book’s mass would be smaller. (c) On Earth, because the book’s weight would be smaller. (d) On Earth, because the book’s mass would be smaller. (e) Same in both places.

5 THE LAW OF FORCE PAIRS: YOU CAN’T DO JUST ONE THING

Figure 12

The forces exerted on a rocket during liftoff. Note that the diagram shows only the individual forces and not the 60,000 N net force. The 60,000 N net force is not an individual force (it is the sum of all the individual forces) and hence is not shown.

How do we know that forces always come in pairs? Try these: Slap a tabletop with your hand. Grasp the edge of a table and pull hard on it. Now push hard on it. Find two balls of any kind; place one at rest on a smooth surface and roll the other one toward it so that they collide. ———Pause. I hope you’re actually doing some of this stuff that I suggest. It keeps your brain awake. When you slap a table, it slaps back, as you can feel when it stings your hand. This slap by the table is a force, because it accelerates your hand (by stopping your hand). When you pull on a table, the table pulls you toward it. And when you push on the table, the table pushes you away. These are forces exerted by the table on you. When the balls collide, the ball you rolled (call it the first ball) exerts a force on the second ball, as you can see from the fact that the second ball accelerates into motion. But the second ball exerts a force on the first ball, too, as you can see from the fact that the first ball’s velocity changes. These experiments indicate that whenever one object exerts a force on a second object, the second exerts a force on the first: Forces always come in pairs, called force pairs. Do things still work out this way even if the two objects are not touching? You could investigate this with a pair of magnets. Place the magnets on a smooth surface and hold them at rest with their poles near each other but not touching. When you release them, they both accelerate (if they don’t then find a smoother surface). So each exerts a force on the other.

Physicists like to think of every force as an interaction between two objects, rather than as something one object does to another. If you think of slapping a table as an interaction between your hand and the table, it’s natural to conclude that each exerts a force on the other.

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Touch your friend’s face. Your hand is touched by your friend’s face. You cannot touch without being touched.5 Slap lightly on a tabletop. Now slap hard. The table slapped back harder the second time, right? This gives us quantitative information about force pairs: When one member of a force pair grows bigger, so does the other. In fact, quantitative experiments show that the two members of any force pair have the same strength. If one of them is, say, 3.71 newtons, the other one will be 3.71 newtons too. The directions of the two forces in a force pair are not the same, however. In fact, our examples show that they are in opposite directions. For instance, when you pull a table toward you, the table pulls you toward it (Figure 13). Newton recognized this idea as a key physical principle. I’ll summarize it as follows: The Law of Force Pairs6 Every force is an interaction between two objects. Thus, forces must come in pairs: Whenever one body exerts a force on a second body, the second exerts a force on the first. Furthermore, the two forces are equal in strength but opposite in direction.

The fact that the two forces always have exactly the same strength is surprising. This says, for example, that a bug hits a car with the same force that the car hits the bug! Surprising—but true. However, these equal forces cause vastly different responses in the bug and the car: The bug feels an enormous acceleration, while the car experiences a barely measurable acceleration. The reason for this difference is Newton’s law of motion, and the vastly different masses of the bug and the car. Figure 14 illustrates an interesting point. Since Earth exerts a gravitational force on an apple, the law of force pairs says that the apple must exert a gravitational force on Earth! Furthermore, the strengths of these two forces must be equal: If Earth exerts a 2 N force on the apple, then the apple must exert a 2 N force on Earth. This might seem surprising. Why haven’t you noticed this force, by apples and other objects, on Earth? Why doesn’t Earth accelerate toward the apple?

Force by table on you

Force by Earth on apple Apple

Force by apple on Earth

Force by table on you

Force by you on table Force by you on table

Earth (a) Pulling on the edge of the table.

(b) Pushing on the table.

Figure 13

When you pull or push on a table, it pulls or pushes on you in the opposite direction. Figure 14 5

6

Thanks to my friend Paul Hewitt, author of Conceptual Physics, 10th ed. (New York: Addison Wesley 2006), for this nice way of putting it. Often called, boringly, Newton’s third law.

Earth and a falling apple: Which one exerts the larger force? (Answer: They are the same).

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The answer is that an apple causes only a slight acceleration of Earth because Earth’s mass is so large. You can’t use an apple to noticeably accelerate a planet. Large astronomical objects, however, can noticeably accelerate a planet. For example, scientists can detect Earth’s acceleration in response to the motions of the moon. CONCEPT CHECK 9 Your hands push a heavy box across the floor. The other member of the force pair is (a) friction pushing backward on the box; (b) gravity pulling downward on the box; (c) the box pushing backward against your hands; (d) the box pushing downward against the floor.

As you can see from Concept Check 9, the two forces in a force pair always act on different objects; your hands exert a force on the box, while the box pushes back against your hands. Similarly, if a rope pulls forward on a water skier, the skier pulls backward on the rope. So both the skier and the rope feel forces. The first force keeps the skier moving forward, while the second force keeps the rope taut. CONCEPT CHECK 10 A big truck and a small car collide head on. Regarding the forces: (a) the truck exerts a larger force on the car than the car does on the truck; (b) the car exerts a larger force on the truck than the truck does on the car; (c) the truck and car exert equally large forces on each other. CONCEPT CHECK 11 Regarding the accelerations in the preceding question: (a) The truck’s acceleration is largest; (b) The car’s acceleration is largest; (c) The truck and the car have equally large accelerations.

6 NEWTON MEETS THE AUTOMOBILE You can’t get anywhere by pulling on your nose. Try it (Figure 15)! You might pull your nose out of joint, but you won’t go anywhere because your nose pulls back on your hand, and both pulls are on your body, so they result in zero net force on your body—they “cancel out.” The same argument shows that you can’t get anywhere by pushing or pulling anywhere on your own body. If you want to accelerate, something outside of you—something in your environment—must exert a force on you. That’s why Newton’s law of motion says that an object’s acceleration is determined by the net force exerted on it by its environment. This presents an interesting dilemma when you consider self-propelled7 objects such as an automobile or an animal that accelerates itself into motion. How can they get themselves going if things cannot push or pull themselves into motion? Try this: Stand up and walk just one step, noting carefully the sensations in your legs, especially along the bottom of the foot that is accelerating you. Your foot pushes backward against the floor.8 You can demonstrate this more convincingly by accelerating rapidly from standing to running on a dusty dirt road. Your feet push dust backward, showing that they exert a backward force on the road. The law of force pairs tells us that when your foot pushes backward on the ground, the ground pushes forward on your foot. Voila! We’ve discovered the force Figure 15

You can’t get anyplace by pulling on your nose.

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7

8

“Self-propelled” means that the energy to propel the object comes from within the object itself. But, as we’ll soon see, the force to propel it comes from the outside. ... if you are barefoot. Otherwise, your foot pushes backward against your shoe, which in turn pushes backward against the floor.

Why Things Move as They Do

that accelerates you forward! It’s the ground pushing forward on your foot. This force arises from friction between the two surfaces (the surface of the ground and the bottom of your foot), as you can demonstrate by accelerating quickly from rest to a fast run on a nearly frictionless surface such as a smooth sheet of ice (be careful!). Automobiles are useful applications of Newtonian principles. What’s more important, automobile technology has drastically reshaped the social fabric of the modern world. Like all powerful technologies, cars have important social pros and cons. They provide unparalleled freedom of movement, have transformed our cities, use most of our petroleum, create much of our pollution, and are the leading cause of death of Americans under 35. They will come in for lots of discussion in this text. Before reading further, try listing or drawing the forces exerted on a car by its environment while traveling along a straight, level road. ———(A pause, for listing or drawing.) As shown in Figure 16, one force is the gravitational force, or weight of the car, exerted downward by Earth on the car. A second force is the normal force, exerted upward by the road on the car. These two forces act vertically. Since a car on a level road has no acceleration in the vertical direction, the net vertical force must be zero. So these two vertical forces must be of equal strength. The three horizontal forces relate more directly to the car’s motion. Two backward resistive forces act on the car: The atmosphere exerts the force of air resistance already discussed in Section 1, and the contact between tires and road creates another backward force known as rolling resistance. Rolling resistance is caused by flattening of the tire where the rubber meets the road. It turns out that the force that the road exerts on the deformed tire acts to slow the tire’s rotation, so the road exerts a retarding (backward) force on the car. This is most pronounced in more flexible, air-filled tires. Hard tires rolling on a smooth, hard surface, such as steel wheels on steel tracks, reduce rolling resistance to a minimum—one reason trains are far more energy efficient than cars. Rolling resistance also explains why underinflated tires reduce your gas mileage. High-mileage cars such as the Toyota Prius use special low-rolling-resistance tires for this same reason. The four forces discussed so far act even on a car that is coasting with its engine shut off. If these are the only forces on the car, then the net force must be backward, so the acceleration is backward and the car must slow down. But when a car is driving instead of coasting, there is an additional force on it. It’s a misconception to think that this force is exerted by the engine, because things cannot accelerate themselves and the engine is part of the car. Instead, the engine causes the drive wheels to turn; the drive wheels exert a frictional force backward against the road; and (because of the law of force pairs) the road in turn exerts a frictional force forward against the drive wheels. If the car moves at an unchanging velocity, there is no acceleration. Newton’s law of motion tells us that the net force must then be zero, which means that the five forces shown in Figure 16 must balance. In this case, the forward force on the drive wheels must equal the sum of the two resistive forces. In order for the car to speed up, the forward frictional force on the drive wheels must be larger than the sum of the two resistive forces; in order for the car to slow down, this forward force must be smaller than the sum of the resistive forces. Most other self-propelled objects are similar to this: A swimmer pushes backward on the surrounding water, and the water pushes forward on the swimmer. A motorboat’s propeller pushes backward on the water, and the water pushes forward on the propeller. An airplane’s propeller pushes backward on the surrounding air,

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Why Things Move as They Do Figure 16 Normal force by road on car

The five forces on an automobile. Air resistance by air on car

Rolling resistance by road surface on car Drive force, a second force by surface of road on drive tires Gravitational force by Earth on car

and the air pushes forward on the propeller. A jet airplane pushes air backward, too. As a jet engine moves through air, the air flowing into its front end heats by combustion with jet fuel, and the heated gas expands and rushes out of the back end at high velocity. One nice thing about space travel is that there are no resistive forces in space, so you don’t need a forward force by the environment on the spaceship to keep going. Your spaceship keeps going because of the law of inertia. But if you want to accelerate your spaceship—by changing its direction for instance—you have a problem. It’s difficult to get the surroundings to exert a force on your spaceship because there’s nothing around to push against! You would have a similar problem if you were stranded in the middle of a smoothly frozen pond. If the ice were absolutely smooth, you could not walk off it because, with no friction, you could not push backward on the ice. How could you get off? You could fan the air, pushing air backward in the way that a swimmer pushes water backward. That would work. But suppose that, as in space, there were no air. What then? Well, suppose you had something with you that you could throw away—your physics book, or a shoe. While throwing your shoe, you would push on the shoe, so it would push in the other direction on you, so your body would accelerate away from the shoe. When you let go of the shoe, you would have acquired a velocity. So you would slide along the pond. This is the principle of rocket propulsion. Rockets take along their own material just to have something to push against. Shoes would work, but they aren’t terribly practical (Figure 17). The rocket fuel for the U.S. space shuttle’s main rocket engines is hydrogen and oxygen, stored as low-temperature liquids. When combined, their combustion produces steam, which accelerates rapidly out the back end of the engine. Thanks to the law of force pairs, this backward push by the shuttle on the steam means that the escaping steam must push the shuttle forward. There are about 1000 large “near Earth asteroids” in our solar system—rocks more than 1 kilometer in diameter that orbit the sun and can cross Earth’s orbit and can therefore hit us, possibly dealing civilization a death blow. People are thinking

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about methods to nudge such rocks off course in case one of them is discovered heading for Earth. One suggested method: Send a space probe to attach itself to the asteroid, scoop up rock, and hurl it away. Thus is just like hurling shoes: The asteroid would react by moving in the opposite direction, deflecting it from its collision course. The law of force pairs comes to the rescue! CONCEPT CHECK 12 A car weighing 10,000 N moves along a straight, level road at a steady 80 km/hr. Air resistance is 300 N, and rolling resistance is 400 N. The net force on this car (a) is 10,000 N; (b) is 9300 N; (c) is 10,700 N; (d) is 700 N; (e) cannot be determined without knowing the strength of the drive force; (f ) is zero. CONCEPT CHECK 13 In the preceding question, the strength and direction of the drive force are (a) 10,000 N forward; (b) 700 N backward; (c) 700 N forward; (d) 400 N forward; (e) zero. Figure 17

Shoe power.

7 MOMENTUM An object’s momentum is defined as its mass times its velocity. It’s conventional to abbreviate it with the symbol “p” (I have no idea why): momentum = mass * velocity p = mv Momentum measures an object’s “amount of motion”—how much mass is moving how fast. It’s useful in connection with any “system” (this word simply means a collection of objects) of two or more objects that interact with each other via “internal” forces—forces exerted by objects in the system on other objects in the system. A good example is two colliding pool balls. When they collide, each ball exerts a brief force on the other during the short time they’re in contact. As a result of these forces, both balls accelerate—usually changing both the magnitude (speed) and direction of their velocity. To keep things simple, suppose the balls collide head-on so that all motion occurs along a single direction, call it the x-axis (Figure 18). One reason momentum is important in physics is that it’s one of nature’s “conserved quantities,” in other words a system’s total momentum remains unchanged throughout collisions such as is shown in Figure 18, despite the changes in both balls’ velocities during the collision. Here’s why:

v1 1

v2 2 x

Figure 18

Two pool balls, moving along the x-axis in the + and – direction, respectively, about to collide.

How do we know momentum is conserved? During the short impact time, which we’ll call ¢t, each ball experiences a change in velocity, which we’ll call ¢v1 and ¢v2. (The symbol ¢ is often used to mean “a change in”). According to the definition of acceleration, the accelerations of ball 1 and ball 2 during the impact are ¢v1>¢t and ¢v2> ¢t . Using Newton’s law of motion and Newton’s law of force pairs and making

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Why Things Move as They Do the important assumption that the only significant forces acting on either ball during the collision are the force by ball 1 on ball 2 and the force by ball 2 on ball 1, a little algebra9 shows that the change in the quantity m1 v1 is equal in magnitude but opposite in direction to the change in the quantity m2 v2. In symbols, ¢ (m1 v1) = - ¢ (m2 v2) The minus sign means “in the opposite direction along the x-axis.” In other words, the change in the first object’s momentum is the negative of the change in the second object’s momentum. But this means that the sum of the two momenta (plural of momentum) doesn’t change at all during the collision.

Momentum has a direction, namely the same direction as the velocity. For motion along a single axis, the direction can be indicated by a + or – sign: A positive momentum is along the +x direction, while a negative momentum is along the –x direction. I showed above that the total momentum p1 + p2 remains unchanged throughout the collision, where total momentum means the sum of the two individual momenta, with the directions (+ or –) included. This important result is known as conservation of momentum. Remember that it only applies so long as there are no external forces (forces other than the internal forces by each object on the other) on the system. For instance, suppose that ball 2 is initially at rest and ball 1, moving with a velocity of +3 m/s (the + emphasizes that it’s in the +x direction), hits it. Then the system’s total momentum before collision is m 1v1 + m 2v2 = m * (3 m>s) + m * 0 = (3 m>s) m, where “m” means the mass of either ball (pool balls all have the same mass). Conservation of momentum says that the total momentum after collision must also be 3 m: p1 + p2 = (3 m>s) m or mv1 + mv2 = (3 m>s) m or (with simple algebra) v1 + v2 = 3 m>s 9

Newton’s law of motion applied to each ball tells us

¢v1> ¢t = F (on ball 1)>m1 and ¢v2> ¢t = F (on ball 2)>m2.

Multiply both sides of the first equation by m1 and both sides of the second equation by m2 to get m1 ¢v1>¢t = F (on ball 1) and m2 ¢v2> ¢t = F (on ball 2)

But Newton’s law of force pairs says that the force on 1 by 2 and the force on 2 by 1 are equal and opposite, in other words F (on ball 1) = -F (on ball 2). It follows that m1 ¢v1 = -m2 ¢v2. But m1 is just a fixed number, so m1 times the change in v1 is the same as the change in m1 v1, and the same goes for ball 2. So the previous equation says that ¢ (m1v1) = - ¢ (m2v2).

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where v1 and v2 now represent the two balls’ final velocities. This result could be useful: If you knew one of the two final velocities, you could find the other. For instance, suppose ball 1 stops when it collides with ball 2. Then v1 = 0 and our result says that v2 = +3 m>s. So ball 2 takes off with the same velocity ball 1 had just before the collision. For another instance, suppose the balls are made of soft clay and that they stick together after collision. Then v1 = v2 and so v1 + v2 = 3 m>s tells us that 2v1 = 3 m>s. Simple algebra then says that v1 = v2 = +1.5 m>s. CONCEPT CHECK 14 If ball 2 takes off with a velocity of 2 m/s, ball 1’s final velocity is (a) 0 m/s; (b) 0.5 m/s; (c) 1 m/s; (d) –1 m/s (in the –x direction); (e) 1.25 m/s.

For yet another oddball (so to speak) instance, suppose both balls are covered with a small amount of gunpowder, in such a way that a small explosion occurs when they collide. And suppose that the explosion sends ball 2 zooming off with a velocity of +10 m/s. Notice that momentum must be conserved even in this situation, because all the significant forces on the balls during the collision/explosion (including the explosive force) are by 1 on 2 and by 2 on 1. If you worked through Concept Check 14, I’ll bet you’ll be able to work through this problem and conclude that v1 = -7 m>s. Both balls are now moving faster than ball 1 was moving before collision. The system gained kinetic energy (energy of motion), and this kinetic energy came from the chemical energy of the gunpowder. For a violent but instructive example involving objects of different mass, suppose that a 30,000 kg “18 wheeler” truck moving at 20 m/s (72 km/hr) collides head-on into a small 1000 kg car moving in the opposite direction at 20 m/s. Suppose that the car and truck become enmeshed in each other, so that they stick together. How fast is the combined wreckage moving just after the collision (before the frictional force by the road has had time to begin to slow the wreckage)? It’s easiest if you let the x-axis be in the direction of the truck’s initial motion, so that the car’s velocity is negative. ——— A pause, for figuring. The result is that the truck is hardly slowed by the collision; it slows from its initial +20 m/s to +18.7 m/s (the speed of the wreckage). But the car changes its velocity from –20 m/s to +18.7 m/s. This is a huge velocity change of +38.7 m/s in a small fraction of a second, and implies an enormous acceleration and hence (see Newton’s law of motion) enormous forces by the windshield, seats, etc., on the car’s occupants. Ouch. Notice that the forces on the car driver’s body would be much smaller if the driver’s velocity change of 38.7 m/s occurred in a much longer time, such as one second instead of a small fraction of a second. This is why air bags and vehicle front-ends designed to crumple slowly upon collision are a good idea. The truck driver, on the other hand, suffers a much more mild velocity change of only 1.3 m/s. This example gives you a feel for the momentum concept. Momentum involves both mass and velocity, so more massive objects possess more of it, and faster objects possess more of it. Despite the equal speeds of the car and the truck, the truck has far more momentum because of its larger mass. Momentum is a measure of the tendency of an object to keep moving despite forces (such as collisions) that act to change the velocity: The truck slows only slightly, while the car changes its velocity radically. Conservation of momentum applies to an amazing variety of situations. Here are a few.

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First, suppose the pool ball collision is a glancing collision, so that the two balls shoot off into different directions (Figure 19). The collision is then “twodimensional,” not along a single line but still on the surface of the pool table. Conservation of momentum is still valid; in fact, it applies to each of the two directions on the table (since in this text we’re staying away from something ominously called “vectors,” I won’t go into exactly what this means). Now suppose there are more than two balls; for instance, suppose the collision is between a cue ball and a rack of 15 pool balls. It’s amazing (at least I’ve always found it amazing) but true that momentum is conserved: The total momentum of all 16 pool balls just after collision equals the momentum of the cue ball just before collision. The 16 balls might be scattering all over the place, but if you add up all 16 momenta10 you’ll find that the result is equal, in both magnitude and direction, to the magnitude and direction of the initial momentum mv of the cue ball! But notice that momentum is conserved only from just before to just after the collision, before any “external forces” such as friction from the table top or bounces from pool table walls have had time to change any of the 16 velocities. The balls could all have different sizes and masses, and momentum would still be conserved. In fact, I allowed for two different masses in arriving (above) at the principle of conservation of momentum for two pool balls. The objects needn’t physically collide (bang against each other) at all. The “collision” could be between two magnets sliding on a frictionless surface, never touching but simply influencing each others’ motion magnetically, or an encounter between two stars that exert gravitational forces on each other and remain millions of miles apart. The two stars’ interaction (“collision” isn’t the appropriate word here) could take years, but so long as external forces (by other stars, for instance) don’t interfere, the total momentum of the two stars remains the same throughout the entire interaction. We’ll find that Newton’s laws are far from absolute. They break down for fastmoving (comparable to the speed of light) objects, for small objects such as individual molecules, and in situations involving strong gravitational forces or distances stretching across many galaxies. Since Newton’s laws break down, it’s natural to question the principle of conservation of momentum in these situations. But surprisingly, physicists have checked a wide variety of such situations and found that momentum is always conserved. In fact, a very broad argument based simply on the notion that the laws of physics are the same everywhere in the universe leads to the conclusion that momentum must be conserved in any system that has no external forces exerted on it, regardless of whether Newton’s laws are valid. Figure 19

y

A cue ball glancing off another ball, viewed from above. The second ball is initially at rest. Arrows show the initial velocity of the cue ball and final velocities of both balls. Arrows point in the direction of the velocity; longer arrows represent greater speeds. (a) Just before collision. (b) Just after collision.

v2 2 1

x (a)

10

110

2

v1

1

x v1

(b)

Since the collision occurs in two dimensions, the 16 momenta must be added using something called “vector addition,” which means roughly that you should add up 16 arrows whose lengths and directions represent the magnitudes and directions of the 16 balls’ momenta.

Why Things Move as They Do

Conservation of momentum seems to be one of the universe’s most fundamental principles. To summarize: The Law of Conservation of Momentum The total momentum of any system remains unchanged, regardless of interactions among the system’s parts, so long as no part of the system is acted upon by forces external to that system.

© Sidney Harris, used with permission.

Contemporary physicists have learned that many of nature’s deepest principles are conservation laws in which some quantity such as momentum remains unchanged over time. We’ll encounter two other conservation laws: conservation of energy and conservation of electric charge.

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Why Things Move as They Do Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions FORCE 1. How can we tell whether a body is exerting a force on another body? 2. Can an object have force? Can an object exert a force? Can an object be a force? Can an object feel a force? 3. List at least six specific examples of forces. Try to list examples that are significantly different. 4. What is a resistive force? Give two examples. 5. Give two examples of forces that act at a distance.

NEWTON’S LAW OF MOTION 6. What does Newton say that forces cause? What does Aristotle say? 7. What do we mean when we say that one object has “more inertia” than another object? 8. When you move an object from Earth to the moon, does its inertia change? Does its weight change? Does its mass change? Does its amount of matter change? Does its acceleration differ while falling freely? Does it respond differently to a net force of 1 N? 9. Forces of 8 N and 3 N act on an object. How strong is the net force if the two forces have opposite directions? The same directions? 10. Is an object’s acceleration always in the same direction as its velocity (its direction of motion)? If not, give an example in which it is not. Is an object’s acceleration always in the same direction as the net force on the object? If not, give an example in which it is not. 11. As you increase the net force on an object, what happens to its acceleration? What if you double the net force? As you increase the mass of an object (for example, by gluing additional matter to it), what happens to its acceleration? What if you double the mass?

16. Draw a force diagram showing the forces on a rocket during liftoff. Which force is largest? What is the direction of the net force? 17. Where is it easiest to lift your automobile: on Earth or on the moon? Where is the automobile’s mass larger?

LAW OF FORCE PAIRS 18. Describe several experiments demonstrating that forces come in pairs. 19. Do you exert a gravitational force on Earth? How do you know? What direction is this force? 20. Describe the other member of the force pair for each of the following forces: the normal force on a book lying on a table, the weight of an apple, the force by a bat against a baseball, the force by a baseball hitting a catcher’s mitt.

THE AUTOMOBILE 21. Describe four examples of forces that propel “self-propelled” objects. 22. Draw a force diagram showing the forces on a car driving along a straight, level road. How would this force diagram be altered if the car were coasting? What if the car were braking? 23. What is the main difference between propeller-driven airplanes and jet airplanes? 24. How does the forward force on a car compare with the resistive forces when the car maintains a constant speed? When the car is speeding up? Slowing down? 25. When a car moves at constant speed along a straight road, is the forward force (the force that moves the car forward) zero? Is the net force zero? Is the acceleration zero? Is the speed zero? 26. What is the main difference between the force that propels a rocket and the force that propels airplanes and automobiles?

MOMENTUM WEIGHT 12. What is weight? Is it the same as mass? If not, what is the difference? 13. Describe a simple way to determine, in a lab, whether two objects have equal masses. Would this method work in distant space? What would work in distant space? 14. Find the gravitational force on a 1 N apple. Would it still weigh 1 N if we took it to the moon? 15. Draw a force diagram showing the forces on an apple at rest on a table. Find the net force on the apple.

27. Conservation of momentum follows logically from two other laws of physics. Which two? 28. Which of the following quantities do you need to know in order to calculate the magnitude of the momentum of an object, and how do you do the calculation: weight, mass, acceleration, velocity, location, length? 29. True or false: Every system’s total momentum is always conserved. (Recall that a “system” is any collection of physical objects). Explain your answer.

From Chapter 4 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Why Things Move as They Do: Problem Set 30. A hunter fires a rif le bullet northward and then spins and fires a second bullet (from the same rif le) southward. Do the two bullets have the same momentum? Explain. 31. Which has greater momentum, a truck at rest or a slowrolling pool ball? 32. What is the magnitude of the momentum of a 7 kg bowling ball rolling at 3 m/s?

Conceptual Exercises FORCE 1. Is any force exerted on you when you speed up along a straight line? When you slow down along a straight line? How do you know? 2. Is any force exerted on you while you move in a circle at unchanging speed? How do you know? 3. A smooth ball rolls on a smooth table. Initially, no horizontal forces are exerted on the ball. Then you bring a magnet near the rolling ball, but you are not sure whether the magnet actually exerts a magnetic force on the ball. How can you tell whether the magnet is exerting a horizontal force on the ball? 4. Does a high-speed bullet contain force? Does a stick of dynamite contain force?

NEWTON’S LAW OF MOTION 5. You place your book on a table and hit it horizontally with a hammer, strongly but briefly. Do not neglect friction. Describe the motion of the book, beginning from just before you hit it with the hammer. Describe the direction and strength of the net force on the book during the entire motion. 6. If you exert a force on an object and then exert three times as strong a force on the same object, what (if anything) can you say about the object’s acceleration during the exertion of each force? Assume no other force acting on the object. 7. A ball weighing 8 N is thrown straight upward. Disregarding air resistance, find the direction and strength of the net force on the ball as it moves upward. What is the direction of the ball’s acceleration? Are the net force and the acceleration in the same direction in this case? Can they ever be in different directions? 8. An object moves with unchanging speed in a straight line. Does it then have no forces acting on it? Explain. Does it have no net force acting on it? 9. An object is at rest. Does it then have no forces acting on it? Explain. Does it have no net force acting on it? 10. When you stand on the floor, does the floor exert a force on your feet? In which direction? Why, then, don’t you accelerate in that direction? 11. You push on a solid concrete wall. Is your push the only horizontal force on the wall? How do you know? What can you say about the net force on the wall? 12. A car starts up from rest, moving along a straight highway with an acceleration of 1 m/s2. A second car comes racing past at a steady 120 km/hr. Which car has the larger net force acting on it?

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13. A 40 tonne truck and a small 1 tonne car maintain a steady speed of 80 km/hr on a straight highway. Which vehicle has a larger net force exerted on it? Which vehicle has the larger normal force exerted on it? 14. A 40 tonne truck and a small 1 tonne car maintain a steady speed of 80 km/hr on a straight highway. Which vehicle has the larger drive force exerted on it? The larger air resistance force? The larger net force? 15. A 3 kg rock rests on the ice. You kick it, briefly exerting a 60 N force. Find the rock’s acceleration, assuming that there is no friction. Still assuming no friction, what will be the rock’s acceleration after your foot is no longer in contact with the rock? Will the rock have a (nonzero) speed at this time? 16. In the preceding question, assume now that a frictional force of 6 N acts on the rock whenever it is moving across the ice. Find the net force on the rock and the rock’s acceleration. What can you say about the net force on the rock after your foot is no longer in contact with it? 17. Ned the skydiver weighs 600 N and has a mass of 60 kg. How large must be the force of air resistance acting on Ned in order for Ned to maintain an unchanging speed while falling through the air? 18. In the preceding question, what would be Ned’s acceleration if there were no air? 19. A car weighing 8000 N moves along a straight, level road at a steady 80 km/hr. The total resistive force on the car is 500 N. Find the net force on the car and the acceleration of the car. 20. In the preceding question, find the drive force on the car.

WEIGHT 21. Roughly, what is your weight in newtons? 22. Which has the greater mass, a tonne of feathers or a tonne of iron? Which has the greater weight? Which has the larger volume? 23. Would you rather have a hunk of gold whose weight is 1 N on the moon or one whose weight is 1 N on Earth—or wouldn’t it make any difference? 24. Would you rather have a hunk of gold whose mass is 1 kg on the moon or one whose mass is 1 kg on Earth—or wouldn’t it make any difference? 25. A standard kilogram in your physics lab weighs (approximately) 10 N, or 2.2 pounds. What are its mass and weight in distant space? 26. Find the strength and direction of the net force on an apple weighing 2 N, neglecting air resistance, in each of the following cases: The apple is held at rest in your hand. The apple is falling to the ground. The apple is moving upward, just after you threw it upward. 27. An apple is accelerated upward by your hand. Which is larger, the apple’s weight or the upward force by your hand? What if you accelerate the apple downward while it is in the palm of your hand? What if you lift the apple at an unchanging velocity? What if you lower the apple at an unchanging velocity? 28. Would it be easier (in other words, would it require less thrust and less rocket fuel) to lift a rocket off the moon’s surface than off Earth’s surface? Why? 29. An astronaut on the moon picks up a large rock. Would it be easier, or harder, or neither for him to pick up the same rock on Earth?

Why Things Move as They Do: Problem Set 30. An astronaut on the moon kicks (horizontally) a large rock. What if she kicked the same rock on Earth? Neglecting frictional effects, would it hurt her foot more, or less, or just as much? 31. Neglecting friction and air resistance, would it be easier to set this book into horizontal motion at 5 m/s on Earth, or on the moon, or in distant space?

LAW OF FORCE PAIRS 32. “Planet Earth is pulled upward toward a falling boulder with just as much force as the boulder is pulled downward toward Earth.” True or false? Why? 33. “Planet Earth is pulled toward a falling boulder with just as much acceleration as the boulder has as it moves toward Earth.” True or false? Why? 34. A large truck breaks down on the highway and receives a push back into town by a small car. While moving at unchanging speed, does the car exert any force on the truck? Does the truck exert any force on the car? If so, is this force weaker or stronger than the force that the car exerts on the truck? 35. A car collides head-on with a large truck. Which vehicle exerts the stronger force? Which has the larger force exerted on it? Which experiences the larger acceleration? 36. When a rifle fires, it accelerates a bullet along the barrel. Explain why the rifle must recoil. 37. A 2 N apple hangs by a string from the ceiling. Describe the two forces on the apple. How strong is each of these forces? Do these forces form a single force pair? If not, then for each force, describe the other member of that force’s force pair. 38. A horizontally moving bullet slows down. Is anything exerting a force on it? How do you know? Is it exerting a force on anything? How do you know? 39. I push you away from me. Do you also push (exert a force on) me? Which force is stronger—or does it depend on which of us is heavier? 40. A pitcher exerts a force on a baseball while throwing it. Describe the other member of the force pair. 41. A rope pulls forward on a water skier. Describe the other member of the force pair. 42. As we know, “weight” is a force, and force is an interaction. In the case of your own weight, name the two objects that are involved in this interaction. 43. Describe the two forces that act on a book that rests in the palm of your hand. Are these two forces equal but opposite to each other? Are these two forces part of one force pair? 44. Continuing the preceding question, suppose you accelerate the book into upward motion. How many forces act on it? Are these two forces equal but opposite to each other? 45. A freely falling apple has a weight of 1 N. Earth’s mass is 6 × 1024 kg. How strong is Earth’s force on the apple? 46. In the preceding question, how strong a force does the apple exert on Earth? 47. Still continuing the preceding question, how big is the apple’s acceleration? Find the acceleration that the apple would cause Earth to have if the apple was the only object exerting a force on Earth.

THE AUTOMOBILE 48. Since the law of inertia states that no force is needed to keep an object moving in a straight line at an unchanging speed, why is a force needed to keep a car moving? 49. While driving your car on a straight, level road, you slam on your brakes. Draw a force diagram of the car during braking. What is the direction of the net force? Draw a force diagram for a car that is coasting without braking. In which of the two cases is the net force stronger? 50. Why is it easier to pedal a bicycle with hard high-pressure tires as compared with soft balloon tires? 51. When you hold your foot on a car’s accelerator pedal, is the car necessarily accelerating? Could it be accelerating? Could it have a forward acceleration? Could it have a backward acceleration? 52. There are three acceleration devices on any car. What are they, and what kinds of acceleration does each one give to the car? 53. If a jet airplane were above Earth’s atmosphere, could it then accelerate? What about a rocket-driven plane? 54. Magnetic forces can levitate railroad trains a short distance above the tracks, making friction practically negligible. Suppose such a “maglev” train runs inside an evacuated (emptied of most air) tunnel from New York City to Chicago. If friction and air resistance are negligible, during what parts of the trip would an external horizontal force act on the train? Discuss the direction of this force during each part of the trip.

MOMENTUM 55. Which has greater momentum, a 1000 kg small car moving 10 m/s or a 1 kg artillery shell shot at 10 times the speed of sound (the speed of sound is 330 m/s)? 56. Which has greater momentum, you steadily jogging one kilometer in 6 minutes or a 5-gram rifle bullet moving at 1000 m/s? 57. Why is it so much harder to stop or turn a moving supertanker than a small speedboat? 58. Can a swarm of flying insects have a total momentum of zero? Explain. 59. A stationary firecracker explodes, breaking into two parts of equal mass. One part is moving north at 20 m/s. What is the velocity of the other part? 60. Explain, in terms of momentum, why guns recoil. 61. Are there any kinds of situations in which momentum is not conserved? Explain. 62. An artillery shell explodes in midair into many fragments. Is the total momentum of the fragments just after the explosion equal to the momentum of the shell just before the explosion? Explain.

Problems NEWTON’S LAW OF MOTION 1. You push on a 2 tonne (2000 kg) vehicle on level pavement with a force of 250 N. Find the vehicle’s acceleration.

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Why Things Move as They Do: Problem Set 2. How large is the acceleration of a 60 kg runner if the friction between her shoes and the pavement is 500 N? 3. In order for a 60 kg runner to accelerate at 8 m/s2, what must be the frictional force between her shoes and the pavement? 4. A 747 jumbo jet of mass 30,000 kg accelerates down the runway at 4 m/s2. What must be the thrust of each of its four engines? 5. What would a skydiver’s acceleration be if air resistance were half as large as the skydiver’s weight? What if air resistance were as large as the skydiver’s weight? 6. How much force must a pitcher exert on a 0.5 kg baseball in order to accelerate it at 50 m/s2? 7. Find the force acting on a 0.01 kg bullet as it is accelerated at 1 million m/s2 (100,000 times larger than the acceleration due to gravity!) down a rifle barrel. 8. A 2 kg flower pot weighing 20 N falls from a window ledge. How large must air resistance be in order that the pot fall with an acceleration of 8 m/s2? 9. An 80 kg firefighter whose weight is 800 N slides down a vertical pole with an acceleration of 3 m/s2. What is the frictional force on the firefighter? 10. A black box and a white box accelerate at the same rate across the floor despite the fact that the net force on the black box is four times larger than the net force on the white box. Which box has the larger mass, and how much larger? 11. A 70 kg runner speeds up from 6 m/s to 7 m/s in 2 s. Find the runner’s acceleration and the frictional force by the ground on the runner during this time. 12. A 1 tonne (1000 kg) automobile experiences 100 N of air resistance and 200 N of rolling resistance. How large a forward force must the road exert on the drive wheels in order for the automobile to accelerate at 0.5 m/s2?

THE LAW OF FORCE PAIRS 13. Wearing frictionless roller skates, you push horizontally against a wall with a force of 50 N. How hard does the wall push on you? 14. In Problem 13, if your mass is 40 kg, then what is your acceleration? 15. Your friend (mass 80 kg) and you (mass 40 kg) are both wearing frictionless roller skates. You are at rest, behind your friend. You push on your friend’s back with a force of 60 N. How hard does your friend’s back push on you? 16. In the preceding question, what is your acceleration? What is your friend’s acceleration? 17. A small car having a mass of 1000 kg runs into an initially stationary 60,000 kg 18-wheeled truck from behind, exerting a force of 30,000 N on the truck. How big, and in what direction, is the force that the truck exerts on the car? 18. In the preceding question, find the car’s acceleration. Is this a “speeding up” or a “slowing down” type of acceleration? Find the truck’s acceleration. Is this of the “speeding up” or “slowing down” variety? 19. You press downward with a 100 N force on a brick weighing 40 N that rests on a table. With what force, and in what direction, does the brick press against your hand? Draw a force diagram similar to Figure 16, with an arrow representing each force acting on the brick.

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20. In the preceding question, how big is the net force on the brick? Find the force (how big and in what direction) that the table exerts on the brick. How hard is the brick pressing down against the table?

MOMENTUM 21. A 1000 kg car moving at 20 m/s slams into a stationary 27,000 kg truck from the rear, and sticks to the rear end of the truck. Assuming the truck is free to roll, how fast is the wreck moving after the collision? 22. A 27,000 kg truck moving at 20 m/s slams into a stationary 1000 kg car. The two stick together. Assuming the car is free to roll, how fast is the wreck moving after the collision? 23. A 50 kg boy and a 30 kg girl are standing on ice skates on a smoothly frozen pond. The boy gives the girl a push and she slides at unchanging speed to the edge of the pond, 20 m away, in 4 seconds. What happens to the boy? 24. A 30 kg girl standing on slippery ice catches a 0.5 kg ball thrown with a speed of 16 m/s. What then happens to the girl?

Answers to Concept Checks 1. The rock accelerates only when the pebble is pushing (tap2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

ping) it; since the rock has a large mass, its acceleration will be small, (c). If you tap them, Newton’s law of motion tells us that the one with the larger mass will accelerate less, (b). (d), because neither car is accelerating, so both cars have zero net force on them. acceleration = force>mass = 10>2 = 5 m>s2, (a). The net force is now 10 - 4 = 6 N, so acceleration = 6>2 = 3 m>s2, (d). (d) The mass must stay the same, but the weight is far less than on Earth, (b). (a) (c) (c) (b) Since the acceleration is zero, the net force must also be zero, (f ). Since the net force is zero, the forward drive force must be equal to the sum of the backward forces, (c). Since v1 = +2 m>s, v1 + v2 = 3 m>s tells us that v1 + 2 m>s = 3 m>s, so v1 = +1 m>s, (c).

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. A force must be exerted on you, both when you speed up and when you slow down, in order to accelerate you. Newton’s law of motion says so.

Why Things Move as They Do: Problem Set 3. If the ball accelerates (speeds up, slows down, or changes 5.

7.

9.

11.

13. 15. 17. 19. 21. 23.

25. 27.

29. 31. 33. 35.

37.

direction), it must have a force on it. Initially, the book is at rest; then it quickly speeds up during the fraction of a second that the hammer is actually in contact with the book. After the hammer is no longer touching the book, the book gradually slows down to a stop. There is no net force on the book before the hammer hits it; then there is a large force in the forward direction while the hammer is in contact with the book. Then there is a smaller force in the backward direction while the book is slowing down. Since gravity is the only force acting on the ball, the net force on the ball is 8 N downward. Thus the ball’s acceleration is also downward, opposite to the ball’s upward velocity, because Newton’s law of motion says that the acceleration is in the direction of the net force. Acceleration and net force are always in the same direction. The object could have several forces on it, adding up to zero net force. For example, an object at rest on a table has two forces on it: weight acting downward, and normal force by the table acting upward. The object has no net force on it. No, your push cannot be the only force on the wall. Because the wall doesn’t accelerate, the net force on the wall must be zero, and so there must be another force (provided by the concrete structure) pushing back in the other direction on the wall. Each vehicle has zero net force on it. The truck has the larger normal force on it. a = F>m = 60 N>3 kg = 20 m>s2. After the kick, the acceleration must be zero (the law of inertia). The rock will have a non-zero speed. 600 N, acting upward, to balance the force of gravity. The net force is zero, because the car is not accelerated. The acceleration is zero. Since 1 N is about 1/4 pound, multiply your weight in pounds by 4 to get your approximate weight in newtons: 100 pounds is roughly 400 N, etc. You would be better off having a hunk of gold whose weight is 1 N on the moon, because it would be a more massive hunk of gold (containing more gold atoms) than one whose weight is 1 N on Earth. Mass = 1 kg, weight = 0. The upward force by your hand must be larger, because the net force on the apple must be upward to provide the upward acceleration. For the downward acceleration, the apple’s weight must be larger. For the unchanging velocity (both lifting and lowering), the upward force by your hand and the downward force of gravity have equal strengths. Harder, because it would weigh more. Same in all three places, because the book has the same mass in all three places, and there are no resistive forces in any of the three places. False. The boulder has a much larger acceleration than does Earth, because the boulder’s mass is much smaller than Earth’s mass. The two vehicles exert equally strong forces on each other, and the two vehicles feel equally strong forces from the other vehicle. The car experiences the larger acceleration, because it has the smaller mass. The string pulls upward on the apple, and Earth’s gravitational force pulls downward. Each force has a strength of

39. 41. 43. 45. 47.

49.

51.

53.

55. 57. 59. 61.

2 N. These do not form a force pair. The other members of the two force pairs are (1) the apple pulling downward on the string and (2) the apple’s gravitational force pulling upward on Earth. Yes, you push on me too. The two forces are equally strong. The backward pull by the water skier on the rope. Your hand pushes upward on the book. Earth’s gravitational force pulls downward on the book. These two forces are equal and opposite, but they are not part of one force pair. Earth exerts a 1 N force on the apple. The apple’s acceleration is 9.8 m>s2. Earth’s acceleration would be a = F>m = 1 N>6 * 1024 kg = 1.7 * - 25 2 10 m>s , which is so small that it is not measurable. The diagram should show a large frictional force, acting backward, in addition to the forces of air resistance (backward), gravity (downward), and the road’s normal force (upward). The net force is backward. For a car coasting without braking, the forces are rolling resistance, air resistance, gravity, and the normal force. The net force is strongest in the case of braking. No, the car could be moving at a constant velocity. However, it could be accelerating. The acceleration could be forward (if the car is speeding up) or backward (if the car is slowing down). No, a jet plane could not accelerate (except for the “natural” acceleration of 9.8 m>s2 downward due to gravity). A rocketdriven airplane could accelerate. For the car, For the shell, p = 104 kg m>s. p = 3.3 * 103 kg m>s. The car’s momentum is larger. The supertanker’s huge mass gives it a much larger momentum than the speedboat. Conservation of momentum says that the two parts must have equal and opposite momenta, and they have the same masses so they must have the same speeds. 20 m/s south. A system having external forces acting on it does not necessarily have an unchanging momentum.

Problems 1. a = F>m = 250 N>2000 kg = 0.125 m>s2. 3. a = F>m, where F is the frictional force. Solving for F, F = ma = 60 kg * 8 m>s2 = 480 N. 5. Air resistance would reduce the downward net force to half of the skydiver’s weight, so the skydiver would accelerate downward at half of the acceleration of gravity: 4.9 m>s2. If air resistance were as large as the skydiver’s weight, the skydiver’s acceleration would be zero, i.e., he or she would be falling at an unchanging speed. 7. Solving a = F>m for F, F = ma = 0.01 kg * 106 m>s2 = 104 N = 10,000 N. 9. The net downward force is F = ma = 80 kg * 3 m>s2 = 240 N. But the net downward force is F = weight - f, where f means “the upward force due to friction.” Thus, weight - f = 240 N. Solving, f = weight - 240 N = 800 N - 240 N = 560 N. 11. a = (change in speed)>(time to change) = (7 m>s - 6 m>s)>2 s = (1 m>s)>2 s = 0.5m>s2 frictional force = net horizontal force = ma = 70 kg * 0.5 m>s2 = 35 N 13. The wall pushes with 50 N.

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Why Things Move as They Do: Problem Set 15. Your friend’s back pushes on you with a force of 60 N. 17. The truck exerts a 30,000 N force on the car, in the backward

direction. 19. The brick presses upward against your hand, with a 100 N force.

Table pushing upward

Gravity pulling downward

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Your hand pushing downward

21. Using kg, m, and s: Initial momentum = 1000 * 20 =

20,000. Final momentum = 1000 v + 27,000 v = 28,000 v. So conservation of momentum says 28,000 v = 20,000. So v = 20,000>28,000 = 0.96 m>s. 23. The initial momentum of the system (boy plus girl) is zero. The girl’s momentum after the push is (30 kg) * (20 m > 4 s) = (30 kg) * (5 m>s) = 150 kg m>s . Since momentum is conserved, the boy must be sliding the other way with this same momentum. So 50 v = 150, from which v = 3 m>s.

Newton’s Universe

From Chapter 5 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Newton’s Universe

And from my pillow, looking forth by light Of moon or favouring stars, I could behold The antechapel where the statue stood Of Newton with his prism and silent face, The marble index of a mind for ever Voyaging through strange seas of Thought, alone. William Wordsworth. P r e l u d e (Book III), 1850

T

he law of inertia might be history’s most fruitful scientific idea. Besides unifying natural motion on Earth and in the heavens, undermining Aristotelian views, promoting the idea of universal natural law, and leading to Newton’s law of motion, it also led Newton to look in a new way at one specific kind of force: gravity. Because gravity is all around us all the time, it’s difficult to even notice it. This has made it difficult for scientists to properly conceptualize it. From Aristotle until Newton, people believed that every solid body had a natural tendency to seek out Earth’s center, in the way that a thirsty person seeks out water. External influences— forces—were not needed to explain why objects fell: They fell because they “wanted” to. But the inertial view is that bodies “want” to maintain their velocity. Descartes first conceived of this new view of motion. It was a conceptual shift comparable to Copernicus’s shift to a sun-centered view. Newton then built on Descartes’s idea. If you believe that bodies have inertia, you must ask why an apple, released above the ground, falls. Newton’s answer applied to more than apples; it demonstrated that the same gravitational forces are at work in the heavens as on Earth. We live in one universe, not two. Section 1 presents the general idea of Newton’s theory of gravity, and Section 2 gives the specifics along with examples. One significant social/cultural development of the past 100 years is our increased scientific understanding of the origin and future of our universe, our planet, life, and humans. I’ll delve into such topics at several points in this text, beginning with Sections 3 and 4. Section 3 applies Newton’s theory of gravity to the birth and death of the sun and Earth. Section 4 tells of the violent gravitational collapse of stars that are more massive than the sun and the exotic objects that result from the collapse. Sections 5 and 6 return to our theme of comparing Newtonian and modern physics: Section 5 looks at broad implications of Newtonian physics, particularly the “mechanical universe.” Section 6 notes the limitations of Newtonian physics in light of modern physics.

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1 THE IDEA OF GRAVITY: THE APPLE AND THE MOON Isaac Newton, age 22, had just completed his bachelor of arts degree at Cambridge University in England. He was invited to remain, but the school then closed for 18 months because of a plague epidemic, so the graduate returned to his family’s farm. But he didn’t just snooze. During those 18 months, Newton laid the foundations for a theory of gravity and a theory of light and, in his spare time, invented calculus. Some say that greatness is partly a matter of timing. Newton lived at a time that was culturally ripe for a new view of the universe. The scientific foundations had been laid by Copernicus, Brahe, Kepler, Galileo, and Descartes. You have seen that the inertial view of Descartes and Galileo leads naturally to Newton’s law of motion. The concepts surrounding the law of motion, plus the astronomy of Copernicus and Kepler, then led Newton to the law of gravity. Newton stood, as he himself said, “on the shoulders of giants.” As Newton recounted it late in life, the central idea of his theory of gravity came to him during his stay on the family farm when an apple fell from a tree while he could see the moon in the sky. Beyond the fact that both are more or less round, it’s difficult to think of two more dissimilar objects than an apple and the moon. One is on Earth, the other in the heavens; one rots, the other seems eternal; one falls to the ground, the other remains aloft. Yet where others saw difference, Newton saw resemblance. Let’s trace Newton’s thinking. Figure 1 shows an apple falling toward the ground, accelerated by Earth’s gravitational pull. The directions of the apple’s velocity, acceleration, and gravitational force are all downward toward Earth’s center, as shown. The moon’s motion is quite different. The direction of its velocity is parallel to Earth’s surface rather than toward its center. But we are interested in the forces on each, and, according to the inertial view, forces cause accelerations, not velocities. So the forces on the two could be similar, despite the dissimilarity of the velocities. How do the forces compare? Aristotle would say that no force is needed to make the moon move in a circle because that is its natural motion, but the inertial view is that in order for the moon to deviate from straight-line motion, a force must act on it. What is the direction of Moon’s velocity is directed along its orbit

One has to be a Newton to see that the moon is falling, when everyone sees that it doesn’t fall. Paul Valery, French Poet and Philosopher, 1871–1945

Figure 1

The apple and the moon.

v

Falling apple: Velocity, acceleration, and force all are directed downward

v, a, F

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Newton’s Universe Figure 2

The moon is held into its orbit by an inward-directed force.

A

Moon’s displacement due to its inertia

C

B

Moon’s motion without gravity

Moon’s displacement due to gravity

Gravitational force Gravitational force

this force? If the moon were at point A in Figure 2 and if no force acted on it, it would move in a straight line toward point B. But instead it moves around to point C. As you can see from Figure 2, the force required to pull the moon inward—so that it arrives at C rather than B—is directed toward Earth’s center, just like the force on a falling apple. Newton hypothesized that this force has the same source as the force that pulls an apple downward: Earth’s gravitational attraction. Newton offered another argument, one that helps us understand why the moon and other satellites stay up. If you throw an apple horizontally, it will follow a curved path as it falls to the ground (Figure 3). If you throw the apple faster, it will go farther before hitting the ground. And if you throw it fast enough, it might “fall” around a large part of Earth’s surface before striking the ground (Figure 4). If the apple is launched at such a high speed that the curvature of its path just matches Earth’s curvature, it will fall all the way around. In other words, it goes into orbit. The required speed is about 8 km/s, or 29,000 km/hr. This is what any orbiting satellite does, except that the required speed is less for higher-altitude satellites because they feel a smaller gravitational pull and so don’t need to move as fast to avoid spiraling down onto Earth’s surface. For instance, the moon’s speed is only about 1 km/s. The force that shapes the moon’s path is gravity—the same gravity that pulled the apple to the ground that day on Isaac Newton’s family’s farm. It was an imaginative leap, in more ways than one. It was difficult to believe that anything at all was pulling on the moon, much less that it could be the same force that pulled on an apple. Most difficult was the notion that the gravitational force could reach across nearly 400,000 kilometers of empty space (the distance was known in Newton’s time). It’s easy to see that things exert forces on one another when they are in direct contact, but a force that acts across so great a distance seems astonishing. Figure 3

If you throw an apple horizontally, the faster you throw it, the farther it will go.

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Slower

Faster

Newton’s Universe Suborbital

Figure 4

Falling around Earth. If you throw an apple fast enough, it will fall around a large part of Earth’s surface or even go into orbit. A diagram like this appears in Newton’s notebook.

Orbital

CONCEPT CHECK 1 A 2 N apple falls from a tree. Neglecting air resistance, while it is freely falling the net force on it is (a) zero; (b) 2 N downward; (c) 2 N upward. CONCEPT CHECK 2 In the preceding question, the apple’s acceleration is (a) zero; (b) impossible to determine from the given information; (c) about 10 m/s2 downward; (d) about 10 m/s2 upward. CONCEPT CHECK 3 Suppose you throw a 2 N apple horizontally, as shown in Figures 3 and 4. Neglecting air resistance, the net force on the apple when it is in the five positions shown is (a) zero; (b) 2 N in the forward direction (along the direction of motion); (c) 2 N downward (toward Earth’s center); (d) 2 N upward (away from Earth’s center); (e) impossible to determine. CONCEPT CHECK 4 The net forces in Concept Checks 1 and 3 are the same in both magnitude and direction. So, what must be the numerical value and direction of the apple’s acceleration in Figures 3 and 4? (a) Zero. (b) Impossible to determine from the given information. (c) About 10 m/s2 in the forward direction. (d) About 10 m/s2 downward. (e) About 10 m/s2 upward.

2 NEWTON’S THEORY1 OF GRAVITY: MOVING THE FARTHEST STAR Since Earth’s gravitational pull holds the moon into its orbit, it’s reasonable to suppose that all satellites—bodies in orbit around larger astronomical bodies—are held in their orbits by the gravitational pull exerted by the larger body. Since the planets are satellites of the sun, Newton’s insight regarding the moon also resolves

1

This theory is usually called Newton’s law of gravity. But the term law, which is always inappropriate in science because every general scientific principle is subject to some doubt, is especially inappropriate here. The reason is that Newton’s theory of gravity has limitations, beyond which the theory isn’t valid. The more accurate theory, having no known limitations, is always called Einstein’s theory of general relativity. It’s ridiculous to call Newton’s theory a “law” while calling Einstein’s theory a “theory.”

What makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit.... The answer is not far from the truth. The only difference is that the angels sit in a different direction and their wings push inwards. Richard Feynman, Physicist

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Newton’s Universe Pick a flower on Earth and you move the farthest star! Paul Dirac, Physicist

Force by apple on book

Force by book on apple

Figure 5

Even ordinary-sized objects exert gravitational forces on one another. Your physics book exerts a force on an apple, and vice versa. It’s a small force, but forces like this have been measured.

the old question of why the solar system moves as it does! The planets keep moving forward because of the law of inertia, and the sun’s gravitational pull bends their orbits into ellipses. Similarly, the moons of the planet Jupiter are held in their orbits by Jupiter’s gravitational pull. But why would gravity act only between astronomical bodies and their satellites? For instance, it seems plausible that there should be a gravitational force between Earth and Mars. Such a force between planets had not been noticed yet in Newton’s day, but Newton realized that this was only because it was so much smaller than the force by the sun on the planets. Likewise, there should be a gravitational force between any two astronomical bodies, even between the farthest stars. But why should gravity be restricted to astronomical bodies? Why shouldn’t a gravitational force be exerted between smaller objects on Earth—oranges, rocks, and so forth? Your physics book, for instance, should exert a gravitational pull on an apple, and vice versa (Figure 5). You won’t notice this force, but that is only because the force between such objects is very small. So Newton reasoned that the gravitational force is universal; it’s exerted between every pair of objects throughout the universe. This is the central idea of Newton’s theory of gravity. Newton understood the importance of quantitative methods. Although his basic insight was qualitative, its expression in a quantitative form led to powerful explanations and predictions. Quantitatively, the gravitational attraction between two objects must be stronger when the objects’ masses are larger, because an apple’s weight is larger when its mass is larger (double the mass, for example, by replacing the one apple by two apples glued together, and you double the weight). And since widely separated objects attract each other only weakly, the gravitational force should get smaller when the distance between the objects gets larger. Newton put all this together (see “How Do We Know Newton’s Theory of Gravity?” later in this section) and came to the following conclusions: Newton’s Theory of Gravity Between any two objects there is an attractive force that is proportional to the product of the two objects’ masses and proportional to the inverse of the square of the distance between them: gravitational force r

(mass of 1st object) * (mass of 2nd object) square of distance between them

F r

m1 * m2 d2

If mass is expressed in kilograms, distance in meters, and force in newtons, this proportionality becomes F = 6.7 * 10 - 11

m1 * m2 d2

For our first example, let’s consider your weight—the gravitational force exerted by Earth on you. Newton’s theory of gravity tells us that this force is proportional to your mass times Earth’s mass, which means that the force is proportional to each of

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the two masses separately. So doubling your mass would double your weight, tripling your mass would triple your weight, and so forth—which certainly makes sense. But the theory also says that if you imagined that somehow Earth’s mass were doubled (without, however, changing its size), this also would double your weight; halving Earth’s mass would halve your weight; and so forth. You can reduce your weight without dieting or exercising: Simply reduce Earth’s mass! What if you altered both masses? For instance, suppose you tripled your mass while simultaneously doubling Earth’s mass. Since the force is proportional to the product of the two masses, this would multiply your weight by 6. What happens when the distance between Earth and you is changed? In fact, exactly what is meant by the “distance between the objects” in a case like this? Does the distance from Earth to your body mean the distance from the near side of Earth (the ground beneath your feet), from the far side, from the center, or from some other point? And to what point in your body should you measure the distance? Newton worked through a lot of mathematics to answer this—in fact, he invented “integral calculus” to answer it. Newton’s answer was that the distance between the “centers” of the two bodies is the correct distance to use when applying the gravitational force formula to two extended bodies. In the case of a body such as Earth that has an obvious center, distance is measured from that center. For other bodies, such as your own, the distance should be measured from the body’s “balance point”—the point at which the body would be balanced under the force of gravity. But because your body is so small compared with the distance from Earth’s center to your body, it matters little which point you choose within your body. Suppose you travel away from Earth. Since the gravitational force is proportional to the inverse of the square of the distance, the increased distance makes the force decrease—another way to reduce your weight! For instance, your weight at the top of Mount Everest, nearly 10 km above sea level, is 0.3% less than at sea level. If your weight is normally 600 N (135 lb), it will be 598 N (134.5 lb) at the top of Mount Everest. Your weight reduction is greater at an altitude of a few hundred kilometers, where low-orbit artificial satellites travel. For example, at a 200 km altitude, your weight would be reduced by 6%, so a person normally weighing 600 N would weigh only 560 N. Now you’re really losing weight (but unfortunately you’re not losing any mass). Moving to still higher altitudes, suppose you are 6400 km—1 Earth radius— above the ground. What is your weight? The proportionalities in the theory of gravity make this an easy question. If you rise 1 Earth radius above the ground, your distance from Earth’s center doubles, so the square of the distance quadruples. Since the force is proportional to the inverse of the square of the distance, the force is divided by 4. Your weight is now one-fourth of normal. Figure 6 is a graph of your weight at various distances from Earth’s center. No matter how far you are from Earth, the gravitational force by Earth on your body never reaches precisely zero. But very far away, the force becomes very small. For example, at 10 Earth radii, your weight is 1% of your normal weight. It’s a good thing for us that the force of gravity declines at larger distances in just the way it does. If the gravitational force declined a little faster, the planets would not move in ellipses but would instead spiral into the sun, and you would not be here to ask about things like gravity. And if the gravitational force declined a little more slowly, the gravity from distant stars would dominate the gravity from Earth, and again you would not be here. It’s something to think about. You can use the theory of gravity to calculate the gravitational attraction between any pair of objects, from apples and books to stars and moons. For example, the force

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Newton’s Universe Figure 6 Normal

Weight

A graph of your weight at various distances from Earth’s center. The same graph applies to the weight of any object.

1/4 of normal weight 1/9 of normal weight 1/16 of normal weight 1/100 of normal weight

2

4 6 8 Distance from Earth’s center measured in Earth radii

10

between a kilogram and another kilogram 1 meter away is found by putting these numerical values into the gravitational force formula. The answer is 6.7 * 10 - 11 newtons, or 0.000 000 000 067 newtons! It’s no wonder that the gravitational force between ordinary objects is difficult to detect. The delicate experiments needed to measure such tiny forces could not be performed until about a century after Newton’s work. When they were performed, they verified Newton’s predictions. The situation inside an orbiting satellite seems paradoxical. Judging from Figure 7, you would feel weightless in an orbiting satellite, at any altitude. But you have seen that if the satellite is in low orbit, your weight is actually only a little less than normal. Why, then, would you feel weightless, even though you are not really weightless? To answer this, let’s imagine a somewhat similar situation (Figure 8): Suppose you are in an elevator and the elevator cable breaks. The elevator is then in free fall, and so are you. After the cable breaks, your feet no longer press down against the floor. If you try to press your feet against the floor, you will simply push yourself away from the floor. A bathroom scale glued to your feet would read zero, because your feet would not press down on it. You are apparently weightless, but, because we have defined weight as the gravitational force on an object, you are not really weightless. Although you have not (I hope) actually experienced a freely falling elevator, you might have experienced a similar “weightless” effect in a roller-coaster while moving rapidly over a crest in the track. You would feel weightless in an orbiting satellite for the same reason that you would feel weightless in a freely falling elevator. As you saw in the preceding section, the satellite falls freely around Earth. You are falling freely around Earth too, regardless of whether you are inside the satellite or outside in space. Since both you and the satellite are just falling around Earth, you have the sensation of weightlessness. Your body behaves as though it were removed from the effects of gravity, but you are not really weightless. How do we know Newton’s theory of gravity? How did Newton verify his theory of gravity? The dependence on mass was not hard to deduce. Because an object’s weight is proportional to its mass (for instance, two identical apples glued together surely have twice the weight of one, so doubling the mass doubles the weight), Newton reasoned that the force of gravity must be proportional to each of the two masses. But what about the dependence on distance? Newton knew that the distance to the moon is about 60 times larger than Earth’s radius (Figure 9). Newton’s theory of gravity then implies that an object at the moon’s distance should experience a force that is 3600 (the square of 60) times smaller than the force on the same object on Earth. So

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NASA Headquarters

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NASA/Johnson Space Center

NASA/Johnson Space Center

(a)

(b)

(c)

Figure 7

Space travelers feel weightless when they are in orbit and at any other time that they are “falling” freely through space. (a) Balancing. (b) Floating. (c) Spacewalking.

(since acceleration is proportional to force) the acceleration of an object at this distance should be 3600 times smaller than the acceleration of an object falling to Earth. In other words, Newton’s hypothesis implies that the moon’s acceleration should be (1>3600) * 9.8 m>s2, or 0.0027 m/s2. But from the known distance to the moon, plus the observed fact that the moon takes 27 days to complete a circle around Earth, Newton could calculate directly that the moon’s acceleration (due to its circular motion) actually is 0.0027 m/s2. Newton’s theory agreed with the observation.

CONCEPT CHECK 5 Suppose that you were in distant space, far from all planets and stars, and you placed an apple and a book at rest in front of you, separated by

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about 1 m, and then moved some distance away in order to observe the apple and book without influencing them. The apple and the book would then (a) very slowly accelerate toward each other; (b) very rapidly accelerate toward each other; (c) move toward each other without accelerating; (d) remain at rest; (e) head for the beach.

I’m apparently weightless.

You’re apparently in trouble.

CONCEPT CHECK 6 When you are in a high-flying jet plane, (a) your weight and mass are both normal (the same as on Earth); (b) your weight and mass are both less than normal; (c) your weight is normal but your mass is less than normal; (d) your weight is less than normal but your mass is normal. CONCEPT CHECK 7 Your weight at an altitude of 2 Earth radii above Earth’s surface is (a) zero; (b) impossible to calculate without knowing Earth’s radius; (c) the same as your weight on Earth; (d) one-third of your weight on Earth; (e) onefourth of your weight on Earth; (f) one-ninth of your weight on Earth.

Figure 8

Falling freely in a freely falling elevator.

The moon is 60 times farther from Earth’s center than is the falling apple

M A K I N G EST I M AT ES

Earth’s mass is about 100 times the moon’s mass, and Earth’s radius is about 4 times the moon’s radius. From this information, use Newton’s theory of gravity to quickly estimate how much more an object weighs on Earth, as compared with its weight on the moon.

3 GRAVITATIONAL COLLAPSE: THE EVOLUTION OF THE SOLAR SYSTEM

Apple

Figure 9

The moon is 60 Earth radii away from the center of Earth.

Like you and me and everything else, stars have a beginning, they go through changes, and they have an ending. The driving force behind this “stellar evolution” is the force of gravity. Stars are made mostly from diffuse (thin) gas, mostly hydrogen atoms, that is spread throughout the universe. In some regions, this material happens to be gathered slightly more densely into great gas clouds that are the spawning grounds for stars (Figure 10). Because of the gravitational pull between all bits of matter, all gas and dust in space tends to aggregate, a process called gravitational collapse. Here is how the sun and Earth were born. Some 5 billion years ago, the atoms that would eventually form the solar system, including every atom in your body, were scattered as cold, diffuse gas and dust over a region far larger than the solar system. Then a blast of radiation and fast-moving particles from a nearby exploding star (more about this later) caused turbulence and clumping in this gas and dust. Such a clump of matter, if sufficiently dense, will gravitationally attract more gas and dust, causing still stronger gravitational forces, pulling even more matter inward, and so forth in a self-reinforcing buildup of matter. As our clump of gas and dust became more massive and more dense, atoms fell at greater and greater speeds toward the center, where they collided and formed a central region of fast-moving atoms. In other words, the center heated up. Every gas cloud spins a little, simply from the net effect of its chaotic flowing and swirling. As our gas cloud contracted, this spinning increased, just as a figure skater spins faster and faster as she brings her outstretched arms into her sides.2 As 2

This is because of something called “conservation of angular momentum,” a sort of rotational version of conservation of momentum.

SO LU T I O N TO M A K I N G EST I M AT ES In the theory of gravity, one of the masses is multiplied by 100 and R is multiplied by 4. Thus F is multiplied by 100/42 = 100/16, or about 6. So your weight is six times larger on Earth than on the moon.

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Newton’s Universe Figure 10

NASA Headquarters

Star birth. These eerie, dark, pillarlike structures are columns of cool interstellar hydrogen gas and dust that are also incubators for new stars. They are part of the Eagle Nebula, a nearby star-forming region in our own galaxy. This region is “only” 7000 light-years away (i.e., it takes light 7000 years to get here from there). The tallest pillar (left) is about 1 light-year long from base to tip—a distance that is about 800 times larger than the distance across our solar system. This is one of the many beautiful and informative photographs taken by the Hubble Space Telescope.

contraction continued, this spinning became rapid enough to flatten the outer regions of the gas ball into a disk, much as a wad of dough can be flattened to make a pizza by spinning it. Some of the gas in the outlying disk rotated fast enough to go into orbit around the larger central ball. Because it was orbiting, this material was left behind as the center collapsed. The outer region continued orbiting while cooling, condensing, and aggregating into clumps that became Earth and the other planets (Figure 11). As the warming sun got hot enough to glow, light streaming outward swept away the dust and gas that had filled the solar system. And then there was light on Earth. The central ball continued collapsing and heating until the center reached million-degree temperatures. New things happen at such temperatures: Atoms collide so violently that their electrons are stripped off, leaving a gas made mostly of bare hydrogen nuclei and electrons. The violently colliding hydrogen nuclei occasionally stick together, a process known as nuclear fusion. Nuclear fusion creates lots of heat, and the pressure from this heat then prevents the ball of gas from collapsing further. When it initiated nuclear fusion nearly 5 billion years ago, our sun turned itself on and became a normal, self-sustaining star. A similar process of star birth is going on all the time, all over the universe. The starry sky is not static. Scientists have recently learned that our sun had a violent birth amidst highenergy radiation and explosions of entire stars. Based partly on material found in meteorites that formed along with the solar system, it’s now known that one or more nearby stars must have exploded during the solar system’s formation. It’s thought

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Newton’s Universe Figure 11

Lynette R. Cook

An imaginary view of the newborn sun during formation of the solar system. Dust partially obscures the sun. As comets streak by, a planet (foreground) begins to form from the dust.

that the sun and some 50 or more other stars all formed at roughly the same time from a single huge region of gas and dust within our Milky Way Galaxy (our galaxy was already some eight billion years old by then). The upper, lit-up, portion of the left-hand “pillar” in Figure 10 is an example of such a star-forming region within our galaxy today. At least a few very massive stars, far more massive than the sun, are likely to form in such regions. Any such massive star “burns” (via nuclear fusion) very hot and therefore very rapidly, and soon exhausts itself in a giant supernova explosion (more on this in the next section). High energy radiation and ejected particles from such massive stars, as well as the blast effects from supernova explosions themselves, then initiated the formation of smaller stars as described in the preceding paragraph, and also helped shape such star formation processes. Once our sun stopped collapsing, it settled into a middle age that has been going on for nearly 5 billion years. The long-term stability of this period made it possible for atoms on one planet to gather and evolve into highly complex forms such as ourselves. Like the rest of the solar system, we came from the universe. But stars eventually die. Over billions of years, our sun’s supply of hydrogen fuel must deplete until, around the year 5,000,000,000 CE, it will no longer support nuclear fusion in its central “core.” Then the sun will enter old age. Although nuclear fusion will cease near the sun’s center, a thin outer shell of hydrogen will continue the fusion process, causing the sun to brighten and expand to three times its present size. The increased energy output will evaporate Earth’s oceans and perhaps cause a runaway greenhouse effect that could make Earth even hotter than Venus’s 500°C. During the following several hundred million years, the sun will become still brighter and 100 times larger, warming Earth to around 1000°C and killing any remaining life. By this time, the central core will have grown hot enough to ignite new, hotter nuclear reactions involving the element helium. The sun will then spend 100 million years as a helium-burning star. Then comes another disaster. After exhausting its helium, the sun will again expand, brighten, and eject its outer layers in a huge shell of glowing gas that will expand outward, engulfing all the planets and drifting outward beyond the solar system into interstellar space. After a million years of this, the sun will have entirely exhausted its energy sources.

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Gravity will assert itself for the final time. Without a nuclear heating source, there will be little to stop the sun from collapsing inward on itself. Certainly the interatomic forces that hold up solid matter against outside pressures on Earth are far too puny to stand up against the enormous inward pull of gravity in the final collapse of a star. The sun will squeeze itself far inside its present boundaries and far inside the volume it would have if it were made of ordinary solid material, squashing its atoms out of recognizable existence until only a solid, tightly packed ball of bare nuclei and unattached electrons remains. At this point, the collapse will be permanently stopped by an effect known as “quantum exchange forces” between the electrons.3 The sun’s burnt-out corpse will be hot, solid, and about Earth’s size, or onemillionth of its present volume! It will be extraordinarily compact, with many tonnes packed into each cubic centimeter. On Earth, even a solid steel platform would be unable to support a mere thimbleful of this material. The sun will warm enormously during its final collapse, but once the collapse ends there will be no further source of heating. This starry remnant will glow brightly for a while and then slowly dim like a dying ember, still orbited by the charred remains of Earth and other planets. A star the size of Earth? When such an object was first discovered in 1862, astronomers thought there must be an error in their observations. But two other such stars were soon discovered, and it’s now known that about 4% of the stars in our galaxy—some 16 billion stars—are of this type. Because of their white-hot glow they are called white dwarfs. How do we know our solar system’s past and future? Detailed quantitative theories predict the scenario just sketched. Observations of stars in the various evolutionary stages described and observations of Earth’s oldest rocks, the moon, moon rocks, meteorites, other planets, other moons, and the sun itself all support these theories. The natural place to look for star births is among thick gas clouds in space. When the Hubble Space Telescope searched the dense gas cloud known as the Eagle Nebula, it found thousands of newly minted stars (Figure 10). Just as the theory predicts, nearly all of these new stars were wrapped in disks of dust and gas, disks that are expected eventually to coalesce into planets.

CONCEPT CHECK 8 Suppose that the sun collapsed tomorrow to become a white dwarf, but without any explosions or expansion that would alter the sun’s mass or directly impact Earth. Which of the following would ensue? (a) Earth’s orbit would be altered. (b) Life on Earth would be radically affected. (c) The gravitational force by the sun on Earth would be radically altered. (d) The sun’s radiation would be radically altered. (e) None of the above.

4 GRAVITATIONAL COLLAPSE: THE DEATHS OF MORE MASSIVE STARS A star’s life cycle is determined primarily by its mass. A star needs at least 10% of the sun’s mass in order to get hot enough to initiate nuclear fusion and become a star in the first place. All stars massive enough to initiate nuclear fusion go through a middle age that is similar to the sun’s present state. Then when the hydrogen fuel 3

My suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose. John B. S. Haldane, British Geneticist, 1892–1964

Quantum exchange forces have no explanation within Newtonian physics. They are far stronger than the ordinary electrical forces that maintain the solidity of normal solid matter.

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in their central cores have been used up, they enter their final phases. Stars having masses up to about 10 times the sun’s mass go through a final phase similar to the sun’s, ending as white dwarfs. But a quite different fate awaits more massive stars. Like the sun, they use up their hydrogen fuel and then contract at the center. But the larger mass makes the contraction stronger, so the center gets hotter. The high temperature initiates a wide range of nuclear reactions that eventually turn the star’s small central core into solid iron. This gets the star into serious trouble. Iron continues forming until the inner core becomes so massive that it cannot hold itself up. The entire solid iron core then abruptly collapses in just one second! As the core collapses, this unimaginably cataclysmic supernova explosion blasts the rest of the star into space. For a brief moment, the dying star glows as brightly as 4 billion suns. No supernova has been seen in our galaxy since 1604, but today astronomers are able to routinely discover them in other galaxies. The nearest of these burst into view in 1987 and was visible to the naked eye (Figure 12). It occurred in a neighboring small galaxy at a safe distance of 150,000 light-years (meaning that light travels from there to here in 150,000 years). A nearby supernova, if it were as close as 10 light-years or so, would produce various radiations that would create a fabulous light show in Earth’s atmosphere, and that would soon kill us. But not to worry: Such an event won’t happen in our corner of the Galaxy, because a candidate star for a supernova must be at least 10 times as massive as the sun and there’s nothing that massive that close. The nearest likely candidate is Betelgeuse, which has been acting unstable for years. But it’s at a safe 430 light-years away. Only 10% to 20% of the original star remains after the explosion. No further nuclear reactions can occur in this remnant, so there is little to oppose the inward pull of gravity. Within one second, this remaining massive core collapses to become one of the two densest things in the universe, a neutron star or a black hole. If the original star had a mass of between 10 and 30 suns, the final collapse is strong enough that electron exchange forces (see the previous section) can’t stop it.

(a)

(b)

National Optical Astronomy Observatories/Science Photo Library/Photo Researchers, Inc. Figure 12

The supernova of 1987, the brightest supernova in 400 years. Its light reached Earth on February 23, 1987. “Before” (a) and “after” (b) photos show the star as it looked before and shortly after the explosion. The supernova is the bright star on the right in figure (b).

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But there’s one remaining force that does stop it, the so-called “neutron exchange force,” a quantum effect similar to the electron exchange force but acting between neutrons. The collapse not only squashes atoms out of existence, it also squashes electrons out of existence by forcing them to merge with protons in the nuclei. This turns each nucleus into a collection of neutrons, and it turns the entire star into an object that resembles a giant nucleus made of neutrons. It’s called a neutron star. Nuclear physicist J. Robert Oppenheimer, who later gained fame as leader of the team that developed the atomic bomb, predicted neutron stars in 1938. None were discovered until 1967, when Jocelyn Bell (Figure 13), a sharp-eyed astronomy graduate student in England, discovered a source of radio waves in space that sent out “beeps” or “pulses” every 1.3 seconds. Some scientists thought at first that she might have discovered a radio beacon from an extraterrestrial civilization. But another was soon discovered, and by now hundreds are known, with a wide range of pulse rates. There’s little doubt that they are neutron stars. A neutron star is pretty impressive. More massive than the sun, the star is only a few kilometers across with a billion tonnes packed into each cubic centimeter! On Earth, a barely visible speck of this material would weigh as much as a large, fully loaded highway truck! The collapsing iron core spins faster and faster during its onesecond collapse, so that the remnant neutron star spins at incredible speeds for such a massive object—up to 700 times every second. The surface of such a rapidly spinning neutron star moves at an incredible 15% of the speed of light. This is staggering when you realize that the star has a mass of some 1027 tonnes. This spinning combines with magnetic effects to create the rapid pulses of visible light and radio signals observed from Earth, the signals that Bell discovered in 1967. As seen from Earth, the entire star appears to flash on and off many times every second. Figure 14 is a sequence of photographs showing two of these visible flashes. The supernova explosion that created this neutron star was seen and recorded on Earth in 1054. For a few days the light from the explosion was brighter than the planet Venus. Today, it’s called the Crab Nebula because the shape of the nebulous halo of gases blown into space by the explosion resembles a crab. Neutron stars pull hard. The star’s radius is only about 10 kilometers, which is 100,000 times smaller than the sun’s radius. Yet the star is more massive than the sun. Newton’s theory of gravity tells us that the weight of an object on the surface of a star is proportional to the inverse of the square of the star’s radius. If a collapsing star’s radius becomes 105 times smaller, an object on its surface becomes 105 * 105 = 1010 (10 billion) times heavier. That’s heavy. What about stars even more massive than those that collapse to form neutron stars—stars having a mass of over 30 suns? When such a star runs out of fuel, the ensuing collapse is so strong that no known force can stop it. According to current theories, it collapses into a single point! Its matter—its atoms and subatomic particles—is squeezed out of existence. The star retains its mass, however, and so it retains its gravitational influence on the space around it. This star pulls really hard. If you had the misfortune to get too close, you could not escape, because gravity won’t let anything escape—not even light itself. That’s why it’s called a black hole. For a typical black hole formed from a collapsing giant star, the distance within which nothing can escape is 10 to 50 km (larger for more massive black holes). You can think of this as the “radius” of the black hole. If you were within this distance of the central point and you wanted to throw an object completely away from the

Robin Scagell/Photo Researchers, Inc.

Newton’s Universe

Figure 13

Jocelyn Bell (later Burnell) discovered the first four neutron stars. Using a radio telescope that she helped build as part of her Ph.D. dissertation, Bell detected a rapid set of pulses occurring at regular intervals. She determined that the position of the unusual radio source remained fixed with respect to the stars, which meant that it was located beyond the solar system. During the course of the next few months, she discovered three more pulsating radio sources. These were later found to be rapidly rotating neutron stars.

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N P 0532

National Optical Astronomy Observatories Figure 14

A sequence of photographs of the neutron star at the center of the Crab Nebula. Portions of a surrounding gas cloud, the remnant of the supernova explosion that created the neutron star, can be seen. This sequence lasts 1/20 second and includes two flashes, the first during frames 3 and 4, and the second during frames 9 and 10.

star, you would need to throw it faster than the speed of light. But objects cannot be thrown faster than light. So nothing can escape a black hole.4 How do we know that black holes exist? Scientists detect black holes by their gravitational influence on things around them. The first black hole, Cygnus X-1, was discovered in 1972. It’s thought to be a double star, two stars orbiting each other. One is a visible giant star, the other an unseen compact (far smaller than a normal star) object. By observing its gravitational effect on the visible star, the compact object’s mass can be deduced to be 10 solar masses.5 Since theories indicate that a compact object of more than 3 solar masses can only be a black hole, astronomers believe that Cygnus X-1 is a black hole. Satellites in orbit around Earth detect X-rays from Cygnus X-1 that further confirm it to be a black hole. Apparently the invisible object’s gravitational pull is drawing gases from the visible star and accelerating them down into and around the black hole, a process that tears apart the gas atoms and causes them to emit X-rays that scientists can observe (Figure 15). Astronomers have now identified about 20 similar objects within our galaxy that are thought to be black-hole remnants of collapsed stars, and they suspect that there might be around one billion of them in our galaxy. Scientists don’t go out of their way to invent bizarre ideas like black holes. To the contrary, they look for the least strange explanation of the data. For example, people once found it strange that Earth could orbit the sun, but astronomers such as Copernicus found that this was the most natural way to account for the data. In the same manner, astronomers today find that a black hole is the most natural explanation for what they observe at Cygnus X-1. If it is not a black hole, then this object is not compact, or it does not have a mass larger than 3 suns, or compact objects of greater than 3 solar masses are not always black holes. Astronomers find it easier or “less strange” to believe that Cygnus X-1 is a black hole than to believe any of these options. There is a second kind of black hole for which the evidence is even more compelling than it is for Cygnus X-1. The centers of most galaxies contain extremely massive black holes. In 1994, for example, the Hubble Space Telescope found a tiny, bright source of

Julian Baum/New Scientist/Science Photo Library/Photo Researchers, Inc. Figure 15

In this artist’s conception, a black hole pulls matter from a companion star and accelerates it into a hot, X-ray–emitting disk before slowly swallowing it.

4

5

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More precisely, quantum theory allows black holes to emit subatomic particles, but this effect is negligible for collapsed stars. This effect is expected to be important for low-mass black holes, although such small black holes have never been observed and may not exist. The original star, before collapse, had a mass of more than 30 suns, but most of this mass blew into space during the collapse.

Newton’s Universe light at the center of a distant galaxy. Detailed analysis of this light showed that nearby gas and stars are orbiting this center so rapidly that gravity can hold them in their orbits only if the bright object has a mass of several billion suns. Given that the central object’s size is only slightly larger than our solar system, it could only be a black hole. The light apparently comes from high-energy processes occurring just outside the black hole. Study of distant galaxies reveals that the centers of most or all of them contain black holes having masses of millions or billions of suns. The distant and powerful objects known as “quasars” are powered by such giant black holes. Observation of a small portion of sky, and extrapolation to the entire sky, leads to an estimate of at least 300 billion giant black holes populating the observable universe! Observations of stars dashing in tight orbits around the center of our own Milky Way Galaxy at up to 1/30th of the speed of light imply a giant black hole lurks there. Despite having a mass of nearly 4 million suns, it is only about 20 times larger than our sun! Such giant black holes radiate X-rays and light as they swallow nearby stars and gas. Their origin is not yet understood.

CONCEPT CHECK 9 If Earth collapsed from its present 6000 km radius to only 6 km, your weight would be (a) unchanged; (b) l/1000th of your present weight; (c) 1/1,000,000 of your present weight; (d) 1000 times your present weight; (e) 1,000,000 times your present weight.

5 THE NEWTONIAN WORLDVIEW: A DEMOCRATIC, MECHANICAL UNIVERSE During the sixteenth and seventeenth centuries, the new sun-centered astronomy and inertial physics ushered in a new philosophical and religious view that I’ll call the Newtonian worldview.6 It is one of the most significant consequences of Newtonian physics. Even though Newtonian scientific ideas have been partly superseded by other more accurate theories, the worldview based on Newtonian physics retains its influence on popular culture. In the Western world, the pre-Newtonian worldview combined medieval Christianity, the Earth-centered astronomy of the ancient Greeks, and Aristotle’s physics. Central to this view was the idea of purpose, or future goals. During the Middle Ages in Europe, popular culture united with religion and science in the belief that there was a purpose for everything and that the universe’s larger purposes were tied to humans, so that humankind was central to all creation. Ancient Earth-centered astronomy and Aristotelian physics, with its goal-directed natural motions, chimed in perfectly with this traditional view. It was well attuned to the era’s hierarchical social structure, comprising a God-ordained king surrounded by a few land-holding nobles surrounded in turn by many land-working serfs and peasants. Astronomy and physics since the Middle Ages have contradicted Earth-centered astronomy and Aristotelian physics. Copernicus removed Earth from the center, Kepler replaced the planets’ “natural” circular orbits with ellipses, and Descartes declared that bodies move not because they have a goal but simply because there is nothing to stop them. The hierarchy of natural places, the notion that Earth is special, the centrality of humankind, and the scientific basis for purpose in the universe—all were swept away. It was not by chance that stirrings for religious and political freedom began at about this time. Once the hierarchical cosmology began to crumble, it was no 6

I am much occupied with the investigation of the physical causes [of the motions of the solar system]. My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork... insofar as nearly all the manifold motions are caused by a most simple, magnetic, and material force, just as all motions of a clock are caused by simple weight. Kepler

Newton, Descartes, and Galileo were among the scientists and philosophers who contributed to this view.

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longer obvious to people that they should follow the old hierarchical cultural habits. The new science established universal natural laws, rather than particular people or religious beliefs, as the ultimate framework for human behavior. Religious reformers such as Martin Luther felt freed to challenge medieval Christian traditions. Political reformer Thomas Jefferson could draw up a Declaration of Independence that threw off the divine rights of the king of England and that was permeated with the concept of “unalienable rights” flowing directly from “the Laws of Nature and of Nature’s God” to all people as the basis for human equality. Thus does our science influence, on quite a deep level, our religion, our social order, and our politics. Galileo sought only to describe how things behave, not why they behave as they do. He was not concerned with a physical phenomenon’s purpose. Analysis—the new technique of separating phenomena into their simplest components and studying those components—was one of his tools. This led to a focus on the simplest and smallest components of matter: atoms. And so atomism—the idea that nature can be reduced to the motions of tiny material particles—underlay the new physics. For example, in a view remarkably similar to Democritus’s view, Newton stated: It seems probable to me that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles... and that these primitive particles being solids are incomparably harder than any porous bodies compounded of them, even so hard as never to wear or break in pieces.... [Men are] engines endowed with wills. Robert Boyle

Now I a fourfold vision see, And a fourfold vision is given to me; ‘Tis fourfold in my supreme delight And threefold in soft Beulah’s night And twofold Always. May God us keep From Single vision And Newton’s sleep! William Blake, 1757–1827, Poet, Painter, Rebel Against the Mechanical Single Vision, or Linear Thinking, of Newton

Newton, Galileo, and Descartes believed firmly in God. What place within the new science could be found for God? Descartes reconciled the new science with traditional religion by assuming that there were two realities, a notion known as dualism. The first reality was the material world, made of matter and operating according to nature’s inflexible laws. Here, the true realities, or primary qualities, were assumed to be impersonal physical characteristics such as the motions of atoms. The second reality was spiritual, the realm of human thoughts and feelings and communication with God. These were assumed to be secondary qualities that were not part of the physical world but were merely reflections of the primary qualities. Thus did science and philosophy relegate human concerns to a shadowy secondary role in a physical universe. This left little room for God in the workings of the material universe. In the traditional view, God is continually and actively present throughout the universe, continually endowing all things with purpose. In the new view, God is, at most, an uninvolved observer. Descartes and Galileo believed that God was needed to establish the laws of nature and to start the universe moving but that once started, the whole thing would run itself.7 A machine, especially a finely tuned machine such as a clock, is an excellent analogy for the Newtonian worldview. Once the owner starts it, a clock runs itself according to its own operating principles. The founding fathers of physics thought of the universe as a clockwork mechanism whose operating principles were the laws of nature and whose parts were atoms. Because of its machinelike quality, I’ll call this view the mechanical universe. In fact, one major consequence of Newtonian physics is that every physical system is entirely predictable, like a perfectly operating clock. For a simple example, Newtonian physics can predict precisely how far a freely falling object will fall 7

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With a few exceptions, Newton also believed that God did not intervene in the universe. On certain occasions, namely in situations for which Newton himself could not find a scientific explanation, he believed that God momentarily intervened. However, this “god of the gaps” view—that every phenomenon that cannot be explained by science requires an intervention by God—becomes less and less tenable as science closes the gaps.

Newton’s Universe

during any specified time. This clocklike predictability has surprising implications. To understand them, imagine a simple, isolated, self-contained collection of atoms that move and interact in accordance with Newtonian physics. Suppose you specify the precise positions and velocities of every atom at one particular time. Then, according to Newton’s theory of motion, the entire future behavior of this system can be precisely predicted, for all time. But the Newtonian view is that the universe itself is just such a collection of atoms. Thus, the future is entirely determined by what all the atoms of the universe are doing right now or at any other time. Furthermore, since humans are entirely made of atoms, it follows that every thought or feeling that enters your head is reducible to the motion of atoms within your brain and elsewhere. Thus, all of your thoughts, feelings, and actions are entirely predetermined and predictable. You never choose to scratch your nose, for example—the laws of nature choose for you. You might believe that you choose, but this, too—this believing that you choose—was chosen for you by the laws of nature. Such a mechanistic universe, the loss of free will, and the absence of a continuously creative God strike many observers as inhuman and cold. For example, German social scientist Max Weber (1864–1920) spoke of the “disenchantment of the world” brought about by Newtonian science. Poet and painter William Blake (1757–1827) wrote disdainfully in a poem “May God us keep/From Single vision/ and Newton’s sleep!” Nevertheless, from the seventeenth into the twentieth century, these ideas influenced many educated people. Newtonian physics was so successful that the associated philosophy was accepted with little question. People absorbed the clockwork universe without knowing they were absorbing it. There are reasons today to question both the Newtonian worldview and Newtonian physics. Nevertheless, it would be surprising if these views were not still influential today. A person’s worldview tends to be absorbed thoughtlessly, as part of the cultural air of the times. It seems likely that the Newtonian worldview remains active, even (or perhaps especially) among people who have never heard of Isaac Newton. It is for you, valued reader, to determine to what extent the Newtonian worldview is valid, whether it retains a significant influence, and what difference it might make.

[It is unbelievable] that all nature, all the planets, should obey eternal laws, and that there should be a little animal, five feet high, who in contempt of these laws, could act as he pleased, solely according to his caprice.

6 BEYOND NEWTON: LIMITATIONS OF

It’s a material world.

NEWTONIAN PHYSICS

Voltaire, French Philosopher and Writer, 1694–1778

They may say what they like; everything is organized matter. Napoleon Bonaparte, 1769–1821

I never satisfy myself until I make a mechanical model of a thing. If I can make a mechanical model I can understand it. As long as I cannot make a mechanical model all the way through I cannot understand. Lord Kelvin, Nineteenth-Century British Mathematician and Physicist

That Man is the product of causes which had no prevision of the end they were achieving; that his origin, his growth, his hopes and fears, his loves and his beliefs, are but the outcome of accidental collocations of atoms;... all these things, if not quite beyond dispute, are yet so nearly certain, that no philosophy which rejects them can hope to stand. Bertrand Russell, Philosopher and Mathematician, 1872–1970

Madonna

Tested repeatedly during the eighteenth and nineteenth centuries, Newtonian physics stood up in quantitative detail to every challenge. In fact, it was so powerful and accurate that scientists began to accept it as true in an ultimate, absolute sense. But science is never absolute. Even though a scientific principle has been confirmed repeatedly, it hangs always by the slender thread of new experiments. Around 1880, experimental results began appearing that couldn’t be reconciled with Newtonian physics. The incorrect Newtonian predictions arose in four extreme situations: at high speeds, for enormous gravitational forces, at huge distances, and at tiny distances. During the first few decades of the twentieth century, Albert Einstein, Werner Heisenberg, and many others invented three new theories to account for these discrepancies: special relativity, general relativity, and quantum physics. To date, at least, scientists have found no exceptions to any of the new theories. Experiments show that Newton’s law of motion and Newtonian views of time and space break down at high speeds. The disagreement is not noticeable at slow

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speeds, but the errors become worse as speeds increase. The non-Newtonian effects are difficult to detect for automobiles, jet planes, or even orbiting satellites moving at some 10 km/s. But at 30,000 km/s (around the world in about 1 second!), Newtonian predictions are off by 0.5%. At 290,000 km/s, nearly the speed of light, typical Newtonian predictions are incorrect by a factor of 4! Scientists didn’t notice these non-Newtonian effects for 200 years because they had never closely studied such fast-moving objects. Special relativity gives correct predictions at all speeds, both low and high. These predictions become indistinguishable from Newtonian physics whenever the speeds are considerably less than the speed of light. Similarly, experiments show that Newton’s theory of gravity, Newton’s law of motion, and Newtonian views about time and space are incorrect for objects subjected to enormous gravitational forces and also over huge distances. For example, non-Newtonian gravitational effects are measurable, but small, for the orbit of the innermost planet, Mercury, which feels strong gravitational forces from the sun. Non-Newtonian gravitational effects are pronounced near neutron stars and black holes and for physical systems that range over large portions of the observable universe. General relativity gives correct predictions for all these situations and is regarded as the correct theory of gravity. The predictions of general relativity become indistinguishable from Newtonian physics whenever gravitational forces are not too strong and distances are not too large. The disagreements between Newtonian physics on the one hand and Einstein’s special and general relativity on the other stem from profoundly different ways of viewing space and time. Newton took the common intuitive view that all of us take in our daily lives: Space is infinite in extent, time is infinite in duration, and both have the same properties everywhere and at all times. Einstein, however, found that space and time are “relative,” or different for different observers, namely observers moving at different speeds. For example, the duration of a process such as the melting of your ice cream cone is different as viewed by you from its duration as viewed by your friend who is moving past you. From this, all sorts of new results emerge, such as that space can “curve,” and time runs differently in different places. Finally, experiments show that Newton’s law of motion and subtle Newtonian views concerning predictability and cause and effect are incorrect for objects of molecular dimensions or smaller. Quantum physics gives correct predictions for objects of all sizes, from microscopic to macroscopic. For macroscopic objects like footballs and apples, quantum theory’s predictions become indistinguishable from Newtonian physics. Although quantum physics represents an even more profound revolution than does relativity, it stems from a seemingly insignificant difference concerning such properties as speed, momentum, and energy. Newtonian physics allows such properties to have any numerical value whatsoever within a continuous range of possibilities, for example, the speed of a particular airplane might be anything between zero and 1000 km/h. But quantum physics states that such properties can only have specific “permitted” values, such as 0.01 km/h, 0.02 km/h, 0.03 km/h, etc. up to 1000.0 km/h.8 The speed of the airplane is said to be “quantized.”

8

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This is an exaggerated example for purposes of illustration; the differences between the permitted values would be much much smaller than this for a real airplane, and so this “quantization principle” isn’t significant for airplanes and other large objects. But for microscopic objects, these “small” differences between permitted values are more important, and so quantization makes a big difference in the microworld.

Newton’s Universe

This turns out to have surprising and profound implications, especially in the microscopic world. Figure 16 is one way of indicating, graphically, these limits of validity of Newtonian physics. The vertical axis represents the speed of individual objects. Special relativity predicts that objects cannot move faster than the speed of light— 300,000 km/s—so these speeds are forbidden. The horizontal axis shows the size of individual objects. Because a principle known as “quantum uncertainty” predicts that objects cannot be both small9 and slow moving, there is another forbidden region in the small-size, low-speed corner of the diagram. The lesson is that it’s not a Newtonian universe. Earth, where Newtonian physics works well for ordinary objects, is an exception in a universe dominated by relativistic and quantum phenomena. The conditions we regard as normal occur only rarely in the universe. Newtonian and intuitive concepts of time, space, matter, and much else are far from correct throughout most of the universe. The “real” universe— the quantum-relativistic universe—is fundamentally different, and far stranger, than our Earth-bound intuitions could have imagined.

Speed Forbidden

Speed of light, 300,000 km/s

Quantum ⫹ special relativity

Special relativity General relativity

10% of speed of light, 30,000 km/s Quantum

Newtonian

Forbidden 10⫺5 m

1022 m

Size or distance

Figure 1610

Newtonian physics is correct for common phenomena on Earth, but breaks down for objects that are very small, very large, or very fast. Newtonian physics also breaks down for strong gravitational forces, such as those near a neutron star or black hole. The quantum and relativity theories apply throughout the entire range of the phenomena observed to date. The diagram is only schematic and approximate. 9

“Highly localized” is a more accurate term than “small.” A particle such as an electron is said to be “localized” within a small region of space when it is observed (or known) to be located within that region. The quantum uncertainty principle implies that a particle that is localized within a very small region must have (on the average) a high speed. 10 Thanks to Douglas Giancoli, the author of several physics textbooks, including Physics: Principles with Applications (Englewood Cliffs, NJ: Prentice Hall, 1991), for suggesting diagrams of this type.

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© Sidney Harris, used with permission.

Newton’s Universe

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Newton’s Universe Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions THE IDEA OF GRAVITY 1. What is the direction of a falling apple’s velocity? Of its acceleration? 2. What is the direction of the moon’s velocity? Of its acceleration? 3. Does Earth exert a force on the moon? What is its direction? How would the moon move if this force suddenly vanished? 4. In what ways are the moon and a falling apple similar? In what ways do they differ?

NEWTON’S THEORY OF GRAVITY 5. Does this book exert a gravitational force on your body? 6. What would happen to this book’s weight if you managed to double Earth’s mass? What if, instead, you doubled the book’s mass? What if you doubled both? 7. In order to use Newton’s theory of gravity to calculate your weight, what data would you need? 8. If you were orbiting Earth in a satellite 200 km above the ground, would you be weightless? Would your weight be as large as it is when you are on the ground? Would you feel weightless? Explain.

GRAVITATIONAL COLLAPSE 9. What caused the sun to get hot? What keeps it hot today? 10. Describe the process that formed the planets. 11. Since gravity pulls inward on the material in the sun and since the sun is made only of gas, why doesn’t the sun collapse? 12. Are there places in our galaxy where stars are being born? 13. Name the process and also the substance that fuels the sun. 14. What will happen to the sun after it runs out of fuel? 15. Name and describe the object into which the sun will evolve after it runs out of fuel. 16. What causes different stars to evolve differently? 17. All stars eventually evolve into one of three types of objects. Name them. What kinds of stars evolve into each of the three types of objects? 18. Describe a neutron star. 19. Describe a black hole. Since nothing can come out of a black hole, how can we detect it?

THE NEWTONIAN WORLDVIEW 20. List some ways in which ancient Greek astronomy and Aristotelian physics support the medieval philosophical and religious worldview.

21. List some of the ways in which Copernican and Newtonian science are less supportive of the medieval worldview. 22. How is Newtonian physics related to democracy? 23. According to the Newtonian worldview, is a red napkin really “red”? Explain. 24. List several ways in which, according to the Newtonian worldview, the universe is similar to a clock.

BEYOND NEWTON 25. For what kinds of phenomena is Newtonian physics incorrect? Why did it take so long to discover such exceptions? 26. List the three theories that give correct predictions for the situations in which Newtonian physics is incorrect.

Conceptual Exercises THE IDEA OF GRAVITY 1. Does Earth’s gravity pull more strongly on a block of wood or on a block of iron having the same size? 2. Which one falls faster when dropped, a block of wood or a block of iron having the same size (neglect air resistance)? 3. What is the magnitude (strength) and direction of the gravitational force on you right now? 4. When you crumple a sheet of paper into a tight ball, does its mass change? Does its weight change? 5. Are you in orbit around (falling around) Earth’s center? Is there anything around which you are in orbit? 6. The moon is in orbit around two objects simultaneously. Which two? (Actually, there is a third—our galaxy’s center.) 7. Do you exert a gravitational pull on people around you? Do they exert a gravitational pull on you? 8. Do you exert a gravitational force on Earth? If so, how large is it, and in what direction is it? 9. How far does Earth’s gravitational influence extend? 10. As a spacecraft travels from Earth to the moon, does it ever entirely leave Earth’s gravitational influence? 11. A low-orbiting satellite weighs 8000 N. How big and in what direction is the gravitational force on it? How big and in what direction is the gravitational force by the satellite on Earth? 12. What would be the approximate orbital period (time for one complete orbit) for an apple placed in circular orbit around Earth at the moon’s distance from Earth? 13. If gravity suddenly shut off right now, what would be the shape of Earth’s orbit? What about the moon’s orbit? 14. What force (if any) keeps the planets moving?

From Chapter 5 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Newton’s Universe: Problem Set

NEWTON’S THEORY OF GRAVITY 15. Which is larger, the gravitational force by Earth on the moon or the gravitational force by the moon on Earth? 16. How strongly and in what direction does Earth pull on a 1 N apple? How strongly and in what direction does the apple pull on Earth? 17. Suppose you went to another planet that was identical to Earth on the surface but that was mostly hollow inside. Would this affect your weight? How? 18. Suppose you went to another planet having a larger radius than Earth but having the same total mass as Earth. Would this affect your weight? How? 19. List at least three bodies that have a detectable (measurable) gravitational effect on Earth’s motion. 20. The giant planet Jupiter is about 300 times more massive than Earth. It seems, then, that an object on Jupiter’s surface should weigh 300 times more than it weighs on Earth. But it actually weighs only about 3 times as much. Explain. 21. If you were in a freely falling elevator and you dropped your keys, they would hover in front of you. Are the keys falling? Are the keys weightless? 22. If gold were always sold by weight, could you make money buying gold at one altitude above the ground and selling it at a different altitude? Where would you want to buy—at a high altitude or a low altitude? 23. Would you weigh more in Denver or in Los Angeles? Why? 24. Is there any net force acting on the moon? 25. Is the moon accelerated? If so, in what direction is the acceleration? In what direction is the moon’s velocity?

26. Suppose that a heavy- and a lightweight satellite are put into low orbits around Earth. Could you tell, by observing the shape or speed of the two orbits, which satellite was the heavy one? 27. Suppose that two satellites are put into orbit, one around Earth and one around the moon, and suppose that the radii of the two orbits (the distance from the center of Earth and the moon) are the same. From the knowledge that Earth’s mass is larger than the moon’s mass, can you make any predictions about the speeds of the two orbits? 28. Communications satellites must be in geosynchronous orbits. That is, they must remain above a fixed point on Earth’s surface, enabling sending and receiving antennas to be aimed at a fixed point overhead. What, then, must be a communication satellite’s orbital period (the time for one complete orbit around Earth)? 29. In the “orbital” case in Figure 4, draw three arrows—labeled f, a, v—attached to the apple that show the direction of the gravitational force on the apple, the direction of the apple’s acceleration, and the direction of the apple’s velocity. 30. Figure 3 shows two possible paths for an apple that has been thrown horizontally. Assume that air resistance is negligible. For each path, draw three arrows—labeled f, a, v—attached to the apple that show the direction of the gravitational force on the apple, the direction of the apple’s acceleration, and the direction of the apple’s velocity. 31. Suppose that the gravitational force between an apple and an orange placed a few meters apart is one-trillionth (10–12) N. What would the force be if the distance were doubled? Halved? Tripled? Quartered?

Figure 4

Suborbital

Falling around Earth. If you throw an apple fast enough, it will fall around a large part of Earth’s surface or even go into orbit. A diagram like this appears in Newton’s notebook.

Orbital

Figure 3

If you throw an apple horizontally, the faster you throw it, the farther it will go.

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Slower

Faster

Newton’s Universe: Problem Set 32. Referring to the previous exercise, what would the force be if the mass of the apple were doubled? Tripled? What if the mass of the apple were tripled and the mass of the orange were quadrupled? 33. Referring to the previous exercise, what would the force be if the mass of the apple were doubled, the mass of the orange were doubled, and the distance between them were doubled?

GRAVITATIONAL COLLAPSE 34. If Earth collapsed to one-tenth of its present radius, how much would you then weigh? 35. If Earth expanded to 10 times its present radius, how much would you then weigh? 36. Find your weight at a distance of 10 Earth radii from Earth’s center. Compare with the preceding question. 37. Will Earth ever collapse to become a black hole? Why? Will the sun? 38. The orbits of all nine planets lie approximately in the same flat plane. Why?

BEYOND NEWTON 39. What theory or theories would be needed to predict the behavior of an atom moving at half the speed of light? 40. According to the most widely accepted scientific theory of the creation of the universe, the observable universe during the first few moments (much less than 1 second) of its existence was extremely hot, was full of densely packed matter, and was very tiny—smaller than an atom. What theory or theories would be needed to explain what was happening during these first few moments?

Problems NEWTON’S THEORY OF GRAVITY 1. What happens to the gravitational force between two planets when the distance between them is decreased to one-third of its previous value? 2. What happens to the gravitational force between two planets when the distance between them is increased to three times its previous value? 3. Earth’s mass is 6.0 * 1024 kg, and its radius is 6.4 * 106 m. Use Newton’s theory of gravity to find the weight of a 1 kg object lying on Earth’s surface. 4. The moon’s mass is 7.4 * 1022 kg, and its radius is 1.7 * 106 m. Use Newton’s theory of gravity to find the weight of a 1kg object lying on the moon’s surface. If you did the preceding problem, then compare the two answers. 5. A certain neutron star has a mass of 4.0 * 1030 kg (twice the sun’s mass) compressed into a sphere of radius only 10,000 m (10 km). Find the gravitational force on a cubic centimeter of water, whose mass is 1 gram, lying on the surface (in reality, this “water” would no longer be in its normal liquid state if it were on the surface of a neutron star!). 6. Find the force by the moon on Earth. Their masses are 7.4 * 1022 kg and 6.0 * 1024 kg, and it is 3.8 * 108 m between their centers.

7. In the preceding question, how large is the force by Earth on the moon? In what direction is the force by Earth on the moon? 8. Find the force by the sun on Earth. Their masses are 2.0 * 1030 kg and 6.0 * 1024 kg, and it is 150 million kilometers between their centers. 9. In the preceding question, how large is the force by Earth on the sun? In what direction is the force by Earth on the sun? 10. Find the force by a 0.1 kg apple on another 0.1 kg apple, if their centers are 2 m apart. 11. MAKING ESTIMATES Estimate the gravitational force, in newtons, that you exert on a person standing near you. Is the answer closer to 1000 N, 1 N, one-thousandth N, one millionth N, or one-billionth N?

Answers to Concept Checks 1. 2. 3. 4. 5. 6. 7. 8. 9.

(b) (c) (c) According to Newton’s law of motion, if the net force is the same, then the acceleration is also the same, (d). (a) Although your mass is unchanged, your weight is reduced because you are farther from Earth’s center, (d). You have multiplied the distance by 3, so you’ve divided the force by 9 (3 squared), (f ). (b) and (d) The radius is reduced to 1/1000th of its previous value. The square of this is 1/1,000,000, and the inverse of this is 1,000,000, (e).

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. Iron. They fall equally fast. 3. The magnitude is your weight (in pounds, or in newtons), and the direction is downward. 5. No, you are not in orbit around Earth’s center (although you are revolving around that center due to Earth’s spinning motion). You are in orbit around the sun. 7. Yes; yes. 9. It extends to infinity (but it is very small at great distances). 11. 8000 N, downward. 8000 N, upward. 13. A straight line; a straight line. 15. The law of force pairs tells us that these two forces are of equal strength. 17. The other planet’s mass would be less than Earth’s mass, so your weight would be reduced. 19. The sun, Earth’s moon, planets such as Mars and Venus. 21. Yes, the keys are falling, along with you. The keys have weight, although they are “apparently weightless.” 23. You would weigh more in Los Angeles, because your distance from Earth’s center would be smaller. 25. Yes, the moon is accelerated in the inward (downward) direction. The moon’s velocity is forward—along the moon’s orbit.

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Newton’s Universe: Problem Set 27. The inward force on the earth satellite would be larger

29. 31. 33. 35. 37. 39.

(because Earth’s mass is larger than the moon’s mass), so the earth satellite would have the larger acceleration (because of Newton’s law of motion). In order to have this larger acceleration, the earth satellite would have to be moving faster. f and a point directly toward Earth’s center, and v points along the path of motion. (1>4) * 10 - 12 N, 4 * 10 - 12 N, (1>9) * 10 - 12 N, - 12 16 * 10 N. The force would be unchanged. 1/100th of your present weight. No. Earth does not have sufficient mass for it to collapse this far. Neither does the sun. Special relativity and quantum theory.

Problems 1. The new force is nine times larger than it was. 3. F = 6.7 * 10 - 11 m1 * m2>d2 = 6.7 * 10 - 11(1 kg)(6.0 * 1024 kg)>(6.4 * 106 m)2 = 9.8 N

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5. F = 6.7 * 10 - 11 m1 * m2>d2

= 6.7 * 10 - 11 (10 - 3 kg) * (4 * 1030 kg)>(104 m)2 = 2.7 * 109 N (2.7 billion newtons!) 7. The answer is the same as in Problem 6, 2.1 * 1020 N. The force by Earth on the moon is directed toward Earth’s center. 9. The answer is the same as in Problem 8, 3.6 * 1033 N, directed toward Earth’s center. 11. A typical mass for a person is about 50 kg. A typical distance from you to a person next to you might be about 1 meter. So the gravitational force by one person on the other is about (6.7 * 10 - 11) * (50) * (50)>(1)2 N = 1.7 * 10 - 7 N, less than one-millionth N.

Conservation of Energy

From Chapter 6 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Conservation of Energy You Can’t Get Ahead—

Energy is the most difficult part of the environment problem, and environment is the most difficult part of the energy problem. The core of the challenge of expanding and sustaining economic prosperity is the challenge of limiting, at affordable cost, the environmental impacts of an expanding energy supply. John Holdren, President Obama’s Science Advisor

E

nergy is physics’ most important concept and, as you can see almost every day in the newspapers, energy is also highly relevant to society. In fact, we define human cultures largely by their use of energy resources. Civilization itself is nearly synonymous with the organized use of solar energy. Humankind’s first permanent villages developed 10,000 years ago because of the needs of trade and agriculture. For centuries, trade was facilitated by solar energy, which drove the winds that pushed the sails of merchant ships, warships, and exploration ships. And agriculture is the organized use of solar energy to grow food. Today, the chemically altered remains of ancient life known as fossil fuels—coal, oil, and natural gas— energize our industrial culture. Energy is one of the four recurring themes of this text. It will be the basis for analyzing all sorts of natural phenomena in this chapter and for discussing many energyrelated social issues. Our goal in this chapter is to understand what scientists mean by energy and to use this concept to understand a multitude of physical processes. Like all powerful scientific ideas, energy explains and unifies a wide variety of phenomena. Unlike Newton’s laws, the principles of energy apply to all phenomena, from subnuclear “quarks” to the cosmos, observed so far. New energy sources have been nearly synonymous with significant social changes. The coal-fueled steam engine stimulated the Industrial Revolution around 1750, with profound economic and social consequences. Because the new industrial machines were large, complex, and expensive, traditional home workshops grew into large centrally located factories run by the rich. Whereas the skills of traditional craftspeople required a long apprenticeship, even unskilled workers and children could tend the new machines. Consequently, nineteenth-century Europe and North America were marked by increased productivity, the capitalistic organization of industry, and a shift of population from farms to cities. Many political ideologies of the twentieth century—facism, communism, capitalism, socialism—grew out of the economics of the Industrial Revolution. Today, the Industrial Revolution is spreading all over the world and to new industries such as computers. The new industries stimulated nineteenth-century scientists to understand the two grand principles of energy. This chapter develops one of these principles, conservation of energy. This principle appears in Section 5, following the development in

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Sections 1 through 4 of the concepts of work and energy. Everything that happens in the universe involves an energy transformation of one sort or another. Section 6 studies several examples of energy transformations. Section 7 looks at a highly useful related idea: power, or the rate (per unit of time) of transforming energy. CONCEPT CHECK 1 We’ve never observed a violation of conservation of energy or the second law of thermodynamics. Thus, these principles of energy are (a) good theories; (b) good hypotheses; (c) certain to be correct in all future observations; (d) facts; (e) absolutely true; (f) hogwash.

1 WORK: USING A FORCE TO MOVE SOMETHING People commonly say that a material system such as a person, a flashlight battery, or a tank of gasoline, has “energy” if it has an inherent capacity to bring about changes in its environment or itself. The physicists’ definition of energy is a refinement of this notion. I’ll say a system has “energy” whenever it has the capacity to do work, where “work” refers to bringing about external or internal changes. In this section, I’ll discuss “work.” The physicist’s definition of work is a refinement of the common notion that you do work whenever you exert yourself to perform a task. In physics, work is done whenever an object is pushed or pulled through a distance. For instance, you do work on a book when you push it across a table. A magnet does work on a paper clip when the magnet pulls the clip toward the magnet. More precisely, object A (a person or any other thing) does work on object B if A exerts a force on B while B moves in the direction of that force (we won’t need to consider situations in which the motion is not in the same direction as the force). You do work on a book when you lift it. Earth does work on the book when the book falls. Notice that work is always done by one specific object on another specific object. And notice that both force and motion are needed in order for work to be done. CONCEPT CHECK 2 Jed leans against a brick wall while Ned pushes hard against it and “works up” a sweat in the process (Figure 1). Is either Jed or Ned doing any work on the wall? (a) Both are. (b) Ned is but Jed is not. (c) Neither one is. CONCEPT CHECK 3 A single electron flies through the vacuum (assume it’s a “perfect” vacuum) inside your TV picture tube, from the back to the front side, where it makes a tiny flash when it strikes the inside of the screen. Neglect the force of gravity on the electron. During the electron’s motion through the tube, (a) only air resistance does work on the electron, causing it to slow down; (b) inertia does work on the electron, causing it to move at an unchanging speed in a straight line; (c) inertia does work on the electron, causing it to slow down; (d) no work is done on the electron, which moves at an unchanging velocity; (e) no work is done on the electron, which slows down.

Anybody who has ever thought about the cost of filling a car’s gas tank, or about a car’s gasoline efficiency, knows that it’s of practical importance to be quantitative about energy. Let’s think about the quantity of work you do in various situations. If the word work is to agree roughly with common language, the work you do in lifting a load should be larger for larger loads. To see how much larger, compare lifting one

Jed

Ned

Figure 1

Is either Jed or Ned doing any work on the brick wall?

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book with lifting two identical stacked books (Figure 2). The effect is twice as big in the second case, so the work done should be twice as big. This means work should be proportional to force. Now compare pushing a book across one table with pushing it all the way across two adjoining tables (Figure 3). Again the effect is twice as big in the second case. So work should be proportional to the distance moved. So work should be proportional to both force and distance. Thus, we define the amount of work done by object A on object B as the force exerted by A on B times the distance that B moves while experiencing that force:1 work = force * distance = Fd Figure 2

It takes twice as much work to lift two books.

For instance, if you push on a book with a 3 newton force while pushing it 2 meters, you have done (3 N) * (2 m) = 6 newton-meters of work. Note that the unit of work is the newton-meter, a unit that’s so widely used that it’s been renamed the joule (J) (rhymes with school), in honor of James Prescott Joule (Figure 4). From the following Concept Checks you can see that the work done in lifting an object is the object’s weight multiplied by the height lifted. CONCEPT CHECK 4 Suppose you slowly, and at constant speed, lift a 12 N book from the floor to a shelf 2 m above the floor. While you are lifting it, the net force on the book is (a) zero; (b) 12 N; (c) 24 N. CONCEPT CHECK 5 The force by your hand against the book in the preceding question is (a) zero; (b) 12 N; (c) 24 N.

Figure 3

It takes twice as much work to push one book twice as far.

CONCEPT CHECK 6 The work done by you on the book in the preceding question is (a) zero; (b) 24 J; (c) 48 J.

2 WORK AND ENERGY: A SIMPLE EXAMPLE

American Institute of Physics/ Emilio Segre Visual Archives

Do this two-step experiment and observe carefully: First, place your book on your outstretched hand on the floor; slowly lift it to some height; hold it there a few seconds; and then slowly lower it back to the floor. Second, repeat the same lifting process to the same height, but this time suddenly remove your hand from the book so that it falls to the floor. You do work on the book when lifting it, but it does work on you when lowering back to the floor, because the book pushes downward against your hand all the way down. So the raised book has a capacity to do work and it actually does this work as it’s lowered. Let’s look more closely at the “capacity to do work.” You just saw that raised objects have this capacity. And so do moving objects. For example, suppose that you throw your book, horizontally, at a wall. One way to get work out of the moving book would be to stick a thumbtack partly into the wall, directly in line with the book’s motion, so that the book will hit the tack and drive it in farther (Figure 5). Any moving object has the capacity to do work.

Figure 4

British physicist James Prescott Joule. His experiments in the 1840s helped unravel the confusion surrounding thermal energy and so led to the first clear understanding of energy.

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This definition assumes that the motion is in a straight line in the direction of the force. If the motion and the force are not in the same direction, then in the formula we must use only that part or “component” of the force that is along the motion. We won’t need this refinement here.

Conservation of Energy

In the second step of our experiment, the book was again raised, and in the raised position it again had the capacity to do work. But then you dropped it so that it simply fell, without doing work on your hand. As the falling book lost height, it gained speed, so it retained an ability to do work. Just before hitting the floor, the book still had a capacity to do work; only now this capacity resulted from the book’s speed rather than from its height. You could get the falling book to actually do this work by sticking a tack partly into the floor and letting the book drive in the tack farther (Figure 6). So you give your book the capacity to do work when you lift it or throw it. The work you do is “stored” in the raised or moving book. You could get this work back at any time, for example by letting the book push your hand down to the floor. Physicists have a word for the capacity to do work. It’s called energy. You’ve seen that both raised objects and moving objects have energy. It’s useful to distinguish these different forms. We’ll say that a raised object has “gravitational energy,” because this energy is caused by Earth’s gravitational pull on the object, and that a moving object has “kinetic energy,” because “kinetic” is related to the Greek word for motion. As you’ll see, there are several other energy forms. As it falls, the book loses gravitational energy but gains kinetic energy and so retains energy. This retention of energy when no outside agent (such as your hand) influences the system is one example of the law of conservation of energy. These are the essential ideas about work and energy, presented in the context of a simple example. The rest of this chapter expands on these ideas.

Figure 5

One way to get work out of a moving book: Allow it to push a thumbtack into a wall.

3 A QUANTITATIVE LOOK AT ENERGY Let’s look quantitatively at the experiment described in the preceding section. Recall that the amount of work you do to lift an object is its weight multiplied by the height raised. Once the book is raised, it has gravitational energy because it can do work in pushing your hand back to the floor. How much gravitational energy does it have? Quantitatively, we define an object’s energy to be the amount of work it can do. It’s measured in joules (or newton-meters), just as work is. The energy of the raised book is the amount of work it can do in slowly pushing your hand back down to the floor, which is just the book’s weight multiplied by the distance to the floor. You can see from this that, for any raised object,

Figure 6

One way to get work out of a falling book.

gravitational energy = weight * height When you slowly lower the book, it uses up its energy while pushing your hand back to the floor. But when the book falls, its gravitational energy is transformed into kinetic energy. How much kinetic energy is possessed by a moving object such as the book? In other words, how much work can a moving object do because of its motion? It’s possible to work out the answer to this question, starting from Newton’s laws, although we won’t work it out here. The answer turns out to be 1 kinetic energy = a b * (object’s mass) * (square of object’s speed) 2 This formula tells us that a more massive object has more kinetic energy and a faster object has more kinetic energy, as we might expect. It makes sense that the formula should involve the object’s mass rather than its weight, because kinetic energy is possible even in the absence of gravity.

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wt ⫻ ht at this point

equals (1/2) ms 2 at this point

Figure 7

An amazing thing: The gravitational energy at the top precisely equals the kinetic energy at the bottom, just before the book hits the ground.

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Now comes an incredible fact, also provable from Newton’s laws: If you neglect air resistance (which I’ll deal with later), the amount of gravitational energy the book has at the beginning of its fall will precisely equal the amount of kinetic energy it has at the end (Figure 7). The book’s total capacity to do work, its total energy, is quantitatively unchanged during the falling process. Its energy is simply changed in form—transformed—from gravitational to kinetic, but its total energy remains the same. As physicists put it, energy is precisely “conserved.” This physics use of the word conserved should be distinguished from the way “energy conservation” is used socially. When the newspapers speak of energy conservation, they mean preserving certain high-value forms of energy such as oil by consuming them less. When physicists speak of the conservation of energy, they mean that the total amount of energy remains unchanged throughout some physical process. Furthermore, since energy is conserved for any distance of fall, it must be conserved at halfway down, at three-quarters of the way down, and at every other point during the fall. The loss in gravitational energy during any portion of the fall precisely equals the gain in kinetic energy during that portion (Figure 8). CONCEPT CHECK 7 How much kinetic energy does a car have when it moves at 100 km/hr, as compared with when it moves at 50 km/hr? (a) The same amount. (b) One-half as much. (c) One-fourth as much. (d) Twice as much. (e) Four times as much. CONCEPT CHECK 8 A bag of groceries having a mass of 6 kg and a weight of 60 N falls from a shelf that is 2 m high. Just as it begins to fall, its gravitational energy (relative to the floor) is (a) zero; (b) 12 J; (c) 120 J; (d) none of the above. CONCEPT CHECK 9 Refer to the preceding question. Neglecting air resistance, just before hitting the floor the bag of groceries’ gravitational energy and kinetic energy are (a) both zero; (b) zero and 120 J, respectively; (c) 120 J and zero, respectively; (d) both 120 J; (e) none of the above. MAKI NG ESTI MATES Estimate your physics book’s energy, relative to the floor,

Figure 8

The total energy is conserved all the way down. The loss in gravitational energy between points 1 and 2 during the fall is precisely balanced by the gain in kinetic energy between these two points. This is true no matter where point 2 might be between the woman’s hands and the floor.

when lifted to arm’s length over your head. Assume the book’s weight is about 10 N. If you dropped it from this height, about how much kinetic energy would it have just before hitting the floor?

4 ENERGY: THE CAPACITY TO DO WORK Now let’s expand on the preceding two sections, extending these ideas to a wide variety of systems. This word system comes up a lot in science. It means a specific part of the universe, such as a particular collection of objects. Any system having the capacity to do work is said to have energy. Quantitatively, a system’s energy is the amount of work it can do. Although work and energy can both be measured in joules, work and energy are not the same thing. A system does

Suppose you can lift to about 2 m above the floor: GravE = wt * ht L 10 N * 2 m = 20 J. If you drop it, it will have nearly this entire 20 J of energy, in the form of kinetic energy, just before it hits the floor. The remaining small amount is converted to thermal energy, due to air resistance. SO LUTION TO MAKI NG ESTI MATES

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work, but it has energy. Work is a process, whereas energy is a property of a system. You can think of energy as stored work. A system’s energy is the amount of work the system could do, regardless of whether it ever actually does this work: A raised boulder has energy, even though it might be tied up and left that way forever and never do any work. The difference between energy and work is similar to the difference between money and spending. Just as energy is the capacity to do work, the money in your bank account represents your capacity to spend. Work is then similar to the act of spending some of that money. Notice that although your bank account and your spending are both measured in dollars, they are different things. There are many forms of energy, because there are many ways to do work. I’ll discuss eight of them, beginning with the two you already know something about. Kinetic energy (KinE) is energy due to motion. It’s the work a system could do while coming to rest. Gravitational energy2 (GravE) is energy due to gravitational forces. It’s the work a raised system could do while Earth (or any other object that can pull gravitationally) pulls it back to its initial position. There’s a quirk about gravitational energy: Its numerical value depends on the level chosen as the initial or reference level, simply because the amount of work you can get from a raised object depends on how far down it must go before you consider it no longer “raised.” For instance, your book’s gravitational energy is only a few joules relative to the floor of your room, but it may be thousands of joules relative to sea level. So we sometimes need to be explicit about the agreed-upon reference level when discussing gravitational energy. If you stretch a rubber band or bend a ruler, it can snap back when released. There’s energy in the deformed system because it can do work while snapping back. For instance, a stretched rubber band can do work in pulling your fingers together. This energy, resulting from the capacity of a deformed system to snap back, is called elastic energy (ElastE). A pot of hot water has more energy than does a pot of cold water of the same size. How do we know? Well, if the hot pot is boiling, it can rattle its lid, and this requires work, so boiling water has the capacity to do work. If the hot pot is below the boiling temperature, one way to get work from it would be to find another liquid that boils at a lower temperature and let the hot pot warm the other liquid to boiling so that this other liquid rattles its lid. This kind of energy that exhibits itself as warmth—as higher temperature—is called thermal energy (ThermE).3 It’s enlightening to look at thermal energy microscopically. As you know, temperature is associated with random microscopic motion, or thermal motion, that’s not visible macroscopically. For example, as water’s temperature rises, its molecules move faster, gaining kinetic energy. Thermal energy is this microscopic energy that cannot be directly observed macroscopically.4 To prevent confusion. I will reserve the term kinetic energy for macroscopic kinetic energy. Put one hand on a warm object and the other on a similar cool object. From a microscopic point of view, you are not really experiencing warmth and coolness, you

2

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4

Also known as “gravitational potential energy.” Some books define “potential energy” as energy resulting from a system’s position or configuration. Gravitational, elastic, and electromagnetic energy are all forms of potential energy. In the interest of brevity, we won’t use the word potential. More precisely, thermal energy (which is sometimes called “internal energy”) exhibits itself not only in a system’s temperature but also in its pressure and other so-called “thermodynamic variables.” More precisely, thermal energy includes all of the microscopic forms of energy that are not directly visible at the macroscopic level. It includes energy that results from the forces between molecules, a point that is important to understanding melting and other “phase transitions.”

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are only experiencing fast and slow molecules. Is that amazing or what? It amazes me. How could the mere motion of microscopic particles, that I can’t even see or feel individually, create the feeling of warmth in my hand? The idea of warmth has been replaced by, or “reduced to,” motion. In a way, the notion of warmth has vanished, a classic example of science’s reduction of a wide assortment of phenomena to a few basics. This reduction of sense impressions to the mechanical motion of atoms is precisely what Democritus was talking about when he proclaimed, “By convention hot is hot and cold is cold.... The objects of sense are supposed to be real—but in truth they are not. Only the atoms and the void are real.” Historically, thermal energy was confusing because it didn’t fit comfortably into the mechanical framework of Newtonian physics and because it is fundamentally different from the other energy forms. During the eighteenth and nineteenth centuries, Joule and others eventually demonstrated that what is experienced as warmth is in fact a form of energy. This was a key step in comprehending what energy really is. I’ll run quickly over the remaining four types of energy, returning to them in more depth later. The energy that results from electric and magnetic forces is called electromagnetic energy (ElectE), or sometimes simply “electric energy” or “magnetic energy.” There is energy in a light beam, as you can tell from the fact that light (sunlight, for instance) can warm things, and you can get work out of warm things. The energy carried by a light beam is one form of radiant energy (RadE). There are other forms of radiant energy, some of them familiar to you: radio, microwave, infrared, ultraviolet, X-ray, and gamma-ray energy. Chemical reactions can do work, as you can see from a wood fire used to boil water. This energy results from the molecular structure of the wood. The energy that results from a system’s molecular structure is called chemical energy (ChemE). Whereas chemical energy results from molecular structure, nuclear energy (NuclE) results from nuclear structure, from the way protons and neutrons are arranged into nuclei. One obtains nuclear energy from nuclear reactions, just as one obtains chemical energy from chemical reactions. CONCEPT CHECK 10 An operating lightbulb transforms ElectE into (a) KinE; (b) ElectE; (c) ThermE; (d) ChemE; (e) RadE. CONCEPT CHECK 11 In the operation of a hydroelectric power plant, the energy to generate the electricity can be traced to (a) GravE in the lake behind the plant’s dam; (b) ChemE in the lake behind the plant’s dam; (c) ThermE in the lake behind the plant’s dam;(d) RadE that comes from the sun; (e) ChemE that comes from the sun; (f) good vibes.

5 THE LAW OF ENERGY: ENERGY IS FOREVER You’ve seen that the gravitational energy lost during any portion of an object’s fall is exactly balanced by its kinetic energy gain, provided you neglect air resistance and observe the object only until just before it hits the floor. The object’s

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overall, or total, energy is conserved all the way down. Newton’s physics predicts this, and experiment confirms it. In fact, it’s possible to prove, still based on Newton’s laws, that any system that experiences only gravitational forces conserves its total energy, just as the falling object does. To a good approximation, our solar system is a system of this type that moves under the influence of gravity alone. It’s remarkable that there should be this rather abstract quantity, energy, that remains unchanged as the eight planets and their moons go through their complex motions. But far more remarkably, experiments show that the energy principle goes far beyond Newton’s physics. Energy is conserved in every physical process yet observed. We call this The Law of Conservation of Energy The total energy of all the participants in any process remains unchanged throughout that process. That is, energy cannot be created or destroyed. Energy can be transformed (changed from one form to another), and it can be transferred (moved from one place to another), but the total amount always stays the same.

This statement is as true as any general rule ever gets in science. It’s correct in every situation yet observed. It holds even when Newton’s physics is not remotely correct, such as near black holes, close to the speed of light, and for subatomic particles. It’s a useful principle, because once you have calculated or measured the total energy at one moment during some process, you automatically know it at any other moment without having to calculate or observe all the messy details of what happened between the two moments. For example, the number of joules of chemical energy consumed from a car’s fuel tank must equal the total number of joules appearing as the car’s kinetic and gravitational energy, exhausted thermal energy, chemical energy of pollutants, and so forth, during that time. The law of conservation of energy says that something, namely a system’s total capacity to do work, or its “total energy,” remains the same throughout any physical process. It’s similar to the law of conservation of momentum, which says that a system’s total motion through space or “total momentum” remains the same. You’ll encounter a third such conservation law: conservation of something called “net electric charge.” Another prominent conservation law, one that I won’t be discussing, states that any system’s total rotational motion or “total angular momentum” remains the same. And there are certain subatomic properties associated with microscopic interactions that are also conserved. Energy conservation is a kind of symmetry principle. Recall that a system has symmetry if it looks the same from various perspectives. Energy conservation says that a system’s energy remains the same no matter at what time you view it. In fact, all conservation principles can be traced to symmetries in nature. There’s a useful alternative way of stating conservation of energy. Whenever work is done, it’s done by some system on some other system. The system doing the work must lose some of its capacity to do work; in other words, it must lose energy. Since total energy is conserved, this energy cannot just vanish but must instead go

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into the system on which work is done. So work is an energy transfer from the system doing the work into the system having work done on it. I’ll call this The Work-Energy Principle Work is an energy transfer. Work reduces the energy of the system doing the work and increases the energy of the system on which work is done, both by an amount equal to the work done.5

How do we know that energy is conserved even in nuclear processes? Early in the twentieth century, nuclear physicists investigated a form of “radioactive decay” known as beta decay, a process in which a nucleus spontaneously creates an electron and spits it out of the nucleus. This alters the original nucleus. If energy is conserved, the nuclear energy of the original nucleus should equal the nuclear energy of the altered nucleus plus the energy of the ejected electron. But measurements showed that the energy was larger before than after! Being reluctant to conclude that energy was not conserved, physicists hypothesized that some undetected particle was also ejected along with the electron. It was thought that when this other particle’s energy was included, the energies would balance. Although the hypothesized particle had not been detected, it was thought that its energy could be directly measured by surrounding the nucleus with a large cylinder of lead. The unseen particle would surely be slowed down and stopped inside a sufficiently thick cylinder and so deposit its energy in the lead, causing a temperature rise in the lead. But there was no measurable temperature increase. Perhaps energy was not conserved in beta decay. This is where the matter stood from 1914 to 1930. By 1929, some physicists, such as Niels Bohr, were suggesting that energy conservation didn’t apply to the nucleus. But others didn’t accept this suggestion, and in 1930 Wolfgang Pauli hypothesized that the new particle was so penetrating that it could pass right through the thick lead without depositing any energy and that energy would be found to be conserved once the elusive particle was found. This set off a search for such a particle. Before long, physicists found other indirect evidence (other than beta decay), and they gave the hypothesized particle a name: “neutrino.” It was finally detected directly in 1956. Experiments showed that, as Pauli had predicted, energy was conserved once the neutrino was included in the balance.

It takes about 8 solid light-years of lead to stop half the neutrinos emitted in a typical nuclear decay. They move like “greased lightning” through matter.... If you make a fist, there are thousands of neutrinos flying through it right now, because the entire universe is filled with neutrinos.... Another proposal, made tongue-in-cheek, is for a neutrino bomb, a pacifist’s favorite weapon. Such a bomb would explode with a whimper and flood the target area with a high flux of neutrinos.... [T]he neutrinos would fly harmlessly through everything. Heinz Pagels, in The Cosmic Code

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x x

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CONCEPT CHECK 12 Farswell Slick (Figure 9) invites your investment in a business venture to manufacture his remarkable “supertranspropulsionizer.” His diagrams show a dazzling array of superconductors, lasers, liquid-helium coolants, and fancy computers. Slick informs you that this ultimate propulsion system will accelerate spaceships to nearly lightspeed for interstellar travel. Amazingly, no fuel supply is needed, either on board or outside the spaceship. The principle involved, he explains, is “bremsstrahlung superconduction” (BS). With BS, the device operates in a continuous cycle that both accelerates the spaceship and “feeds back” some of its laser light to maintain, for as long as may be desired, the operation of the transpropulsionizer itself. Should you invest?

Figure 9

“A remarkable device,” Farswell Slick remarks. Would you buy a supertranspropulsionizer from this man?

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There is another worklike process by which energy can be transferred, called “heating.” Heating is thermal energy transfer due to a temperature difference and can be thought of as microscopic work. When expanded to include not only ordinary work but also heating, the work–energy principle is called “the first law of thermodynamics.” We won’t need the first law of thermodynamics in what follows.

Conservation of Energy

6 TRANSFORMATIONS OF ENERGY Everything that happens can be described as an energy transformation. This section describes the energy transformations involved in some familiar processes. Once again, drop your book to the floor (it’s coming in for a lot of rough treatment in this chapter!). You’ve studied this process up until its impact with the floor. Where is the energy after impact? Conservation of energy says it can’t just vanish. Going through our eight forms of energy, there’s only one plausible candidate: thermal energy. The impact must warm the book or the floor. This temperature rise is hard to detect, but you can demonstrate the same effect by driving a nail into a board with a hard hammer blow. Feel the nail before and after the blow. Try several blows. We can summarize the energy transformations in the following way: GravE (at the top) : KinE (just before impact) : ThermE (after impact) Let’s add the effects of air resistance. Since air resistance slows the book, the falling book has less kinetic energy than it did before. But this energy is not lost— you can’t lose energy. It must be transformed into thermal energy. The air and book must warm a little as the book falls. Until the work of Joule and others around 1850, scientists had long believed that the work going into forces such as air resistance and friction, work that produces warming, was lost. Thus, it was believed that energy tended to decrease in most systems, rather than being conserved. The key to uncovering conservation of energy was discovering that warming represented an energy increase in a then-unknown form of energy, namely, thermal energy. How do we know that energy is conserved even when thermal energy is involved? I asserted above that air warms when you stir it with a falling object. James Prescott Joule (Figure 4) did an experiment like this in the 1840s, using water instead of air. He placed a paddle wheel in a tub of water, stirred the water with the paddle wheel, and measured the temperature rise in the water.6 He quantified the experiment by allowing a falling weight attached to a cord to turn the paddle wheel. The weight’s energy loss is then its weight multiplied by the distance fallen. Joule found that the water’s temperature rise was precisely proportional to the gravitational energy lost. This showed that the lost gravitational energy went directly into a temperature rise, in other words into thermal energy. Energy concepts were murky in Joule’s day because scientists didn’t understand that warmth (thermal energy) actually is a form of energy. Joule clarified matters by showing that work is precisely convertible to thermal energy. This breakthrough showed that the principle of energy conservation extends to processes involving thermal energy, a microscopic form of energy that lies outside of Newtonian physics.

Joule showed that a particular amount of work, about 4200 J, produces a 1°C rise in the temperature of 1 kilogram of water. This amount of energy is the dietitian’s Calorie.7 Although the Calorie is often used to measure thermal energy, Joule’s work showed that it is really a general energy unit, equivalent to 4200 joules.

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When you stir hot water in open air, the water cools because of evaporation. In Joule’s experiment the stirring occurred inside a closed container that prevented evaporation. The dietitian’s Calorie is always spelled with a capital C. Physicists use “calorie” (lowercase c) to denote the energy needed to raise 1 gram of water by 1°C.

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Conservation of Energy ThermE (air)

GravE Kin

E

ThermE (impact)

Figure 10

Energy flow diagram for a falling book, with air resistance. The “pipe” widths correspond to the amounts of energy involved in various parts of the process. Since energy is conserved, the pipe widths match up at each intersection.

Figure 11

What energy transformations occur when you briefly push a book and then let it slide?

ThermE (body) ChemE Ki

nE

ThermE (table and book)

Figure 12

Energy flow diagram for a book that is given a quick push and allowed to slide across a surface while coming to rest. The pushing process is very inefficient, with most of the initial chemical energy going into warming your body rather than into the book.

Back to the falling book (it’s remarkable what you can learn just by thinking carefully about a falling book): Just before impact, all the energy has been converted to kinetic energy of the book and thermal energy of the air and book. Since air resistance has only a small effect on the motion, thermal energy must form only a small fraction of the total. Finally, the impact converts the pre-impact kinetic energy to thermal energy of the floor and the book.8 As a helpful way to visualize energy transformations of all sorts, I’ll use energy flow diagrams. For example, Figure 10 shows the energy of the falling book transforming as though it were water flowing through pipes, beginning as gravitational energy, then transforming into kinetic energy and a little thermal energy of the air (note the smaller pipe), and finally transforming entirely to thermal energy. Since energy is conserved, the pipe widths match up at each intersection. Now give your book a quick hard push so that it slides across your tabletop, sliding to rest (Figure 11). Where does the energy come from for this process, and in what form is it? (...Time out, for thinking.) ... It comes from your body, in the form of chemical energy. Figure 12 shows the energy flow diagram for this process. Most of the initial chemical energy used to push the book turns into thermal energy in your body. The small amount that goes into the book then winds up as thermal energy produced while the book slides to rest. You might have noticed how frequently the various forms of energy transform into thermal energy. Energy transformations in animals provide many interesting examples. The energy that enables you to do useful work comes from foods and is stored in your body as chemical energy. Dietitians measure this stored chemical energy in Calories. For example, a 70 Calorie slice of bread gives you 70 Calories of stored chemical energy that can then provide 70 Calories of work and thermal energy. When animal chemical energy is used to do work, only a small fraction actually transforms into useful work. We say that such a process is “inefficient.” A “highly efficient” process, on the other hand, is one in which most of the initial, or “input,” energy is transformed into useful “output” energy and the wasted fraction is small. Quantitatively, the energy efficiency of any energy transformation is the fraction of the input that appears as useful output: useful output energy energy efficiency = total input energy It is usually expressed as a percentage. The energy efficiency of typical human muscular activities is only about 10%. Energy being one of this text’s four major themes, you will encounter many more energy transformations and energy flow diagrams in future texts. CONCEPT CHECK 13 The energy transformation during photosynthesis is (a) KinE : ThermE; (b) ThermE : KinE; (c) KinE : ChemE; (d) ElectE : ChemE; (e) RadE : ChemE; (f) ChemE : RadE.

8

156

And you can hear the impart. A small fraction of the energy is transformed into the energy of sound, a form of kinetic and elastic energy of the air.

Conservation of Energy

CONCEPT CHECK 14 While a wooden matchstick burns, the energy transformation is (a) ThermE : ElectE + RadE; (b) ElectE : ThermE + RadE; (c) KinE : ChemE + RadE; (d) ChemE : KinE + RadE; (e) ThermE : ChemE + RadE; (f) ChemE : ThermE + RadE. CONCEPT CHECK 15 Robin Hood shoots an arrow from his bow. Beginning just before he draws the bow, the energy transformation is (a) ChemE : ElastE : KinE; (b) ThermE : ElastE : KinE; (c) ElastE : ChemE : KinE; (d) ChemE : KinE : ElastE; (e) ElastE : KinE; (f) ThermE : ElastEly.

7 POWER: THE QUICKNESS OF ENERGY TRANSFORMATION What’s the difference between running and walking up a flight of stairs? Your gravitational energy increases by the same amount in both cases. So the work you do is the same. And yet your body knows there’s a difference between running and walking upstairs. The difference is that you do the work in less time when you run. There’s a word for this notion of how quickly work is done. It’s called power. Quantitatively, power is the work done per second—in other words, the work done divided by the time to do it: power =

work done time to do it

Because work is an energy transformation, power can be thought of as the rate of transforming energy. Suppose you run up one flight of stairs and then walk up a second, identical flight of stairs in twice the time it took to run up the first flight. You do the same work for each flight, but your power output during the first flight is double your power output during the second flight. The difference is a power difference, not an energy difference. The unit of power is the joule per second (J/s). It differs by an all-important “per second” from the unit of work or energy. Power is such a popular concept that its unit is given a special name. The joule per second is called the watt (W), in honor of the eighteenth-century developer of the steam engine, James Watt. The kilowatt (kW) is 1000 watts, and the megawatt (MW) is 1 million watts. Think of several everyday devices: automobile, lightbulb, electric blender, toaster, and so forth. These can be understood as energy transformers; they transform energy from one form to another form that you can use. An important feature is often the rate at which the energy is converted. For example, to get a certain lighting level from a lightbulb, the bulb must convert a certain number of joules per second to visible light. So lightbulbs and other devices must be rated in power units (watts) rather than energy units. A popular power unit for automobile engines and other heat engines is the horsepower, equal to about 750 watts. Table 1 gives the power transformed by typical household electrical appliances. The numbers give the power consumed only during the time that the appliance is turned on. Total electric energy consumption, perhaps over one day or over one year, often tells quite a different story. For example, one ordinarily uses a toaster for only a short time each day, so its daily energy consumption is low even though its 1200 W power consumption is high. And refrigerators, although their power consumption

157

Conservation of Energy MAKI NG ESTI MATES What’s your power output while running up a flight of stairs? If the energy efficiency of this process is 10%, what’s your (chemical) power input, in watts and in Calories/second?

can be as low as 300 W, are leading household energy consumers because they operate for so many hours every day. Although home energy use over the course of a year determines how much energy a power plant must deliver, energy use during so-called peak times has a special impact on the need for new plants. Each electric plant has a maximum power output, its power rating, usually one hundred or more megawatts. The plant’s actual power output is largest at times such as hot afternoons when many people are running air conditioners. If the power peak approaches the plant’s rating, the plant will cut its output by reducing the supply to all customers, causing lightbulbs and other appliances to dim in a “brownout.” Because industrial societies waste enormous amounts of energy, there are countless opportunities today for electric power companies to save customers’ money, enhance company profits, and protect the environment, all with no reduction in services, by finding ways to avoid building expensive new power plants. Pressures Table 1 Power consumption of household appliances while the appliance is turned on and consuming electric energy Power (W)

Appliance

Cooking range

12,000

Clothes dryer

5,000

Water heater

4,500

Air conditioner, window

1,600

Microwave oven

1,400

Dishwasher (incl. hot water)

1,200

Toaster

1,200

Hair dryer

1,000

Refrigerator, frostless

600

Refrigerator, not frostless

300

TV, color

350

Stereo set

100

SOLUTION TO MAKING ESTIMATES Suppose that you weigh 500 newtons (110 pounds), the vertical height of one flight of stairs is 4 meters (measure it!), and you run up one flight in 5 seconds (try it!). The work done to lift yourself and the power output are

work = weight * height = 500 N * 4 m = 2000 J power = work , time = 2000 J , 5 s = 400 W This is a large power output for a human being, as you will discover if you do the experiment. To produce this output at 10% efficiency, you must convert chemical energy at a rate of 4000 W, or 4000 J/s. In Cal>s (1 Cal = 4200 J), this is 4000>4200 L 0.95 Cal>s, your metabolic rate in this example. If you could maintain this rate for an hour (3600 s), you would “burn up” (transform) 3600 * 0.95 = 3400 Calories, the energy content of a big steak.

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Conservation of Energy

for new plants arise when existing plants can no longer provide the electricity needed during periods of peak demand. Energy-efficient devices can provide the same services (the same amount of light, for example) with less energy. Because such efficiency measures are usually far cheaper than the cost of building new plants, many power companies are actively seeking and providing new energyefficiency opportunities. For example, because it’s usually far cheaper to warm a house with additional insulation than with additional electricity, many power companies provide services and low-cost loans to encourage customers to insulate their homes. Everybody wins: the customer, who gets a warmer home cheaper; the power company, for which the insulation is cheaper than a new plant; and the environment, which benefits from reduced resource consumption and less pollution. Pricing is another way to reduce the need for new power plants. If electric companies charge higher rates at peak power times, balanced by reduced rates during offpeak periods, people have an incentive to switch their power use from peak to off-peak times. This reduces the need for new power plants, and the resulting financial savings can reduce customers’ electrical bills while increasing company profits. The most useful energy unit for measuring your home’s electric energy consumption is the kilowatt-hour, the amount of energy transformed when a power of 1 kilowatt operates for 1 hour. Since 1 kilowatt is 1000 joules/second and 1 hour is 3600 seconds, 1 kilowatt-hour = 1000 J>s * 3600 s = 3.6 * 106 J If a known power in kilowatts operates for a known number of hours, it’s easy to figure the number of kilowatt-hours of energy consumed: Just multiply the number of kilowatts by the number of hours. Electricity costs about 10 cents per kilowatt-hour. That sounds pretty cheap for 3,600,000 joules. How cheap? For instance, how far could 3,600,000 joules lift a 1000 newton (225 pound) person? Since the work done in lifting an object is the object’s weight times the distance through which it’s raised, the 3,600,000 J must equal 1000 N times the distance. So the distance is 3,600,000 J divided by 1000 N, or 3600 meters—nearly 12,000 feet! That’s a lot of lifting for just a dime. Electricity is phenomenally cheap, and we use a lot of it. The average U.S. household consumes about 1.4 kilowatt-hours of electric energy every hour! Society’s use of energy is a crucial topic today for many reasons, including global warming, pollution, national security, declining energy resources, nuclear power issues, and environmental destruction. CONCEPT CHECK 16 You press a 500 N weight from your shoulders up to arms’ length, a distance of 0.8 m, during a period of 2 seconds. How much work did you do? (a) 800 W. (b) 800 J. (c) 400 W. (d) 400 J. (e) 200 W. (f) 200 J. CONCEPT CHECK 17 In the preceding question, your power output is (a) 800 W; (b) 800 J; (c) 400 W; (d) 400 J; (e) 200 W; (f) 200 J.

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© Sidney Harris, used with permission.

Conservation of Energy

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Conservation of Energy Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions 1. What type of energy technology fueled the Industrial Revolution? 2. What is a fossil fuel? Name three kinds.

WORK 3. Is work done whenever a force is exerted? Explain. 4. Is work done whenever an object moves through a distance? Explain. 5. To what two quantities is work proportional? 6. You slowly lift a 3 N grapefruit by 2 m. How much work did you do and on what object? 7. A 3 N grapefruit falls 2 m to the floor. Was work done during the fall? By what object on what other object?

19. An apple in a tree has 90 J of gravitational energy (relative to the ground). It falls. If you neglect air resistance, what can you say about the amount of kinetic energy the apple has just before it hits the ground? What if you do not neglect air resistance?

POWER 20. Explain the difference between energy and power. 21. Choose the correct answer(s): The (watt, newton per second, joule, calorie, joule per second, meter per second, horsepower, kilowatt-hour) is a unit of power. Which are units of energy? 22. You lift a 2 N rock by 4 m in 3 s. What is your work output? Your power output? 23. Which do you pay for in your monthly electric bill, energy or power?

ENERGY 8. 9. 10. 11.

Explain the difference between energy and work. List eight physical types of energy. Explain thermal energy from a microscopic point of view. Give one example of each of these energy forms: elastic, thermal, chemical, kinetic, radiant, gravitational. 12. If you double the speed, how is the kinetic energy affected? If you double the height, how is the gravitational energy affected? 13. Choose the correct answer(s): One joule is the same as one (watt-meter, newton-meter, meter per second squared, newton-second, kilowatt-hour).

THE LAW OF ENERGY AND ENERGY TRANSFORMATIONS 14. Has the law of conservation of energy been found to be correct in all situations observed so far? Is the same true of Newton’s laws? Explain. 15. Choose the correct answer(s): For a system that returns to its initial state, during one complete cycle you can’t get more (acceleration, force, energy, power, speed) out of the system than was put in. 16. What energy transformations occur when this book falls to the floor? When you lift this book? 17. Give an example of each of these energy transformations: kinetic energy : thermal energy, kinetic energy : elastic energy, elastic energy : kinetic energy. 18. What do we mean when we say that the energy efficiency of a lightbulb is 10%?

Conceptual Exercises WORK 1. Does Earth do gravitational work on you as you walk downstairs? 2. In order for you to get out of bed with the least amount of work, would it be better for your bed to be on the floor or about a meter high? Explain. 3. Describe some work you could do that would produce elastic energy. Repeat for gravitational energy. For kinetic energy. 4. Describe some work you could do that would produce thermal energy. 5. Your left hand lifts a 2 N apple by 1.5 m, and your right hand lifts a 4 N grapefruit by 0.5 m. Which hand did the most work? Which hand exerted the largest force? 6. MAKING ESTIMATES About how much work would it take to lift the U.S. population by 1 km?

ENERGY 7. Which of the eight physical types of energy was the basis for the earliest human culture? Which was the basis for the industrial revolution? Which other types might have been used by early cultures, and which other types are used today? 8. Name the type of energy possessed by each of the following: Jill at rest at the top of a sliding board, Jill sliding off the bottom of the sliding board, sunlight, coal, hot air.

From Chapter 6 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Conservation of Energy: Problem Set 9. Name the main type of energy possessed by each of the following: dynamite, water at rest behind a high dam, a bow about to release an arrow, a wooden match, food. 10. Name the type of energy possessed by each of the following: a raised book, gasoline, a stretched spring, sunlight, a speeding train, hot steam. 11. Does your body contain any kinetic energy when you are sitting still? Explain. 12. Give an example of a system that has both kinetic and gravitational energy. 13. A rubber ball is thrown upward and bounced off a ceiling. What kinds of energy does it have at its highest point (when it hits the ceiling)? 14. Name two kinds of energy that are produced as a result of a typical explosion, such as exploding dynamite. What kind of energy is used (or transformed)? 15. You lift a brick and put it on top of a wall. What quantities could you measure in order to determine how much work you did? 16. You throw a baseball. What quantities could you measure in order to determine how much work you did during the throw? 17. Explain, in microscopic terms, why the air pressure inside a tire increases on a hot day. If the air in a balloon is warmed, will the balloon expand or contract? Why? 18. (a) Where would an apple have greater gravitational energy, at 100 km high or at 1000 km high? (b) Would the gravitational energy of an orbiting satellite be increased or decreased by moving it from an orbit that is 6000 km high up to an orbit that is 12,000 km high (see Figure 13)? (c) At which point, 6000 km high or 12,000 km high, does a satellite have the larger gravitational force on it?

6000 km Before

12,000 km After

THE LAW OF ENERGY AND ENERGY TRANSFORMATIONS 23. Give an example in which kinetic energy transforms into gravitational energy. 24. Give an example in which kinetic energy transforms into thermal energy. 25. Give an example in which chemical energy transforms into kinetic energy. 26. What is the main energy transformation (input and useful output) when an automobile speeds up? When a bicycle speeds up? 27. Neglecting air resistance, does the energy of a falling rock increase, decrease, or remain the same? What happens to its kinetic energy? Its gravitational energy? 28. Including air resistance, does the energy of a falling rock increase, decrease, or remain the same? What happens to its kinetic energy? Its thermal energy? 29. What is the main energy transformation (input and useful output) in the operation of an electric blender? A toaster? A lightbulb? 30. You squeeze an elastic spring and clamp it in the squeezed position. You then drop the clamped spring into acid, dissolving the spring. What happened to its elastic energy? 31. What energy transformation occurs when you climb a rope? 32. You throw a baseball horizontally, and Jill catches it. Neglect air resistance. Describe the energy transformation that occurs (a) during the throw (while the ball is in your hand) and (b) during the catch. 33. You throw a ball upward and then catch it at the same height. How does the ball’s final speed compare with its initial speed, (a) neglecting air resistance and (b) including air resistance? Defend your answers. 34. Does an automobile use more gasoline when its lights are on? When the air conditioner is on? (Note: The battery does not run these devices while the engine is running.) Defend your answer. 35. Imagine a 100% efficient automobile. Would it emit any exhaust? Would its engine be hot? 36. Figure 14 is a graph of a roller coaster’s height above the ground versus the length of track it covers. The coaster is powered up to its high point at 100 m from the starting point. From the high point, the coaster coasts freely all the way to the end. Assume that the coaster starts from rest at the high point and encounters no friction or air resistance. Between 200 m and the finish, where is it moving slowest? Fastest?

What happens to the gravitational energy of the satellite when it is moved to a higher orbit?

19. If you triple your altitude above the ground, how is your gravitational energy (relative to the ground) affected? 20. If you halve your altitude, how is your gravitational energy (relative to the ground) affected? 21. If you triple your speed, how is your kinetic energy affected? 22. If you halve your speed, how is your kinetic energy affected?

Elevation, meters

Figure 13 40 30 20 10 0

0

200

400 600 800 Track distance, meters

Figure 14

Elevation versus track distance for a roller coaster.

162

1000

1200

Conservation of Energy: Problem Set 37. Referring to the preceding exercise, is the roller coaster moving faster at 1000 m or at 1100 m? Describe how the coaster’s speed changes during the last 300 m (900 m to the end).

POWER 38. You start a bowling ball rolling by swinging it with your arm and releasing it. Then you start a second identical bowling ball rolling, at the same speed as the first, by hitting it sharply with a sledge hammer. Which process, the arm swing or the hammer blow, imparts more kinetic energy to the ball? Which process has the greater power output? 39. You lift bricks, one at a time, onto a table. After a while, you begin to lift bricks slower and slower. As you slow down, does the energy you put into lifting each brick increase, decrease, or remain the same? What about the power you put into lifting each brick? 40. What other unit(s) could automobile engines be rated in, instead of horsepower? 41. Why are electricity rates often higher in summer than in winter? 42. Which process has the largest power output: 2 J of work performed in 0.1 s, or 1000 J of work performed in an hour? Which has the largest energy output? 43. An automobile travels 60 km in 50 minutes, doing 30 × 106 J of work against outside forces (air resistance and rolling resistance) in the process. What is the automobile’s average power output in watts? 44. Referring to the previous question, if the auto’s energy efficiency is 10%, what is its power input (its rate of converting the gasoline’s chemical energy into other forms) in watts? How many 100 W lightbulbs could this light up? 45. A cyclist delivers 150 W of power to her bicycle, while her metabolic rate is 1000 W. What is her body’s bicycling energy efficiency? 46. How much does it cost to run a nonfrostless refrigerator for a month? Use Table 1. Assume the refrigerator consumes

power for 8 hours each day, and the electricity costs 10¢ per kilowatt-hour. What if it is frostless? 47. If one load takes 30 minutes to dry in an electric dryer and you dry 16 loads per month, how much does one month’s drying cost at 10¢ per kilowatt-hour? 48. A clothes dryer is equivalent to how many 100 W lightbulbs? Use Table 1. 49. MAKING ESTIMATES Use Table 1 to estimate the number of kilowatt-hours of electric energy a typical single-family home consumes in one month. Don’t forget lightbulbs (they aren’t in Table 1).

Problems WORK 1. Your gasoline engine has a limited supply of gasoline, able to do 5000 J of work. If you weigh 800 N, how high can this engine lift you? 2. A jumbo jet has four engines, each having a thrust of 30,000 N. How much work do the engines do during a 1500 m takeoff run? 3. If you do 20 J of work lifting a rock weighing 30 N, how far will you lift it? 4. If an airplane does 40 million joules of work during a takeoff run that is 1000 m long, what must be the total thrust of its engines?

ENERGY 5. You slam on your automobile brakes, sliding 40 feet with locked brakes. How much farther would you slide if you had been moving twice as fast? 6. You slam on your automobile brakes, sliding 40 feet with locked brakes. About how far would you have slid if you had been moving half as fast?

Table 1 Power consumption of household appliances while the appliance is turned on and consuming electric energy Appliance

Power (W)

Cooking range

12,000

Clothes dryer

5,000

Water heater

4,500

Air conditioner, window

1,600

Microwave oven

1,400

Dishwasher (incl. hot water)

1,200

Toaster

1,200

Hair dryer

1,000

Refrigerator, frostless

600

Refrigerator, not frostless

300

TV, color

350

Stereo set

100

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Conservation of Energy: Problem Set 7. Ned the skydiver weighs 600 N and is falling at terminal speed. That is, he has sped up until air resistance has built up so much that he has reached a final unchanging speed. How much work does Ned do on the air as he falls through 200 m at this terminal speed? 8. In the preceding question, into what type of energy does this work go? From what type of energy did it come? 9. How much more gravitational energy (relative to the water’s surface) does a diver have when she stands 7 m above the water, as compared with when she stands 2 m above the water? 10. How much more kinetic energy does a runner have when dashing at 7 m/s as compared with jogging at 2 m/s? 11. MAKING ESTIMATES Estimate the energy (relative to the water) of you standing on a high diving board, 3 m above the water. 12. MAKING ESTIMATES Estimate your jogging speed, in m/s (Hint: 1 mile = 8/5 km), and then estimate the kinetic energy (in joules) of you jogging. 13. Use your answer to the preceding question to estimate the kinetic energy of you walking, assuming that your walking speed is half of your jogging speed.

Answers to Concept Checks

THE LAW OF ENERGY AND ENERGY TRANSFORMATIONS

10. 11.

14. Jack, who has a mass of 30 kg and weighs 300 newtons, sits in a child’s swing. You pull the swing back so that it is 2 m above its low point, and release it. What form of energy, and how much energy, does Jack have when he is pulled back and held at rest? 15. In the preceding question, what form of energy, and how much energy, does Jack have as he swings through the low point (neglect air resistance and friction in the moving parts)? How fast is Jack moving at this point? 16. In a crash test, a 1000 kg automobile moving at 10 m/s crashes into a brick wall. How much energy goes into demolishing and warming the wall and the auto? 17. Referring to the previous question, from how high a cliff would the automobile need to fall in order to sustain the same amount of damage upon hitting the ground? (Note: A 1000 kg automobile weighs (on Earth) about 10,000 N.)

POWER 18. You do a pullup, lifting yourself by 0.5 m in 2 s. If your weight is 600 N, how much work did you do, and what was your power output during lifting? 19. How much energy does a 75 W lightbulb use while running for 30 minutes? 20. How long must a 100 W lightbulb run in order to use a million joules of electrical energy? 21. Find the power output of a 60 kg runner who accelerates from 0 to 10 m/s in 2 s. 22. Compare the amount of metabolic energy used by a typical person in running up a flight of stairs to the energy required to light a 100 watt bulb for 1 minute.

164

1. (a) 2. Neither Jed nor Ned is moving the wall, so the work done is

3. 4. 5. 6. 7. 8. 9.

12.

13. 14. 15. 16. 17.

zero, (c). However, at the microscopic level, Ned’s muscles do work on one another. This microscopic work is done only within Ned’s body. It causes Ned to sweat, but does not result in any work being done on the wall. (d) Except for starting and stopping the book, it moves at unchanging velocity (zero acceleration), so the net force is zero, (a). (b) (b) (e) (c) Its height above the floor is now zero, so there is no gravitational energy. Conservation of energy tells us that the total energy is still 120 J, so this must be the amount of kinetic energy (since no other kind of energy is produced during the fall), (b). (c) and (e) The high water level in the lake creates the water pressure that presses water through the turbines, (a). This water level can be further traced back to the evaporation of water that lifted it so that it could fall as rain. The sun’s radiation caused this evaporation, (d). Don’t invest. And don’t bother investigating his design. BS violates conservation of energy. Work must be done by the transpropulsionizer to accelerate the spaceship, so the device puts out energy. Conservation of energy then tells us that the device must also consume energy, so it needs a consumable fuel supply. You can’t get something for nothing. (e) (f) (a) work = force * distance = 500 N * 0.8 m = 400 J, (d). power = work>time = 400 J>2 s = 200 W, (e).

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. Yes, because Earth exerts a downward force on you as you walk downward. 3. Examples: Stretch a rubber band, lift a rock, throw a baseball. 5. Left hand does 2 * 1.5 = 3 joules of work, right hand does 4 * 0.5 = 2 joules of work. So the left hand does the most work, although the right hand exerts the largest force. 7. Possible energy forms used by early cultures: chemical (food), thermal (fire), gravitational (falling water), elastic (bow and arrow), radiant (warmth from sun). Used today: all of the eight types discussed in Section 4, prominently including chemical (fossil fuels). 9. Chemical, gravitational, elastic, chemical, chemical.

Conservation of Energy: Problem Set 11. Yes, it contains the kinetic energy of moving blood, a beating 13. 15. 17.

19. 21. 23. 25. 27. 29. 31. 33.

35. 37.

39. 41.

43. 45.

heart, moving lungs, etc. Elastic and gravitational (but not kinetic). The weight of the brick and the distance to the top of the wall. The air inside gets hotter, so the molecules move faster and hit the inside wall of the tire harder, so the pressure against the inside wall is larger. The balloon will expand because of the increased air pressure inside the balloon. Tripled, because gravitational energy is proportional to height. Multiplied by 9, because kinetic energy is proportional to the square of the speed. One example: A ball moving upward after being thrown upword. Examples: A bomb exploding, the operation of a gasolinefueled vehicle. It remains the same; its kinetic energy increases; its gravitational energy decreases. Electric to kinetic; electric to thermal; electric to radiant. Chemical energy (of human body) to gravitational energy. (a) The speed would have to be the same, since the ball’s energy has not changed. (b) The speed would have to be less, since the ball lost some of its energy in warming the surrounding air. No, it would not emit exhaust, and its engine would not be hot because no energy would go into heating anything. Faster at 1000 m, because its elevation is lower. During the last 300 meters: It speeds up and then slows down from 900 to 1000 m, it speeds up then slows down between 1000 to 1100 m, and it speeds up then slows down then travels a short distance at constant speed between 1100 to 1200 m. The energy remains the same; the power decreases. Because everybody runs their air conditioner in the summer, there is a larger total load on electric plants, so the electric company charges more in order to hold the load down (and also in order to pay for higher-cost “peaking power” that might need to be added). Fifty minutes is 3000 seconds. So the power output is 30 * 106 J>3000 s = 10,000 W. 150>1000 = 15%.

47. 5 kW * 8 hr = 40 kW # h

40 kW # h * $0.10>kW # h = $4

49. To answer this, perform a calculation similar to Exercise 47

for each appliance used in your home, getting the number of kW # h of electric energy consumed by each appliance during one month. Then add up all of the appliances. Problems 1. W = Fd. Solving for d, d = W>F = 5000 J>800 N = 6.25 m. 3. W = Fd. Solving for d, d = W>F = 20 J>30 N = 0.667 m. 5. At twice the speed, you would slide four times as far. Here’s why: Your car has four times as much kinetic energy, so it will do four times as much work in coming to rest, thus it must exert its sliding frictional force over four times as much distance (remember W = Fd) in coming to rest. 7. The force of air resistance on Ned must be 600 N, in order to maintain his unchanging speed. So the force by Ned on the air must be 600 N (law of force pairs). Thus, W = Fd = 600 N * 200 m = 120,000 J. 9. Since GravE is proportional to height, she has 7/2 (3.5) times as much. 11. My weight is about 160 pounds, or about 700 N. At 3 m high, my gravitational energy is wt * height = 700 N * 3 m = 2100 J. 13. At half the speed, you have 1/4th as much kinetic energy. So the preceding answer is divided by four. 15. All 600 J of the original gravitational energy is now kinetic energy. KinE = (1>2)ms2. Solving for s, s = 2(2 * KinE>m) = 2(2 * 600 J>30 kg) = 240 = 6.32 m>s. 17. The initial energy should be the same as the result found in the preceding question, namely 50,000 J. GravE = wt * ht. Solving for ht, ht = GravE>wt = 50,000 J>10,000 N = 5 m. 19. P = W>t. Solving for W, W = Pt = 75 W * (30 * 60 s) = 135,000 J. 21. Assume that all the work goes into speeding up the runner (i.e., into kinetic energy). KinE = (1>2)ms2 = (1>2) 60 kg * (10 m>s)2 = 3000 J P = W>t = 3000 J>2 s = 1500 W.

165

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Second Law of Thermodynamics

From Chapter 7 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

167

Second Law of Thermodynamics —And You Can’t Even Break Even

Many times I have been present at gatherings of people who . . . are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: “Have you read a work of Shakespeare’s?” C.P. Snow, British Scientist and Author

Y

ou might have noticed that the energy going into many processes eventually turns into thermal energy. This tendency of nonthermal forms of energy to end up as thermal energy is an important general feature of the universe, known as the second law of thermodynamics or “second law” for short. It’s our focus in this chapter. The big breakthrough in understanding energy was the discovery that “heat” (thermal energy) is a form of energy, in other words that thermal energy can do work, just like other forms of energy. This breakthrough showed the validity of the law of conservation of energy even in processes involving thermal energy. Because of the central role of thermal energy in understanding the general principles of energy, the study of energy is called thermodynamics, and a reformulated version of the law of conservation of energy is often called the first law of thermodynamics. These laws of thermodynamics have no known exceptions and are among the most general scientific principles known. There are three different ways of stating the second law. In its most straightforward form, it is a familiar observation about thermal energy flow (Section 1). Like many common observations, it has profound consequences. Section 2 discusses one of these, namely, another form of the second law that highlights the special nature of thermal energy. Unlike other energy forms, thermal energy can be transformed into other forms only with limited efficiency. This leads to discussion of a socially significant device: the heat engine (Sections 2 and 3). Section 4 presents the third way of stating the second law, known as the law of increasing entropy. Its intriguing philosophical implications include the direction of time, the ultimate fate of the universe, and why there is a second law of thermodynamics. Sections 5, 6, and 7 study the physics and social implications of two significant heat engines: the automobile and the steam–electric power plant. Because these topics bring up energy resource issues, it’s natural at this point to discuss exponential growth and its implications for resource depletion (Section 8).

168

Second Law of Thermodynamics

1 HEATING Touch a piece of ice (Figure 1). Energywise, what happens? Since your hand cools and the ice begins to melt, thermal energy must have flowed from your hand to the ice. Now touch a hot cup of coffee (Figure 2). Your hand warms while the coffee cools, so thermal energy must have flowed from the cup to your hand. Notice that in each case, thermal energy flowed from the high-temperature object to the low-temperature object. There is a general principle operating here, a principle that you experience whenever you touch an object that feels hot or cold: Thermal energy flows spontaneously (without external assistance) from hot to cold. Any such flow of thermal energy from a higher to a lower temperature is called heating. Heating is a one-way affair: Thermal energy flows spontaneously from higher to lower temperature, but not from lower to higher temperature. Like a lot of simple ideas, this one-wayness of heating has profound consequences. It’s one way of stating the second law of thermodynamics: The Second Law of Thermodynamics, Stated as the Law of Heating Thermal energy flows spontaneously from higher to lower temperature, but not from lower to higher temperature.

Higher temperature Low temperature Thermalenergy flow

Figure 1

When you touch a piece of ice, thermal energy flows from your higher-temperature hand to the lower-temperature ice, so the ice gets warmer and your finger gets colder.

Lower temperature

High temperature

Now we need to be more quantitative. Temperature is a quantitative measure of warmth. An object’s temperature is related to the microscopic motion of its molecules. Most materials expand as they warm, because more rapidly moving molecules take up more space than slow-moving molecules, for the same reason that jitterbugging couples need more space on a dance floor than slow-dancing couples. So it’s not surprising that most materials expand as they warm. Such an expansion can be used as the basis for a temperature-measuring device, or thermometer. One choice is liquid mercury, which expands inside a glass tube. The standard (metric) temperature unit is the degree Celsius (°C). The Celsius scale assigns 0°C and 100°C to the freezing and boiling points of water. The United States still uses the antiquated Fahrenheit scale, where water freezes at 32 and boils at 212 (Figure 3). Temperature and thermal energy are related but different. For instance, a cool lake contains far more thermal energy than does a hot cup of coffee, even though the lake has a lower temperature, because the lake is so much larger. A single cup of lake water has less thermal energy than a hot cup of coffee.

Thermalenergy flow

Figure 2

When you touch a hot cup of coffee, thermal energy flows from the high-temperature cup to your lower-temperature hand, so your finger gets warmer and the coffee cup cools.

CONCEPT CHECK 1 The temperature of a nice day is about (a) 10°C; (b) 75°C; (c) 40°C; (d) 55°C; (e) 25°C. (Hint: See Figure 3.) CONCEPT CHECK 2 Suppose you grasped a cold doorknob and found, surprisingly, that this warmed your hand and further cooled the doorknob. This would violate (a) conservation of energy; (b) the second law of thermodynamics; (c) both conservation of energy and the second law; (d) neither conservation of energy nor the second law although it would violate other physical laws; (e) no known physical laws.

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Second Law of Thermodynamics C

F

2 HEAT ENGINES: USING THERMAL ENERGY TO DO WORK

100 200

150 50 100

50 0

0

Drop a book on the floor. Slide it across a table. Smack it with your hand. Imagine tearing out a page and burning it up. Thermal energy is created during each of these processes! Creating thermal energy is easy—almost inevitable. What about processes or devices that convert thermal energy to other forms? Can you think of any? ———(Keep thinking.) One example is an automobile engine, which operates in regular, repeated cycles to use the thermal energy from burning gasoline to do work. Another would be a steam engine, which operates in repeated cycles to use hot steam to do work. Any such cyclic device that uses thermal energy to do work is called a heat engine.1 Most heat engines, for example the automobile engine, are based on the expansion of a gas when it’s heated. The expanding gas pushes against a movable surface, the “piston,” that causes a car or other device to move. One significant feature of an automobile engine is that, in addition to doing work, it ejects lots of unused thermal energy through its radiator and its tailpipe. So not all the thermal energy created in the engine is actually used to do work. This turns out to be true for every heat engine. A heat engine’s ejected thermal energy is called its exhaust. So the energy transformation for any heat engine is ThermE (input) ¡ Work (which could then produce any form of energy) + ThermE (exhaust)

Figure 3

The Celsius and Fahrenheit scales compared.

Heat engine Work output

See Figure 4. The energy efficiency of any device is its useful energy output divided by the total energy put into the device. Since we usually consider the work done by a heat engine to be “useful” and the exhaust to be “not useful,” the energy efficiency of any heat engine is energy efficiency =

ThermE input ThermE exhaust

Figure 4

Energy flow for a heat engine. Heat engines consume thermal energy and turn part of it into work, which could then produce any from of energy.

work output thermal energy input

As you can see from Figure 4, the energy efficiency of any heat engine must be less than 1, in other words less than 100%. This fact, that you cannot entirely consume (or use) thermal energy but must always have some left over as exhaust, has been found to be true every time anybody has checked. It is, in other words, a fundamental principle of nature. But as we will see, it turns out not to be another new principle of nature. It has a one-way quality about it that’s reminiscent of the law of heating. Perhaps it’s not surprising, then, that it turns out to be the second law of thermodynamics, only put into new words. I’ll call it The Second Law of Thermodynamics, Stated as the Law of Heat Engines Any cyclic process that uses thermal energy to do work must also have a thermal energy exhaust. In other words, heat engines are always less than 100% efficient at using thermal energy to do work.

1

170

Some noncyclic devices use thermal energy to do work. For example, a hot air balloon uses heated air to lift a balloon. I won’t call such devices heat engines.

Second Law of Thermodynamics

How do we know that no heat engine can be 100% efficient? One reason we accept the law of heat engines is that if it were not true, then it would be possible to make thermal energy flow from cold to hot, in violation of the law of heating. The following argument shows this by using an imagined “thought experiment”—a type of argument used frequently in science. Let’s temporarily suppose that (in violation of the law of heat engines) there is a heat engine that can convert thermal energy entirely to work. We could then use that heat engine to extract thermal energy from, say, a pot of warm water and convert this energy entirely to work. This work could then produce thermal energy (by frictional heating of a piece of metal, for example) at a higher temperature. The net result would be to transfer all of the thermal energy from a lower to a higher temperature, without any other change taking place. But this is exactly what the law of heating says we cannot do. In other words, any violation of the law of heat engines would imply that the law of heating can be violated. But we know directly from experiment that the law of heating cannot be violated. So it follows that the law of heat engines cannot be violated either.

2

3

High-temperature object

Heat engine

Thermal energy flow

Heat engines depend on the spontaneous flow of thermal energy from hot to cold. In fact, a heat engine may be described as a device that makes practical use of the natural hot-to-cold flow of thermal energy by shunting aside some of the flowing thermal energy to do work (Figure 5). Since heat engines are driven by thermal energy flowing from hot to cold, you must have a temperature difference before you can have a heat engine. The ocean, for example, contains a lot of thermal energy, but you can’t use it to do work unless you have a colder system into which the ocean’s thermal energy can flow.2 Heat engines always operate between two systems with different temperatures, as Figure 5 shows. How energy efficient can the very best heat engine be? It’s an important question, because most of the world’s energy passes through heat engines, mainly transportation vehicles and steam–electric power plants. Since a heat engine operates because of thermal energy flowing from hot to cold, we expect its energy efficiency to be influenced by both its hot input temperature at which thermal energy is put into the engine and its cooler exhaust temperature. Since temperature differences drive heat engines, we expect a higher efficiency for larger temperature differences between input and exhaust. Nineteenth-century physicists found a quantitative formula that predicts the best possible efficiency of a heat engine operating at any predetermined input and exhaust temperatures.3 As examples, Table 1 lists the best possible efficiencies predicted for several specific types of heat engines, along with the actual efficiencies obtained by these heat engines in practice. Table 1 shows how important the second law is to society. It says that, for practical heat engines, conversion efficiencies of thermal energy into other forms are only 60% even in the best, or “perfect,” case. Friction and other imperfections reduce this further, so that less than half the thermal energy fed into these heat engines goes into work. But it’s important, in these energy-conscious times, to note that the remaining thermal energy, the exhaust, needn’t be wasted. There’s a myriad of direct heating uses for this lower-temperature thermal energy. This dual use of thermal energy to simultaneously produce both work (usually in the form of electricity generation as

Work

Low-temperature object

Figure 5

Heat engines use a portion of the thermal energy that flows naturally from a high to a low temperature and convert it to work.

This means that temperature differences between different depths of ocean water could be used to run a heat engine. Here’s the formula: efficiency = (Tin - Tex)>Tin. In this formula, the temperatures must be measured in degrees Kelvin (K), a new temperature scale. The temperature in K is found by adding 273 to the temperature in °C. A temperature of 0 K (equal to −273°C) is known as absolute zero, because it is the lowest possible temperature—the temperature at which all microscopic motion is at its absolute minimum.

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Second Law of Thermodynamics Table 1 Heat engine efficiencies. Typical temperatures, best possible efficiencies, and actual efficiencies. Efficiency (%) Tin(°C)

Tex(°C)

Best possible

Actual

Gasoline automobile/truck

700

340

37

20

Diesel auto/truck/locomotive

900

340

48

30

Steam locomotive

180

100

20

10

Fossil fuel

550

40

60

40

Nuclear fuel

350

40

50

35

Solar powered

225

40

40

30

25

5

7

???

Engine type

Transportation

Steam-electric power plants

Ocean-thermal (solar)

described in Section 7) and useful heat is called co-generation. It can save society enormous amounts of fossil fuel and other energy resources. For example, many European, and a few American, communities locate electric power plants near large residential neighborhoods and use the plants’ “waste” thermal energy to heat their homes, saving enormous amounts of natural gas or electricity that would otherwise be used for home heating. Communities are beginning to install smaller electrical generation systems, running perhaps on burning trash from a housing development, and piping the “waste” thermal energy to the houses for home heating. Given the expense of transmitting and distributing electricity over large areas, there’s a lot to be said for such small, local electrical generation facilities. U.S. law allows co-generators at such community facilities to sell their excess electricity to electrical power companies at reasonable prices. For another example, steam heating plants at universities and other institutions often produce such high steam temperatures that the steam can be used to generate electricity at reasonable efficiencies. The “waste” thermal energy from electricity generation is then used to heat the university. Universities can often generate a significant fraction of their electricity consumption in this fashion, saving energy resources while getting electricity and heating for roughly the operating costs of the heating alone. If your college or university isn’t doing this, maybe it should. Table 1 shows the importance of “burning hot” and “exhausting cool.” For example, fossil, nuclear, and solar generating plants have progressively lower input temperatures. As you can see, efficiencies decline as the difference between Tin and Tex declines. Ocean-thermal generation of electric power uses some of the ocean’s thermal energy by exploiting temperature differences between different ocean depths. In the tropics, the ocean’s temperature drops from 25°C at the surface to 5°C at 300 m down. This small temperature difference could be used to run a heat engine with an efficiency of 7%. Because the energy resource would be free—sunlight falling on the ocean—the low energy efficiency would be of little concern. CONCEPT CHECK 3 Assuming that the energy flows in Figure 5 are proportional to the width of each “pipe,” this engine’s efficiency is closest to (a) 1/3; (b) 1/2; (c) 1/10; (d) 2/3; (e) 1; (f) 2.

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Second Law of Thermodynamics

CONCEPT CHECK 4 An engine that consumes 400 J of thermal energy while exhausting 300 J has an efficiency of (a) 133%; (b) 100%; (c) 75%; (d) 66%; (e) 33%; (f) 25%. CONCEPT CHECK 5 A typical large coal-fired electric-generating plant burns about 1 tonne (1000 kg) of coal every 10 seconds. According to Table 1, how much of the tonne actually goes into producing electric energy? (a) 600 kg. (b) 60 kg. (c) 500 kg. (d) 400 kg.

3 ENERGY QUALITY: THINGS RUN DOWN Thermal energy is special. A moving bullet or a raised rock can easily use nearly 100% of its kinetic or gravitational energy to do work, but the second law places strict limits on the percentage of thermal energy that can be converted to work. Thermal energy is thus less useful, or “of lower quality,” than other forms. Whenever you transform other energy forms into thermal energy—say by friction or combustion—you reduce the energy’s usefulness even though the total amount of energy is conserved. So there is a one-wayness, an irreversibility, about any process that creates thermal energy. Once a system creates thermal energy, that system will never by itself be able to return to its previous condition. To return, it would have to convert all the created thermal energy back to its original form, and the second law prohibits this. The system can return to its initial state only with outside help. Think of a rock swinging back and forth on a string tied to a hook (Figure 6). Air resistance and friction (between the string and the hook) gradually bring the rock to rest. Although the complete system (rock, string, hook, and surrounding air) loses no energy, it can’t return to its initial condition because thermal energy is created, and this cannot be entirely reconverted to kinetic or gravitational energy. Something is permanently lost when systems run down like this, but it cannot be energy because energy is conserved. Instead, energy quality is lost. When we use Earth’s energy resources, we don’t reduce Earth’s total energy. Instead, we degrade energy from highly useful forms such as the chemical energy of oil to less useful forms, usually thermal energy. Thus one of the two great laws of energy says that the quantity of energy is conserved, and the other says that the quality of energy runs down. You can’t get ahead, and you can’t even break even.

Figure 6

A rock swinging on a string tied to a fixed point overhead. The rock “dies down,” illustrating the irreversibility of natural processes implied by the second law.

4 THE LAW OF ENTROPY: WHY YOU CAN’T BREAK EVEN Suppose you put a box of hot gas and a box of cold gas into contact so that thermal energy (but not the gases themselves) can flow between them. The law of heating predicts that the hot box will heat the cold one, and that this will continue until there is no longer a temperature difference between the boxes. Figure 7 views this process microscopically, showing just a few molecules. The hot box’s molecules are moving faster on the average. The exchange of thermal energy causes the molecules of the hot gas to slow down and the molecules of the cold gas to speed up until both gases come to some intermediate temperature. Now the average speeds in the left- and right-hand boxes are the same, so the fast molecules and slow

173

Second Law of Thermodynamics High temperature Low temperature

(a)

Intermediate temperature

(b)

Figure 7

Microscopic view showing just a few molecules in a box of hot gas and a box of cold gas (a) at the instant they are put into contact and (b) after there has been time for the boxes to come to the same temperature. Figure (b) shows less organization than (a), because the faster molecules are no longer separated from the slower molecules.

molecules are no longer separated from each other.4 From the microscopic point of view, the system is less organized. Microscopic disorganization (a mouthful—sorry!) has increased. This turns out to be the general situation, no matter whether the materials are gases or anything else. When thermal energy flows from hot to cold, microscopic disorganization always increases. In fact, the universal increase of microscopic disorganization turns out (although we won’t prove it here) to be equivalent to thermal energy always flowing from hot to cold. In other words, this is another way of stating the second law. Physicists have found a quantitative measure of the microscopic disorganization of any system. It’s called entropy. For example, the entropy of 1 kg of water is greater than the entropy of 1 kg of ice, because the molecules of water are not organized into a regular crystal pattern as are the molecules of ice. We don’t need to delve into the precise definition of entropy here. Suffice it to say that entropy can be precisely defined and it can be measured entirely macroscopically by measurements of temperature, thermal energy, and a few other quantities such as volume. So we have yet a third way of stating the second law: The Second Law of Thermodynamics, Stated as the Law of Entropy The total entropy (or microscopic disorganization) of all the participants in any physical process cannot decrease during that process, but it can increase.

The law of entropy is similar to the law of conservation of energy. Both place restrictions on natural processes: The total energy of all the participants in any process must remain unchanged, and the total entropy must not decrease. The law of entropy predicts that most processes are irreversible—they cannot proceed in the opposite direction. For example, our hot and cold boxes of gas can 4

174

If the boxes contain different kinds of gases, then the average kinetic energies, rather than the average speeds, of the individual molecules of the two gases become equal when a common intermediate temperature is attained.

Second Law of Thermodynamics

come to the same temperature spontaneously, but they cannot start from the same temperature and evolve to different temperatures unless they have outside help (a heater on one side and a refrigerator on the other). Processes must go in the direction of increasing, not decreasing, entropy. In fact, except for a very subtle effect at the subatomic level,5 the second law is the only principle of physics that distinguishes between the forward and backward directions of time. So if it weren’t for the second law, everything could just as well run backward. For example, a book resting on a table could spontaneously leap into the air by converting some of its thermal energy into kinetic and gravitational energy. This is the reverse of a book falling onto a table. It might appear to violate such principles as Newton’s law of motion, but if viewed at the microscopic level, there is no violation: It is possible, although highly improbable, for the randomly moving molecules in the book to all just happen to be moving upward at the same instant, with sufficient speed to cause the book to leap from the table. Similarly, water could run uphill. Thermal energy could flow from cold to hot. And people could grow younger instead of older. Perhaps you have seen a movie run backwards. The only law of physics that would be violated if these backward events occurred in real time is the second law of thermodynamics. The law of entropy suggests a deeper reason behind the second law. As you can see from the cartoon at the end of this chapter, increased disorganization is common in everyday life. For example, if you start with a partly organized deck of cards having all the spades collected together and then shuffle the deck, you’ll almost certainly disorganize the deck further. This is simply because there are so many more ways to disorganize the deck than there are ways to further organize it. It’s easy to disorganize things but it takes effort—or lots of luck—to organize them. So the second law arises for simple statistical reasons. Like a deck of cards, molecular systems are much more likely to evolve toward greater disorganization than toward greater organization, simply because there are many more ways for a molecular system to become disorganized than to become organized. This leads to a fascinating point about the long-term fate of the universe. Applied to the universe as a whole, the second law says that its natural evolution must be toward greater disorganization. Its long-term fate would then be a state of maximum disorganization, in which no further macroscopic developments would occur. Such a state would be really boring. All the stars would have burned out and no new ones could form because all nuclear reactions would have run their course. Life could not exist because of the lack of sunlike stars and because all chemical reactions would have run their course. This has been called the “heat death” of the universe. Being hundreds of billions of years in the future, it’s not exactly our most pressing issue. More importantly, there is a huge speculative element here, because it’s always risky to assume that the principles of physics are so precisely understood that they can be applied to the entire universe for all time. Looking toward earlier instead of later times, the law of entropy implies that the universe must have had lower entropy—greater organization—in the past. In fact, physicists agree that the universe began in a highly organized, low-entropy state at the time of the big bang and that its entropy has been increasing ever since. The big bang

5

For some reason, the universe at one time had a very low entropy for its energy content, and since then the entropy has increased. So that is the way toward the future. That is the origin of all irreversibility, that is what makes the processes of growth and decay, that makes us remember the past and not the future, remember the things which are closer to that moment in the history of the universe when the order was higher than now, and why we are not able to remember things where the disorder is higher than now, which we call the future. Richard Feynman, Physicist

There is indirect but compelling evidence that certain types of subatomic particles distinguish between the forward and backward directions of time in processes involving the “weak force,” one of nature’s four fundamental forces. It’s not known whether this discovery is in any way related to the second law or to our sense of a forward direction in time.

175

Second Law of Thermodynamics A living organism . . . feeds upon negative entropy. Thus the device by which an organism maintains itself at a fairly high level of orderliness (= fairly low level of entropy) really consists in continually sucking orderliness from its environment. Erwin Schroedinger, Physicist, in What Is Life?

Radiant energy from sun at 5500

98% is reradiated at 25 : large increase in entropy

ThermE out

ThermE in

2% is converted to low-entropy chemical energy: smaller decrease in entropy

Figure 8

Energy flow through a leaf. The leaf is similar to a heat engine. A growing leaf illustrates how Earth has become more organized, despite the universe’s trend toward increased entropy.

was the source not only of energy and matter but also of the organization we see in the universe today. Biological systems provide interesting examples. For example, a growing leaf manufactures complex and highly organized glucose molecules out of less organized CO2 and H2O molecules. The leaf must create this organization. How does it manage to produce this decrease in entropy, in apparent violation of the second law? The answer is that the leaf had help. The second law says that the total entropy of all the participants in any process cannot decrease. In the growth of a leaf, the other vital participant is the sun. Solar radiation has a temperature, the 5500°C surface temperature of the sun. When this radiation is absorbed by a leaf, only about 2% of the energy is converted to chemical energy. The remaining solar energy is reradiated out into space, at the 25°C temperature of the leaf. So most of the solar energy flows from 5500°C to 25°C. The large entropy increase of this thermal energy flow allows the remaining solar energy to be organized into low-entropy chemical energy (Figure 8), without violating the second law, because the total entropy increases. The sun both energizes and organizes life on Earth! In view of the second law, it seems paradoxical that life on Earth could have evolved on its own from the simple organisms that existed shortly after Earth formed billions of years ago into the highly organized plants and animals of today. But like the leaf, biological evolution had help from the sun. Evolution is assisted by sunlight flowing through plants from higher to lower temperature, representing a great entropy increase that compensates for the decrease that occurs when plants evolve. And animals, which do not use solar energy directly, reduce their entropy by eating highly organized food—another form of outside help. Thus, biological evolution does not contradict the second law. Your brain is one result of this long evolution toward greater organization. As an information-storage device, the human brain is the most highly organized form of matter on Earth. It could even be the most organized form of matter in the Milky Way galaxy. It’s remarkable that, in the human brain, nature has finally created a self-aware collection of molecules, molecules so well organized that they are capable of knowing they are a collection of molecules! Nature has spent billions of years of evolution getting to this point. So please, my friend, take good care of yourself, and of all of us. CONCEPT CHECK 6 When a tablespoon of salt mixes with a quart of water, does entropy increase? (a) Yes. (b) No. (c) Sometimes. (d) Only on Fridays.

5 THE AUTOMOBILE You now have the physics background needed for four energy-related social topics that will occupy the remainder of this chapter. Few technologies shape our society as strongly as the automobile. It brings new freedom while affecting our quality of life, family structure, self-perceptions, physical environment, health, work, community structure, resource use, economy, and even war and peace. For most Americans, the environmental effects of their automobile use far outweigh the environmental effects of any other individual activity. Transportation consumes much of the USA’s energy (Figure 9) and most of its oil (Figure 10), most of it going into cars and trucks (Figure 11).

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Second Law of Thermodynamics Residential 22%

Industrial 32%

Transportation 28%

Commercial 18%

Figure 9

The fraction of total U.S. energy consumed by each economic sector. (U.S. Energy Information Administration, 2007)

Residential 4% Industrial 23%

Transportation 68%

Electricity generation 3% Commercial 2%

Water 5% Air 7%

Pipeline 2%

Rail and bus 3%

Autos 63%

Trucks 20%

Figure 10

Figure 11

The fraction of U.S. petroleum consumed by each economic sector. (U.S. Energy Information Administration, 2003)

The fraction of U.S. transportation energy consumed by each transportation mode. (U.S. Bureau of Transportation Statistics, 2007)

MAKI NG ESTI MATES Experiments show that upon combustion, 1 liter (0.26 gallons) of gasoline releases (converts to thermal energy) about 32 * 106 J of energy. Use this figure to estimate the rate, in watts, at which a typical car consumes chemical energy. (Hints: Typical gasoline consumption is 10 km/liter (25 miles/ gallon), and moderate highway speeds are about 80 km/hr (50 mi/hr).)

The preceding Making Estimates question shows that a typical car driven at moderate speed without acceleration consumes its gasoline’s chemical energy at a rate of 70 kW. This is equivalent to the electric power going into 700 continuously burning 100 W bulbs! As another comparison, 70 kW is about the average electric power consumption of 50 households. And if the car is accelerating, you can multiply these figures by about 5. Cars are powerful energy consumers. Most transportation fuel goes into heat engines, where it burns to produce thermal energy that is then partially transformed into useful work. Most cars and trucks are powered by internal combustion engines that burn a fuel–air mixture. The mixture’s high combustion temperature gives it a high pressure, so that the hot gases push strongly on a piston, a movable metal plate connected to a rod (Figure 12). The piston does the work that turns the drive wheels. Combustion is “internal” because it occurs directly inside the gases that do the work, in contrast to external combustion, which occurs in a fuel that then provides thermal energy to a second substance, such as steam, that does the actual work. Figure 13 shows the energy flows (measured in kW, or thousands of joules of energy transformed per second) in a typical gasoline-fueled automobile. An average of 1 kilowatt’s worth of fuel evaporates into the atmosphere, where it contributes to chemical pollution. The remaining 69 kW go to the engine, which produces about SO LU T I O N TO M A KI N G ESTI M ATES During 1 hour at 80 km/hr, a car travels 80 km and so consumes 8 liters. The chemical energy in 8 liters is 8 * 32 * 106 = 2.6 * 108 J. Since this is consumed in 1 hour (3600 s), the chemical energy consumed per second is

Cylinder walls

Piston Expanding gas Work

Figure 12

Cross-section of a single cylinder in an automobile’s engine, showing the conversion of thermal energy to useful work using a piston.

2.6 * 108 J>3600 s L 70,000 J>s, or 70,000 watts, or 70 kW

177

Second Law of Thermodynamics Evaporation

Thermal energy of exhaust, 55 kW

28 kW

Out of tailpipe

27 kW Water pump, etc. Friction, etc.

Figure 13

Typical energy flow rates in an unaccelerated gasoline-fueled car at a moderate highway speed.

Engine

1 kW

3 kW 5 kW Air resistance

Work, 14 kW Transmission and drive train

5 kW Rolling resistance

Waste, 60 kW

Into engine, 69 kW

Removed by radiator

Work, 10 kW

From fuel tank, 70 kW

1 kW

14 kW of work and exhausts the remaining 55 kW as thermal energy and unused chemical energy. About half of the exhaust energy is removed by the radiator, and the other half goes out through the exhaust pipe as polluting gases. Gasoline is a hydrocarbon, made of hydrogen (H) and carbon (C) atoms. Both H and C combust with oxygen from the atmosphere, so the exhaust gases are mostly CO2 and H2O. Although the H2O is harmless, CO2 from fossil fuel combustion is the main source of global warming. The tailpipe exhaust carries various other molecules, mainly CO, NO, NO2, and unburned hydrocarbons. These are toxic pollutants, and their unused chemical energy also represents an energy inefficiency. The carbon monoxide and unburned hydrocarbons are the result of incomplete combustion of the fuel. The two oxides of nitrogen—collectively called NOx—form from atmospheric oxygen and nitrogen, which combine under the influence of the engine’s high temperatures. Cars and trucks produce two-thirds of America’s CO pollution, one-third of its hydrocarbon pollution, and half of its NOx pollution. The automobile’s main loss of useful energy occurs in the engine, as a consequence of the second law. The engine’s theoretically ideal efficiency is 37% (Table 1), and its actual efficiency is 14 kW/69 kW = 20%. Several losses combine to bring the engine’s efficiency from the 37% allowable by the second law down to 20%: incomplete combustion, the formation of NOx, friction in the engine, and thermal losses through the engine’s wall. Of the 14 kW of work produced by the engine, 1 kW goes to internal devices like the water pump and air conditioner, and the rest goes to the transmission and drivetrain that couples the engine to the drive wheels. This coupling is about 75% efficient, so 10 kW finally arrive at the drive wheels. About half of this goes into overcoming air resistance, while the other half goes into overcoming rolling resistance. The overall energy efficiency of the entire automobile (not just the engine) is 10/70 = 14%, or one-seventh. Because the problems of scarce oil resources and dangerous global warming pollution loom ever larger, and because the automobile is a chief source of both, alternative transportation modes (next section) and alternative automobile fuels

178

Second Law of Thermodynamics

AP Photo/The Canadian Press, Jeff McIntosh

are increasingly important. Table 2 and Figures 14 through 17 present several alternative automobile fuels. Several alternatives to the standard internal combustion engine are now on the market or under development. Electric vehicles (EVs, Figure 15) are powered by large batteries that use stored chemical energy to create electricity. They are “zeroemission vehicles” because they have no tailpipes and emit no chemical pollutants

Figure 14

University of Michigan team members run with their solar car, driven by Brooke Bailey, as it crosses the finish line to win the 2008 North American Solar Challenge. Race drivers had to obey posted speed limits while driving 3800 km from Texas to Calgary, Canada, powered only by the sun and what could be stored in the car’s battery. The race has been described as the “Tour de France of engineering.” Table 2 Fuels for automobiles and trucks Fuel

Source of fuel

Description

For internal combustion Gasoline or diesel fuel

petroleum

liquid, widely used

Compressed natural gas

natural gas

high-pressure gas

Liquefied natural gas

natural gas

low-temperature liquid

Methanol (wood alcohol)

wood, natural gas, coal

liquid

Ethanol (grain alcohol)

grain, sugar, trash

liquid

any source of electricity

hydrogen produced from water

any source of electricity

recharge electrically

hydrogen, methane, other

hydrogen can be produced from water

Hydrogen

a

For noncombustion Storage batteriesa Fuel cell a

a

Since electricity for hydrogen production, for storage batteries, or for hydrogen fuel cells can come from any source, the ultimate energy resource can be wind, hydroelectric, nuclear, fossil fuel, photovoltaics, etc.

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Vince Bucci/Getty Images

Second Law of Thermodynamics

Figure 15

The Tesla Roadster, billed as the world’s first highway-capable all-electric car, runs on 900 pounds of rechargeable lithium ion batteries (shown). For safety, the car has a cooling system to prevent overheating the batteries. Its range is 380 km (240 miles) on one battery charge, a full recharge requires 3.5 hours, and its top speed is electronically limited to 200 km/h (125 mi/h). The high cost of the batteries makes it an expensive purchase. The company plans to produce 15,000 cars per year by 2011.

AP Photo/Atsushi Tsukada The Toyota Prius debuted in 2001. It’s a gasoline-electric hybrid vehicle that runs on a battery and on a small gasoline engine that energizes the battery and provides additional power when needed. Thus the car combines the convenience of gasoline with much of the environmental advantage of electric vehicles, while achieving 80 km (50 miles) per gallon. The front end is shown; the electric “inverter” that converts the battery’s DC to the engine’s AC and vice-versa is on the right, and the gasoline engine is on the left. The nickelmetal-hydride battery pack, warranted for 10 years or 240,000 km (150,000 miles), is in the back end.

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Chevrolet Corporation

Figure 16

Figure 17

The chassis of the Chevrolet Volt. A small gasoline engine-generator is on the far side (passenger side) of the front end, an electric motor is on the near side (steering wheel side) of the front end, and a long lithium-ion battery can be seen extending through the center of the car; the 180 kg (400 pound) battery then branches out in the rear end into a T shape (the top of the T is between the rear wheels and cannot be seen in the photo). The car should go on the market in 2011.

Second Law of Thermodynamics

while in operation. But the electricity to charge the battery must come from somewhere. If it comes from a fossil-fuel electric generating plant, then the vehicle causes plant emissions that create pollution and global warming. If the electricity comes from a less polluting source such as solar cells or wind, the vehicle becomes more environmentally benign. Typical EVs require several hours of recharging from a wall socket, the batteries are heavy, and EVs are expensive today. However, intense research is in progress to reduce these problems. Gasoline-electric hybrid vehicles (Figure 16) have recently achieved unprecedented energy efficiencies. “Hybrid” vehicles are fueled by gasoline, which runs a small gasoline engine that maintains a constant low power level. The engine does not directly drive the car, but instead drives an electric generator that provides electricity to energize a storage battery, which in turn drives the car just like an electric vehicle. Since the engine runs at a steady low power, it’s very energy efficient. Since the storage battery is continuously recharged, it needn’t store large amounts of energy, so it can be far smaller and lighter than an EV’s battery. Additional efficiencies are obtained by making the car of lightweight but strong (for crash safety) materials, streamlining (to reduce air resistance), and employing a braking mechanism that recovers kinetic energy losses during deceleration. Several hybrid passenger vehicles have entered the market. Because of recent and probable future rises in the price of oil, the Toyota Motor Corporation is betting that hybrids will become much more popular and has announced plans for all of its vehicles to eventually be run by hybrid engines. The plug-in hybrid car is an important newer variant. It’s a cross between the all-electric car (Figure 15) and the “conventional” hybrid vehicle (Figure 16). The Chevrolet Volt is a good example and will probably be the first plug-in hybrid on the market, in 2011. Figure 17 shows the Volt’s chassis, so you can see a little of how it works. While hybrids carry a battery that is continuously recharged on-board by a small gasoline engine, the Volt carries a battery that drives the car solely on electricity for the first 40 miles, after which the battery is assisted by a small gasoline engine for another several hundred miles. To re-charge the battery after use, it must be plugged in for several hours. So plug-in hybrids, like electric cars but unlike conventional hybrids, get energy for their battery from the electrical grid. The advantage of this is that no gasoline is needed for trips under 40 miles, so on average the car uses far less gasoline than other cars, replacing the gasoline with electricity and energy efficiency. This in turn reduces CO2 emissions, the main contributor to global warming, enormously: By year 2050, plug-in hybrids could reduce U.S. oil consumption by 4 million barrels a day (a 20% reduction), and reduce U.S. CO2 emissions by half a billions tons per year (a 33% reduction in vehicular emissions). Fuel cell vehicles are fueled by hydrogen obtained from fossil fuels or water. Even though the universe is made mostly of hydrogen, this light-weight element escaped from Earth’s gravitational hold long ago and is nearly absent on Earth in its free form (the H2 molecule), uncombined with other elements. It must be obtained from fossil fuels by chemical processes or from water by using electricity to split H2O molecules. It can be stored in the vehicle as a compressed gas, or as a liquid at extremely low temperature, or it can be chemically inserted into certain metals. Instead of being burned, the fuel is fed into a device called a fuel cell that converts the hydrogen’s chemical energy directly into electricity. It works by exploiting hydrogen’s tendency to combine chemically with oxygen—like a battery with hydrogen at one electrode (or pole) and oxygen at the other. The difference between

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a battery and a fuel cell is that batteries store chemical energy during the life of the battery, whereas fuel cells feed a chemical source through the cell when needed. So batteries must be recharged, while fuel cells must be refueled. Compared with batteries, fuel cells are less massive and operate continuously over longer times. But there are many scientific and practical barriers to the widespread use of fuel cells, and they are quite expensive today. Unlike gasoline engines, fuel cells convert chemical energy directly into electrical energy, so they are not heat engines and the second law does not affect the essentials of their operation. Thus, typical efficiencies of obtaining electrical energy from the hydrogen fuel’s chemical energy are 60% or more, far greater than the 20% that’s typical for gasoline engines. Environmental benefits depend on the source of the hydrogen. If the hydrogen is produced from water using solar-generated electricity, fuel cells are one of the most environmentally benign transportation technologies. Since the fuel cell combines the hydrogen fuel with oxygen to yield water as the only emission, there’s no pollution. There’s no fossil fuel consumption, and no global warming gas emissions. But there are several barriers to practical hydrogen fuel cells for cars: Today it’s expensive to produce the hydrogen from water, the fuel cells themselves are expensive, it’s not easy to store sufficient hydrogen in an automobile, and a new fuel distribution system will be needed for the hydrogen. Several companies have small numbers of experimental fuel cell vehicles on the road today. Hydrogen fuel cells might be on the market within ten years but if they are marketed that soon they’ll probably get their hydrogen from natural gas instead of from water and so they’ll lose two of their major advantages—they will consume fossil fuels, and they will emit global-warming gases. Some experts have dubbed the arrival of fuel cells in the marketplace—when and if it happens—as the dawning of a new energy age, the “Hydrogen Age.” CONCEPT CHECK 7 Which of the following uses a heat engine? (a) Gasolinefueled car. (b) Diesel-fueled car. (c) Electric car. (d) Fuel cell car running on methane. (e) Fuel cell car running on hydrogen. (f) Hybrid car.

6 TRANSPORTATION EFFICIENCY

Mary Ellen Scullen Figure 18

The two-person Smart Car. Its reduced size, weight, and engine give it a gasoline efficiency of 16 km/liter (37 mi/gal).

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My previous definition of energy efficiency—work output divided by total energy input—doesn’t really capture the automobile’s purpose. Although its purpose is to move people, its energy goes mostly into moving the car itself rather than people. Gasoline mileage, the common measure of automobile efficiency, suffers from the same defect. Neither of these measures captures people-moving efficiency, in other words, transportation efficiency. Gasoline mileage is, however, useful for comparing different cars with each other (Table 3). Most of the high-efficiency vehicles listed use hybrid engine technology, but the two-person Smart Car (Figure 18) instead achieves high efficiency by means of size and weight reductions. An appropriate measure of people-moving efficiency is passenger-kilometers per unit of energy. For example, if a bus moves 20 passengers a distance of 3 km, it has delivered 20 passengers * 3 km = 60 passenger-km. Similarly, the appropriate measure of freight-moving efficiency is the tonne-kilometer per unit of energy. For example, if a truck moves 5 tonnes a distance of 80 km, it has delivered 400 tonne-km. Table 4 compares passenger-moving efficiencies. For walking and bicycling, the table uses the “gasoline equivalent” of the required number of food calories.

Second Law of Thermodynamics Table 3 Fuel efficiencies of passenger vehicles km/liter

mi/gal

National averages: All U.S. passenger vehicles (cars, SUVs, pickups)

9

22

New U.S. passenger vehicles

12

27

New European Union passenger vehicles

14

34

New cars: Ford Expedition, SUV

6

16

Honda Accord

11

25

Ford Escape, SUV hybrid

13

32

Nissan Altima, hybrid

14

34

Smart Car

16

37

Honda Civic, hybrid

18

42

Toyota Prius, hybrid

21

50

Table 4 U.S. passenger-moving efficiencies of several human transportation modes passenger-km per liter

passenger-mi per gal

passenger-km per MJ

Human on bicycle

642*

1530*

18.0

Human walking

178*

425*

5.0

Intercity rail

60

144

1.7

Carpool auto (occupancy = 4)

36

88

1.0

Urban bus

33

80

0.9

Commercial airline

21

50

0.6

Commuting auto (occupancy = 1.15)

11

25

0.3

*For walking and bicycling, the table uses the “gasoline equivalent” of the required number of food calories.

Pierre Verdy/AFP/Getty Images

Walking and bicycling come out far ahead because no energy goes into moving a heavy vehicle, and because no heat engine is employed so there is no loss due to the second law of thermodynamics. Bicycling is more efficient than walking because wheels keep rolling and thus take maximum advantage of the law of inertia. Walking requires you to start and stop your legs with every step, and these accelerations require a force (Newton’s law of motion), which means that work is done. Trains (Figure 19) are far more energy efficient than other vehicles because they can more efficiently overcome air resistance and rolling resistance and because one train can carry many passengers. Because a train presents a small frontal area relative to its large load, its air resistance per kilogram of load is far below that of cars and trucks. And because a train rolls on inflexible steel wheels, there is little rolling resistance. Table 5 compares freight-moving efficiencies. Again, the advantage of rail is obvious. Since one freight train can carry the loads of 500 trucks, this is not surprising. You can see from Figures 9, 10, and 11 that transportation is a major consumer of energy in general and of oil in particular. This consumption poses huge problems

Figure 19

The French TGV (Train à Grande Vitesse or “train of great velocity”) has been in service since 1972. About 350 TGV “trainsets” are in service today. Like most of Europe’s high-speed trains, the TGV routinely travels at about 300 km/hr (186 mi/hr). Trains are far more energy efficient than other vehicles because they can carry several hundred passengers, because they present a small frontal area relative to their large load, and because they roll on inflexible steel wheels that experience little rolling resistance.

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Second Law of Thermodynamics Table 5 Freight-moving efficiencies of three transportation modes kg-km tonne-km per MJ per liter

2900

100

Truck (heavy)

720

25

Air (freight)

145

5

Rail (freight train)

Table 6 Mass-moving efficiencies of animals and machines, in kilogramkilometers per megajoule Human on bicycle

1100

Salmon

600

Horse

400

Human walking

300

Typical bird

200

Intercity rail

100

Urban bus

55

Hummingbird

50

Carpool auto

40

Commercial airline

25

Fly, bee

20

Commuting auto

12

Mouse

5

Adapted from S. Wilson, “Bicycle Technology,” Scientific American, March 1973.

for the United States: increasing energy costs, declining domestic and foreign sources of oil, and two Middle Eastern wars that were partly due to oil-related security problems. Tables 3, 4, and 5 suggest many ways to reduce oil consumption: • Enact automobile efficiency standards to bring new U.S. cars up to at least the efficiency of new European cars. • Use incentives to promote hybrid vehicles. • Promote carpooling. • Encourage mass transit while discouraging cars. • Move freight by train instead of by truck. • Plan cities that encourage walking, bicycling, and transit and discourage driving. For fun, let’s compare the entire realm of locomotion by animals and all forms of human technology. Which animal or human transportation mode is the most efficient transporter of mass: fruit flies? trains? horses? jet planes? humans walking? bicycling? To compare fruit flies, horses, and humans fairly, we must incorporate the fact that the horse’s energy goes into moving a lot more mass. So the useful output should be measured as the animal’s (or the human’s plus the machine’s) total mass times distance moved. Table 6 gives several such mass-moving efficiencies, in kilogram-kilometers per megajoule of energy. Again, bicycles come out far ahead, because animals don’t have wheels so they can’t take advantage of rolling and because bicycles are about the only major human transportation technology that is not a heat engine. In other words, bicycles come out on top because of the law of inertia and the second law of thermodynamics! As an avid bicycle commuter, I think that’s cool. This reasoning implies that animals with wheels would have a big energy advantage. In fact, some salamanders and other animals use this energy advantage by rolling their bodies into hoops and rolling down hills. One could speculate that on another planet with surfaces created by smooth lava flows, wheeled animals might evolve and be abundant! CONCEPT CHECK 8 You wish to move 125 tonnes of freight a distance of 200 km. How many liters of gasoline are needed to move it by truck, and by rail? (a) 125 liters by truck and 500 liters by rail. (b) 500 liters and 125 liters. (c) 250 liters and 1000 liters. (d) 1000 liters and 250 liters.

7 THE STEAM–ELECTRIC POWER PLANT

Every time I see an adult on a bicycle, I no longer despair for the human race. H. G. Wells, English Historian, Sociologist, Author of War of the Worlds and Other Novels.

184

Let’s turn now to another heat engine that has transformed modern society: the steam–electric power plant. Figure 20 is a schematic diagram of the operation of a coal-burning electric power plant. Coal is the most widely used energy source for electricity. Other plants that use oil, natural gas, nuclear energy, or solar energy to turn water into steam in a boiler operate much as a coal-burning plant does in producing electricity from steam. The coal combusts externally in a furnace, and its thermal energy is transferred to water inside a boiler. Most combustion products escape through the stack but some pollutants are removed first. The boiler produces high-pressure steam at over 500°C, far above the normal boiling temperature. The steam moves through pipes to a large rotating device called a steam turbine that turns when it feels a higher pressure on the front (upstream) side than on the back. Like a car’s piston, the steam

Second Law of Thermodynamics Electricity

Stack gases

Generator

Hot steam Turbine

Cooler steam

Hot steam Condenser

Cold water Hot water

Boiler Pump

Furnace

Water

Lake or cooling tower Fuel

Figure 20

A schematic diagram showing the operation of a coal-fueled steam–electric generating plant.

turbine is the key device that transforms thermal energy into work. The turbine turns an electric generator that creates electricity. The rotating turbine converts some of the hot steam’s thermal energy into work. The second law tells us that this is possible only if the remaining thermal energy flows to a cooler temperature. To maintain the required temperature difference, the exhaust side of the turbine is cooled by an external stream or lake or by evaporative cooling in the atmosphere. To obtain the greatest efficiency, the exhaust is cooled sufficiently to “condense” the steam back into liquid water, because this greatly reduces the pressure against the back side of the turbine. The steam is then sucked forcefully through the turbine, from very high pressure on one side to near-vacuum on the other. Once condensed, pumps move the water back around to the boiler, where the cycle begins again. As you can see from Figure 20, the plant is a heat engine. Thermal energy flows in at the boiler and out at the condenser, and work is done by the turbine (compare Figure 5). Figure 21 shows the energy flow (more precisely, the energy per second in megawatts). A large plant generates about 1000 MW of electric power, enough for a

185

Second Law of Thermodynamics Stack emissions, 300 MW

From coal, 2500 MW

To turbine, 2200 MW

Turbine exhaust, 1200 MW (removed by condenser) Work, 1000 MW

Electric energy, 1000 MW Turbine (heat engine)

Waste, 1600 MW

Lost in transmission, 100 MW

To user, 900 MW

Useful work, 900 MW

Figure 21

Energy flow in a typical 1000 MW coal-fueled electric generating plant.

large city. Because a typical plant’s efficiency is 40%, this electrical output requires 2500 MW, or 2500 million joules of energy input every second, requiring 100 kilograms of coal every second! Of the 2500 MW input, 300 MW go out through the stack, accompanied by oxides (oxygen compounds) of nitrogen, oxides of sulfur, carbon dioxide, and small incombustible particles called “ash.” Oxides of nitrogen and sulfur cause acid rain, and carbon dioxide is the main source of global warming. Modern plants remove most of the sulfur oxides, some of the nitrogen oxides, and nearly all of the ash, which then presents a significant solid-waste disposal problem. Since coal is made mainly of carbon, CO2 is by far the predominant stack gas. None of it is removed today, although in the future it may be possible to remove it and inject it in gaseous form into the ground. The turbine converts the thermal energy of steam to useful work, which in turn drives a generator that creates 1000 MW of electric power. The plant’s biggest loss in useful energy, the 1200 MW exhausted (Figure 21), is an unavoidable consequence of the second law. This exhaust goes into the condenser’s cooling water. If the cooling water comes from a lake or river, the exhaust warms the water, an effect called thermal pollution. Many plants use the atmosphere as the coolant by employing large evaporative cooling towers (Figure 22). Finally, an average 100 MW of the generated electricity is lost as thermal energy during transmission over electric power lines, and 900 MW gets to the user. Grapes/Michaud/Photo Researchers, Inc. Figure 22

Cooling towers (on the right) and stacks at a coal-fueled generating plant. Cool air is sucked into the bottom of the cooling towers, where it cools hot water from the plant. The tower’s shape promotes a rapidly rising hot-air column.

186

CONCEPT CHECK 9 Judging from Figures 20 and 21, the efficiency of the boiler in converting the coal’s chemical energy into steam is (a) 100%; (b) 12%; (c) 88%; (d) 36%; (e) 10%. CONCEPT CHECK 10 A particular coal-burning generating plant consumes 8000 tonnes of coal per day. Assuming that the coal is pure carbon, which of the following is closest to the amount of carbon dioxide that this plant injects into the atmosphere every day? (a) 8000 tonnes. (b) 3200 tonnes. (c) 4800 tonnes. (d) 24,000 tonnes. (e) 16,000 tonnes. (f) 48,000 tonnes.

Second Law of Thermodynamics M A K I N G EST I M AT ES At 100 kg/s, estimate the amount of coal (in tonnes) that

a typical plant uses in one day. How many traincars full of coal is this, at about 100 tonnes per traincar?

8 RESOURCE USE AND EXPONENTIAL GROWTH6

Value of account in dollars

The social implications of energy raise many growth-related issues. As our growing numbers and environmental impact begin to affect the entire natural world, it is important to understand the long-term effects of growth. Suppose you invest $100 at a rate of return or growth rate of 10% per year. When will you double your money? You earn $10 during the first year, so you have $110. In the second year you earn 10% of $110, or $11. Now you have $121, so you earn $12.10 during the next year. Each year you earn more than you did the previous year. Figure 23 graphs your account and compares it with a second graph that increases by a fixed $10 every year. The second graph illustrates linear (straightline) growth. As you can see, there’s a big difference between $10 per year and 10% per year. When a quantity grows by a fixed percentage in each unit of time, its growth is said to be exponential.7 If you continue adding 10% each year, your account will about double to $200 after 7 years. What will happen after another 7 years? Since the same arithmetic applies, it will double again, to $400. In the next 7 years it will double again, to $800. And so forth. Exponential growth has an unchanging doubling time.

2000 1800 1600 1400 1200 1000 800 600 400 200

The arithmetic of growth is the forgotten fundamental of the energy crisis. Albert Bartlett

Growth at 10% per year: exponential growth

Growth at $10 per year: linear growth

0

2

4

6

8

10

12

14 16 Years

18

20

22

24

26

28

30

Figure 23

The 10% investment account. The initial investment is $100. Would you rather have exponential growth at 10% per year or linear growth at $10 per year?

6 7

This section draws on the work of University of Colorado physicist Albert A. Bartlett. It is called “exponential” because the account is worth 100 * 1.1 after 1 year, 100 * 1.12 after 2 years, 100 * 1.13 after 3 years, and so forth. The number of years is in the exponent.

The number of seconds in a day is 60 s>min * 60 min>hr * 24 hr>day ' 105 s>day. 100 kg enter the plant every second, so the amount entering in a day is 105 * 100 = 107 kg, or 10,000 tonnes. This is 10,000/100 = 100 traincars every day! SO LUTION TO MAKI NG ESTI MATES

187

Second Law of Thermodynamics MAKI NG ESTI MATES You take a job requiring you to work every day for 30 days and your employer offers you just one cent for the first day and then a doubled salary every day after that. Will this be a good salary for the month? Pressures resulting from unrestrained population growth put demands on the natural world that can overwhelm any efforts to achieve a sustainable future. If we are to halt the destruction of our environment, we must accept limits to that growth. From World Scientists Warning to Humanity, a Declaration Signed by Nearly 1700 Leading Scientists from 71 Countries, Including 104 Nobel Laureates

In a population of animals, the number of newborns each year is roughly proportional to the number of potential parents in the population that year. So if the population doubles, the number of newborns should also double. So the percentage increase—the number of newborns divided by the total population—should be roughly the same from year to year. This unchanging percentage increase means that population growth is roughly exponential. CONCEPT CHECK 11 Bacteria reproduce themselves by simply dividing. If you start with 1 bacterium, it will divide into 2; they will divide into 4, then into 8, and so forth. Since each population doubling occurs in the same time interval, this is an exponential process. Suppose that some strain of bacteria has a dividing time of 1 minute. You put 1 bacterium into a bottle at 11 A.M., and at noon you note that the bottle is full of bacteria. The bottle was half full at (a) 11:30; (b) 11:40; (c) 11:50; (d) 11:55; (e) 11:58; (f) 11:59.

As Concept Check 11 shows, when you consume a finite resource exponentially, it’s easy to use nearly all of it before you realize there’s a problem (Figure 24). Continuing with Concept Check 11, suppose that at 11:56 A.M. (when the bottle was only 1/16 full, or 94% empty!), some visionary bacteria, realizing they have a problem, launch an all-out search for new bottles. By 11:58 A.M., this program has been successful in discovering a vast new reserve: three new bottles! It took the bacteria an entire hour to fill the first 1

Fraction of jar that is filled with bacteria

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 Number of minutes past 11 A.M.

Figure 24

Bacterial growth in a jar. A finite resource, consumed exponentially, runs out surprisingly rapidly toward the end. On the scale of this graph, growth is imperceptible until 11:53 A.M. when the bottle is 1% full. You will earn $5.12 on day 10. To simplify matters, round this to just $5. Now continue doubling. On day 20, your earnings will be $5120. Round this to $5000, and continue. On day 30 alone, your earnings are more than $5 million!

SO LUTION TO MAKI NG ESTI MATES

188

Second Law of Thermodynamics

bottle. When will the new bottles be full? The answer is at 2 minutes past noon. Continued exponential growth eventually overwhelms all attempts to expand the resource base. There is a simple and useful quantitative relation for exponential growth. Any increase in the growth rate must decrease the doubling time, so we might expect to find a relation between these two. It turns out that they are inversely proportional. The relation is, approximately, doubling time = T =

70 growth rate 70 P

where T stands for the doubling time and P is the growth rate (the percentage growth per unit time, expressed in percent). This can be turned around to read 70 P = T Either quantity, the doubling time or the growth rate, is equal to 70 divided by the other quantity. For instance, the 10% savings account has a doubling time of T = 70>P = 70>10 = 7 years For a historical example, consider the growth of U.S. electric power. As you can see by examining Figure 25, production grew exponentially between 1935 and 1975, doubling about every 10 years for a percentage growth rate of P L 70>T = 70>10 = 7% per year. What if this growth rate had continued past 1975? In 1975, all electric energy could have been provided by about 400 large plants. If the 10-year doubling time had continued, 800 plants would have been needed in 1985, 1600 in 1995, 3200 in 2005, and 6400 in 2015. Sixty-four hundred power plants would mean some 125 in every U.S. state, with everybody living within a few miles of a large power plant! Obviously, expansion at a fixed growth rate is unsustainable. In fact electric power production increased by only 3% per year during 1973 to 1988, by 2% per year during 1988 to 2000, and by 1% per year during 2000 to 2010. U.S. oil production illustrates what happens when a finite resource is consumed exponentially. Like many industries, oil production grew exponentially during its early years, maintaining 8% annual growth during 1870–1930. But this could not be maintained, because recoverable oil resources would be gone by now. The growth 4000 Electricity generated in one year, in billions of kilowatt-hours

3500 3000

Figure 25

2500 2000 1500 1000 500 0 1900

10

20

30

40

50

60

Year

70

80

90

2000 2010

The history of electric power production. The annual electric energy produced, in billions of kilowatt-hours, is graphed from 1900 through 2010. The growth was roughly exponential between 1935 and 1975. (U.S. Energy Information Administration)

189

Second Law of Thermodynamics

rate declined, and then around 1970 U.S. oil production in the 48 contiguous states began to drop, as in Figure 26. This bell-shaped curve is typical for a nonrenewable resource—a resource that cannot be readily replaced within a human lifetime. U.S. oil production is following this pattern and is in region C on the graph. World oil production is probably in region B. Resource depletion is inevitably driving the world toward the end of the oil age, although other problems such as global warming might end it even sooner. Renewable resources, such as wood or solar energy, follow a different history (Figure 27). In their early stages, renewable and nonrenewable resource use rises exponentially. But renewable resources can be sustained indefinitely, assuming they are consumed at less than the replacement rate, so the graph levels off as shown. Finally, consider world population growth (Figure 28). It took 6 million years for the human population to grow to its first billion in 1825. It reached its next billion only a century later in 1930, and its third billion in 1960. The sixth billion was reached in 1999. Population experts estimate that the population in 2050 will be around 9 billion. The actual outcome will be strongly dependent on human fertility between now and then. We can find the approximate current rate of growth from the population doubling during 1960 to 2000: P = 70>T = 70>40 = 1.75%. As you can see from the graph, this is a large—in fact explosive—rate.

Region C: exhaustion and decline Region A: exponential growth

Region B: reduced growth, leveling off

Yearly consumption

Yearly consumption

CONCEPT CHECK 12 U.S. oil production grew at 8% per year during 1870–1930. Its doubling time was roughly (a) 6 years; (b) 9 years; (c) 12 years; (d) 15 years; (e) 30 years.

Years

190

Region C: no growth, sustainable level

Region A: exponential growth

Region B: reduced growth, leveling off

Years

Figure 26

Figure 27

A typical bell-shaped curve showing the life history of consumption of a nonrenewable resource. Exponential growth must slow as the resource is depleted. Consumption eventually levels off and declines as the resource nears exhaustion.

A typical life history of a renewable resource such as hydroelectric power. Exponential growth slows as consumption reaches its natural limits. Consumption eventually levels off at some sustainable value.

Second Law of Thermodynamics 9

About 9 billion in 2050

8

6

6 billion in A.D. 1999

5

4 Third billion in 1960 3 Second billion in 1930 2 First billion in 1825 1

Figure 28 0 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 200 180 160 140 120 100 800 600 400 200 0 200 400 600 800 100 120 140 160 180 200 0 0 0 0 0 0 0 0 0 0 0 0 Year

The population explosion: faster than exponential. Note the resemblance to Figure 24.

© Sidney Harris, used with permission.

World population, in billions

7

191

192

Second Law of Thermodynamics Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions HEATING 1. What is heating? 2. In your own words, state the law of heating. 3. Give an example showing that thermal energy and temperature are really two different things.

HEAT ENGINES AND ENERGY QUALITY 4. Give an example of each of these energy transformations: kinetic to thermal, gravitational to thermal, thermal to kinetic. 5. In your own words, state the law of heat engines. 6. What properties of the input and the exhaust are the most important to determining a heat engine’s efficiency? Describe the manner in which the efficiency depends on these properties. 7. Is the actual overall energy efficiency of an automobile closest to 98%, 90%, 40%, 10%, or 2%? What about a steam–electric power plant? 8. If a heat engine operated entirely without friction, would it then be 100% efficient? Explain. 9. In terms of energy, what happens when the motion of a rock swinging from a string “dies down”? In what sense is this behavior irreversible?

THE LAW OF ENTROPY 10. What is entropy? 11. In your own words, state the entropy form of the second law. 12. Which law or laws of physics distinguish between forward and backward in time? 13. When a growing leaf increases its organization, does it violate the second law?

THE AUTOMOBILE AND TRANSPORTATION 14. Name two general types of heat engines of major social importance. 15. Why do we call it an internal combustion engine? Describe in your own words how it works. 16. Name two alternative fuels (not gasoline or diesel fuel) for the automobile. 17. What is the source of the largest inefficiency in an automobile’s operation?

18. Describe two different ways of measuring a transportation mode’s efficiency and give an appropriate measurement unit for each. 19. Why are trains more efficient than other transportation vehicles?

THE STEAM–ELECTRIC POWER PLANT 20. In a steam–electric power plant, what is the purpose of the turbine? Generator? Condenser? Cooling tower? Stack? 21. What is thermal pollution? 22. What is the source of the most important inefficiency in a steam–electric power plant’s operation? 23. What part of a steam–electric power plant is analogous to the piston in an automobile?

EXPONENTIAL GROWTH 24. What is the difference between exponential and linear growth? 25. Your savings account grows at 7% per year. What is its doubling time? 26. Draw a typical life-history graph for a nonrenewable resource. Is any part approximately exponential? 27. Repeat the previous question, but for a renewable resource.

Conceptual Exercises HEATING 1. How would you describe the weather on a day when the temperature was –3°C? +3°C? 22°C? 35°C? 2. Give the approximate temperature, in °C, of each of the following: your body, water boiling in an open pot, ice water, a nice day. 3. Which is larger, a Celsius degree or a Fahrenheit degree? 4. How does the flow of thermal energy through a closed window illustrate the second law? Which direction is this flow when it is cold outside? Hot outside? 5. Try to think of at least one technological device that causes thermal energy to flow “uphill,” from colder to hotter. Does this device violate the law of heating? Explain. 6. In the operation of a refrigerator, does thermal energy flow from hot to cold, or is it from cold to hot? Does this happen spontaneously, or is outside assistance required?

From Chapter 7 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

193

Second Law of Thermodynamics: Problem Set

HEAT ENGINES 7. Is it possible to convert a given quantity of kinetic energy entirely into thermal energy? Is it possible to convert a given quantity of thermal energy entirely into kinetic energy? In each case, either give an example or explain why it is impossible. 8. Is it possible to convert a given quantity of chemical energy entirely into thermal energy? Is it possible to convert a given quantity of thermal energy entirely into chemical energy? In each case, either give an example or explain why it is impossible. 9. Which are not heat engines: natural-gas-burning power plant, hydroelectric power plant, ethanol-fueled automobile, bicycle, solar–thermal electric power plant, steam locomotive? 10. Which of the following are heat engines: nuclear power plant, diesel locomotive, electric locomotive, geothermal power plant, wind turbine (windmill for generating electricity), solar hot water heater? 11. What does the second law tell us about the efficiency of heat engines? 12. Can you think of any way to drive a ship across the ocean by using the ocean’s thermal energy without violating the second law? 13. Farswell Slick approaches you with plans for a revolutionary transportation system. He has noticed that when he drives an automobile without accelerating, all the input energy eventually shows up as thermal energy. Slick proposes to use this thermal energy to drive the car at a constant speed. The car will still need fuel, but only for accelerating. It will be possible to travel cross-country on only a few gallons of gasoline. He describes his scheme as a “computerized advanced-technology exhaust feedback afterburner.” Should you invest in Slick’s scheme? Explain. 14. On the Fahrenheit scale, what are the freezing and boiling points? Use your answer to calculate the number of Fahrenheit degrees in one Celsius degree. 15. Use the result of the preceding question to convert 10°C to Fahrenheit. Convert 30°C to Fahrenheit. 16. In one cycle of its operation, a heat engine does 100 J of work while exhausting 400 J of thermal energy. What is its energy input? Its efficiency?

Residential 22%

Industrial 32%

Transportation 28%

Commercial 18%

Figure 9

The fraction of total U.S. energy consumed by each economic sector. (U.S. Energy Information Administration, 2007)

194

17. In one cycle of its operation, a heat engine consumes 1500 J of thermal energy while performing 300 J of work. What is its efficiency? How much energy is exhausted in each cycle?

ENERGY QUALITY AND THE LAW OF ENTROPY 18. When your book falls to the floor, is this a thermodynamically irreversible process? Is energy conserved? Does entropy increase? 19. When we say that the motion of a rock swinging on a string is irreversible, do we really mean that it is impossible to get the rock back to its starting condition? Explain. 20. When a block of wood slides down a sliding board, is this a thermodynamically irreversible process? Does this mean that it is impossible to make a block of wood slide up a sliding board? Explain. 21. As an egg develops into a chicken, its contents become more ordered. In light of what you have learned about the second law of thermodynamics, do you expect that this process violates the law of increasing entropy? Explain. 22. A pan of liquid water freezes when you place it outside on a cold day. Liquid water has greater molecular disorder than ice does. Is the freezing process then an exception to the law of entropy? Explain. 23. When orange juice and grapefruit juice are mixed, does entropy increase?

THE AUTOMOBILE AND TRANSPORTATION 24. Describe the energy input for walking and bicycling. How do walking and bicycling illustrate the second law? 25. Suppose an automobile’s fuel could be made to burn hotter without harming the engine’s operation (for instance, without cracking the engine). Would you still get the same amount of useful work from each gallon of gasoline? 26. Suppose an automobile could run on hard wheels that were not squeezed by the weight of the car on the road. Would this alter the car’s efficiency? How might this affect the gas mileage? What kind of wheels and road might you suggest? 27. According to Figures 9, 10, and 11, which of the three main sectors of the U.S. economy (industry, residential– commercial, transportation) consumes the most oil?

Residential 4% Industrial 23%

Transportation 68%

Electricity generation 3% Commercial 2%

Water 5% Air 7% Rail and bus 3%

Pipeline 2% Autos 63%

Trucks 20%

Figure 10

Figure 11

The fraction of U.S. petroleum consumed by each economic sector. (U.S. Energy Information Administration, 2003)

The fraction of U.S. transportation energy consumed by each transportation mode. (U.S. Bureau of Transportation Statistics, 2007)

Second Law of Thermodynamics: Problem Set 28. One car has twice the gasoline mileage efficiency of a second car. Compare the amounts of pollution they produce when they both travel the same distance. 29. Out of every 100 barrels of gasoline, about how many actually go into driving a typical car down the road? 30. A bus carries 30 people 200 km using 300 liters of gasoline. Find its passenger-moving efficiency.

THE STEAM–ELECTRIC POWER PLANT 31. Which type of generating plant would you expect to be more energy efficient, steam–electric or hydroelectric? Defend your answer. 32. Would it be more energy efficient to heat your home electrically or to heat it directly using a natural gas heater, assuming that the electricity comes from a steam–electric plant? 33. Which method of fueling your car is likely to be more energy efficient, and why: gasoline used in a standard car engine or electricity taken from a coal-fueled generating plant and stored in lightweight car batteries? Assume that the batteries convert electricity to work at 100% efficiency. 34. Out of every 100 tons of coal fed into an electric generating plant, roughly how many tons produce the electricity you can use at your home and how many go into waste energy? Use the approximate energy flows indicated in Figure 21. 35. For every 100 kilograms of coal entering a generating plant (recall that this much enters every second), about 15 kilograms of sulfur oxides and ash are removed, producing a significant solid-waste disposal problem. For a typical 1000 MW plant, how much of this solid waste is produced every day? Express your answer in tonnes (1 tonne = 1000 kg). 36. How would the annual pollution from two coal plants compare if the first plant is twice as energy efficient as the second? Assume that they both produce the same amount of electric power.

EXPONENTIAL GROWTH AND RESOURCE USE 37. A lily pond doubles its number of lilies every month. One day, you notice that 2% of the pond is covered by lilies. About how long will it be before the pond is entirely covered? 38. On June 1 there are a few water lilies in a pond, and they then double daily. By June 30 they cover the entire pond. On what day was the pond still 50% uncovered? 39. Company X increases its profits every year by $50 million. Is its growth in profits exponential? Company Y increases its profits by 1% every year. Is its growth in profits exponential? 40. According to Figure 25, did electric power production grow exponentially between 1910 and 1935? Estimate the number of kilowatt-hours produced in 1935, 1945, 1955, 1965, and 1975, and verify that production grew approximately exponentially between 1935 and 1975. 41. Which of the following are renewable energy resources: coal, firewood, nuclear power, wind, water behind a dam. 42. What is the original source of energy in each of the following energy resources: oil, firewood, wind, water behind a dam, geothermal, ocean-thermal electricity? Which of these are renewable resources? 43. The most recent world population doubling, to a total population of about 6 billion, has occurred in about 40 years. Making the (unrealistic!) assumption that this rate of population will continue for two centuries, what would the world’s population be two centuries from now? Such unrealistic assumptions are often useful in projecting future trends because they give us a sense of what is likely or unlikely. For example, this exercise shows us that it is very unlikely that our present population growth will continue for two more centuries. 44. The most recent world population doubling has occurred in about 40 years. Suppose that the next doubling occurs also in 40 years, but that a new agricultural “green revolution” manages to also double food production. Then how many people will be starving 40 years from now, as compared to the number starving now?

Stack emissions, 300 MW

From coal, 2500 MW

To turbine, 2200 MW

Turbine exhaust, 1200 MW (removed by condenser) Work, 1000 MW

Electric energy, 1000 MW Turbine (heat engine)

Waste, 1600 MW

Lost in transmission, 100 MW

To user, 900 MW

Useful work, 900 MW

Figure 21

Energy flow in a typical 1000 MW coal-fueled electric generating plant.

195

Second Law of Thermodynamics: Problem Set 4000

Figure 25

The history of electric power production. The annual electric energy produced, in billions of kilowatt-hours, is graphed from 1900 through 2010. The growth was roughly exponential between 1935 and 1975. (U.S. Energy Information Administration)

Electricity generated in one year, in billions of kilowatt-hours

3500 3000 2500 2000 1500 1000 500 0 1900

10

20

30

40

50

60

70

80

90

2000 2010

Year

45. Is the graph of Figure 29 an exponential curve? Explain.

Time

Figure 29

Is this an exponential curve?

Problems HEAT ENGINES (You will need footnote 3 to solve some of these problems.) 1. If a heat engine’s efficiency is 30% and its work output is 2000 J, how much thermal energy must have been put into it? 2. If a heat engine’s efficiency is 20% and 1.5 million joules of energy are put into it, how much work does it do? 3. The steam entering the turbine in a coal-burning power plant is heated to 500°C. The steam is cooled and condensed to water at 80°C. Find the best possible efficiency of the power plant. Remember that you must convert Celsius temperatures to degrees Kelvin before using the formula. 4. A solar-heated steam–electric generating plant heats steam to 250°C. After passing through the turbine, cooling towers cool the steam to 30°C. Calculate the best possible efficiency of this power plant. Remember that you must convert Celsius temperatures to Kelvins before using the formula in footnote 3. 5. In the preceding question, suppose that for every 1000 J of thermal energy going into this plant, the cooling towers remove 750 J as exhaust. What is the actual efficiency of this power plant? 6. A coal-burning steam locomotive heats steam to 180°C and exhausts it at 100°C. During 1 s of operation, it consumes 500 million J of energy from the burning coal. According to

196

Table 1, how much work can be obtained from this locomotive during 1 s of operation under ideal conditions (no friction or other imperfections)? 7. In the preceding question, how much work can be obtained under actual conditions?

THE AUTOMOBILE AND TRANSPORTATION 8. You travel alone from New York to Los Angeles, about 2800 miles. Working from Tables 3 and 4, how many gallons of gasoline will you use if you travel by car (assuming your car gets average gasoline mileage for U.S. automobiles)? How many gallons if you travel by air? By bus? By train? 9. A 100-car freight train hauls 16,000 metric tonnes of freight from New York to Los Angeles, about 5000 km. How many trucks would be needed for this load, assuming that each truck carries 32 tonnes of freight? Working from Table 5, how many liters of gasoline are saved if this load is carried by train rather than by truck?

THE STEAM–ELECTRIC POWER PLANT 10.

MAKING ESTIMATES. In the United States, solar energy strikes a single square meter of ground at an average rate (averaged over day and night and over the different seasons) of 200 watts (200 joules/second). At what average rate does solar energy strike a football field (about 100 m by 30 m)? 11. Continuing the preceding question, a typical U.S. home consumes electricity at an average rate of 1 kW. How much surface area would be needed to provide this electric power, assuming a 10% conversion efficiency? What dimensions would a squareshaped photovoltaic collector need to cover this area?

EXPONENTIAL GROWTH 12. How much electric energy would have been produced in 1985 if the exponential growth of 1935–1975 had continued for another 10 years beyond 1975? If this growth had contin-

Second Law of Thermodynamics: Problem Set Table 3 Fuel efficiencies of passenger vehicles km/liter

mi/gal

National averages: All U.S. passenger vehicles (cars, SUVs, pickups)

9

22

New U.S. passenger vehicles

12

27

New European Union passenger vehicles

14

34

New cars: Ford Expedition, SUV

6

16

Honda Accord

11

25

Ford Escape, SUV hybrid

13

32

Nissan Altima, hybrid

14

34

Smart Car

16

37

Honda Civic, hybrid

18

42

Toyota Prius, hybrid

21

50

Table 4 U.S. passenger-moving efficiencies of several human transportation modes passenger-km per liter

passenger-mi per gal

passenger-km per MJ

Human on bicycle

642*

1530*

18.0

Human walking

178*

425*

5.0

Intercity rail

60

144

1.7

Carpool auto (occupancy = 4)

36

88

1.0

Urban bus

33

80

0.9

Commercial airline

21

50

0.6

Commuting auto (occupancy = 1.15)

11

25

0.3

*For walking and bicycling, the table uses the “gasoline equivalent” of the required number of food calories.

Table 5 Freight-moving efficiencies of three transportation modes kg-km tonne-km per MJ per liter

2900

100

Truck (heavy)

720

25

Air (freight)

145

5

Rail (freight train)

ued, roughly how many power plants would have been needed in 1985, as compared with 1975? 13. During 1985–1990, annual U.S. population growth was 0.8% per year, for Mexico it was 2.2%, and for Kenya (the highest) it was 4.2%. At these rates, how long does it take for the populations of each of these countries to double?

14. World population is now about 7 billion. The growth rate has been roughly 2% per year since the end of World War II (1945). If a 2% per year growth rate continued, when would world population be 14 billion? 15. Centerville, with a growth rate of 7% annually, is using its only sewage treatment plant at maximum capacity. If it continues its present growth rate, how many sewage treatment plants will it need 40 years from now? 16. During the 1980s, U.S. car and truck miles traveled increased by 4% per year, but the length of highway increased by only 0.1% per year. Find the doubling time for vehicle miles traveled and for miles of highway. 17. Continuing the preceding problem, suppose that these rates are maintained in the future. While vehicle miles double (a 100% increase), by roughly what percentage will the amount of highway increase? Roughly, how much worse will traffic congestion be at that time?

197

Second Law of Thermodynamics: Problem Set

Answers to Concept Checks 1. (e) 2. (b) 3. The “work” pipe appears to be about 1/3 as wide as the “ther4. 5. 6. 7. 8.

9. 10.

11. 12.

mal energy flow” pipe coming out of the high-temperature source, (a). The work done must be 400 J - 300 J = 100 J, so the efficiency is 100 J/400 J, (f). Table 1 tells us that the actual efficiency is about 40%, so the amount that goes into producing electricity is 0.4 * 1000 kg = 400 kg, (d). Salt and water molecules are randomly mixed together, so microscopic disorganization (entropy) increases, (a). (a), (b), and (f ) 125 tonnes * 200 km = 25,000 tonne-km. Table 5 tells us that a truck can transport 25 tonne-km on 1 liter of gasoline, so the number of liters needed for 25,000 tonne-km is 25,000>25 = 1000 liters. The number of liters needed for a train is 25,000>100 = 250 liters, (d). 2200>2500 = 88%, (c) According to the periodic table, carbon and oxygen atoms have roughly equal masses (oxygen is actually 33% more massive). Thus the CO2 molecule is roughly three times as massive as the C atom, so the CO2 emitted is roughly three times as massive as the coal. 3 * 8000 tonnes = 24,000 tonnes, (d). (f ) T = 70>P = 70>8 L 9, (b)

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. Cold, below freezing; cold, above freezing; a nice day; a hot day. 3. A Celsius degree. 5. Refrigerator; it moves thermal energy from inside the refrigerator to outside. It does not violate the second law, because the second law states that thermal energy cannot spontaneously flow from cold to hot. In a refrigerator, the flow is not spontaneous (it is assisted by the operation of the refrigerator, which draws thermal energy out of the inside). Another example: air conditioner. 7. Yes; an example is dropping a book onto a table. No, because the second law prohibits it. 9. Hydroelectric power plant, bicycle. 11. The efficiency must be less than 100%. 13. Don’t invest. Slick’s scheme violates the second law of thermodynamics, because it purports to convert thermal energy entirely into the work needed to drive the automobile. 15. According to the preceding question, there are 1.8 Fahrenheit degrees in each Celsius degree. 10°C is 10 Celsius degrees above freezing, which is 10 * 1.8 = 18 Fahrenheit degrees above freezing, or 32°F + 18°F = 50°F. Similarly, 30°C is 30 * 1.8 = 54 Fahrenheit degrees above freezing, or 32°F + 54°F = 88°F. 17. Efficiency = 300 J>1500 J = 20%. 1200 J are exhausted in each cycle. 19. No. We can push the rock back to its starting condition. The precise process (without the push) cannot be reversed.

198

21. This does not violate the law of increasing entropy,

23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43.

45.

because the egg is not an isolated system; it has outside help in the form of energy (thermal energy that is transferred into the egg). Microscopic disorganization increases, so entropy increases. You should get more useful work, because the efficiency would tend to be higher due to the higher input temperature. Transportation. Assuming an efficiency of 13%, about 13 barrels go into getting the car down the road. Hydroelectric, because it is not a heat engine and so is not subject to the inefficiency implied by the second law of thermodynamics. Electricity from a coal-fired generating plant, because such a plant is about 40% efficient while a car engine is only 10%–15% efficient (Table 1). (15 kg>s) * (3600 s>hour) * (24 hr>day) = 1,300,000 kg>day = 1300 tonnes per day. Between 5 and 6 months: 4% in 1 month, 8% in 2 months, etc. No; yes. Firewood, wind, water behind a dam, ocean thermal. World population is about 6 billion now. 200 years is 5 doubling times, so the population would have increased by a factor of 2 * 2 * 2 * 2 * 2 = 32. So the population would be 32 * 6 billion = 192 billion (of course, this will not happen because the doubling time will not be maintained). No, because part of it is a straight line.

Problems 1. eff = Work output>ThermE input, so ThermE input = Work>eff = 2000 J>0.3 = 6670 J. 3. Input temp = (500 + 273) K = 773 K, eff = (input temp - exhaust temp)>input temp = 420 K>773 K = 54%. 5. The actual efficiency is (work output)>(total energy input) = (1000 J - 750 J)>1000 J = 250 J>1000 J = 25%. 7. From Table 1, actual eff = 0.1. eff = Work out>ThermE in, so Work = ThermE in * eff = (500 * 106 J) * 0.1 = 50 * 106 J. 9. Number of trucks = 16,000 tons>32 tons = 500 trucks. The number of tonne-km is 16,000 tonnes * 5000 km = 8 * 107 tonne-km. By truck, the gallons of gasoline required is 8 * 107 tonne-km>25 tonne-km>liter = 3.2 * 106 liters. By train, the gallons of gasoline required is only 1/4 as much (because it’s four times more efficient, according to Table 5), or 0.8 * 106 liter. 11. To receive 1 kW, a home would need to use all of the solar energy striking an area of 5 m2. But since the conversion efficiency is only 10%, the receiving area would need to be 10 times larger, or 50 m2. A square-shaped collector would need to be about 7 m on a side. 13. T = 70>P = 70>0.8 = 80 yr (U.S.), T = 70>2.2 = 32 yr (Mexico), T = 70>4.2 = 17 yr (Kenya). 15. T = 70>P = 70>7 = 10 yr. So 40 years is 4 doubling times. Centerville’s population will increase by a factor of 16 during this time. It will need 16 treatment plants. 17. At a 4% rate of increase, the number of vehicle miles traveled per year will double in T = 70>4 = 18 yrs. During this time, the number of miles of highway will increase by about (0.1%>yr) * 18 yrs = 1.8%, a very small increase. Thus, congestion will be about twice as bad.

Electromagnetism

One day, Sir, you may tax it. Michael Faraday, Co-Discoverer of Electromagnetism, When asked by the Chancellor of the Exchequer about the Practical Worth of Electricity

I

n this chapter, we are on the trail of light: What is light? How does it behave? How is it related to the basic physical forces such as electricity? Are phenomena such as “X-rays” and “infrared rays” related to light, and if so, how? How do these various kinds of rays affect our planet? Light is all around us, yet it’s not easy to say what light is. How are you able to see this page? Do your eyes emit invisible rays that then move toward the page, as Plato and Euclid thought? Does the page send out or reflect a stream of particles that is received by your eyes, as the Pythagoreans and Isaac Newton thought? Is light a wave, as Newton’s contemporary Christian Huygens thought? The nature of light is one of science’s oldest questions, a question that led to both of the great modern (post-1900) theories, namely relativity theory and quantum theory. Light and other “electromagnetic radiations” are also crucial for understanding social topics like solar energy and global warming. This chapter studies electromagnetism, meaning the combined effects of electric and magnetic forces. It’s a topic that’s essential for understanding light, for understanding modern physics, and for discussing ozone depletion and global warming. Sections 1 and 2 study the electric force and the electric, or planetary, model of the atom. Section 3 looks at electric current and other implications of the electric atom. Section 4 studies electric circuits. In Section 5, you’ll meet a radically new concept: force fields or simply “fields.” “Radical” is not too strong a term, for this text has so far been based on the Newtonian view of a universe made of particles moving mechanically in empty space, and fields are in many ways the opposite of particles. Physicist Michael Faraday first conceived of force fields in 1831 in connection with magnetism, and they soon became essential for understanding electromagnetic phenomena. Today, fields are essential for understanding most of modern physics, especially electromagnetism, gravity, and quantum physics. Section 6 uses fields to present the full electromagnetic force.

From Chapter 8 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

199

Electromagnetism

1 THE ELECTRIC FORCE

Figure 1

Two rubbed transparencies exert electric forces on each other.

Gordon R. Gore Figure 2

A really bad hair day: electric hair. The source of this effect is the “charged” metal sphere.

Find a couple of plastic transparencies that teachers sometimes use on overhead projectors. Rub them vigorously with tissue paper and hold them from one edge in separate hands, parallel and just a few centimeters apart (Figure 1). They should repel each other. The force weakens at larger separations, but careful observation would show that this force still exists even when the transparencies are far apart. Now spread out the tissue on a level surface and hold a rubbed transparency directly above it. The tissue is pulled to the transparency. You may experience similar forces when you take clothes from a dryer or brush your hair. Figure 2 shows an extreme example. This force is a little like gravity: It can act across a distance, and it weakens as the separation widens. But there are reasons why it cannot be gravity: The transparencies repel each other, whereas gravity attracts: this new force is far stronger than gravity could possibly be between relatively low-mass objects such as transparencies and tissues; and this force depends on whether the transparencies are rubbed, and it’s hard to see how rubbing could affect gravity. This force is called the electric force. An “electrified” object such as the rubbed transparency or tissue is said to be electrically charged or just “charged,” and any process that produces this state is called charging. The experiments show that when we electrically charge two identical objects in identical ways, they repel each other. But the charged transparency and the charged tissue attract each other. So the transparency must be charged differently from the tissue. After experimenting with lots of charged objects, one finds that they all fall into just two categories: those that repel the transparency but attract the tissue, and those that attract the transparency but repel the tissue. These two categories are named positive and negative. Don’t attach much significance to the names—they could just as well be called red and blue, or charming and revolting. During the eighteenth century, French physicist Charles Coulomb measured the changes in the electric force between two objects (that is, the force by either object on the other) as he varied the amount of charge on them and the distance between them. He found that the force is proportional to the amount of charge on either object: If you double the charge on the first object, the force doubles; if you triple the charge on the second object, the force triples, etc. This is pretty much what you’d expect. But the way the force varies with distance is more interesting. As the transparency experiment shows, the force gets smaller as the distance gets larger. Working with charged objects that were small compared to the distance between them, Coulomb found that if the distance doubles, the force falls to one-fourth the initial value; if the distance triples, the force falls to one-ninth; if the distance quadruples, the force falls to one-sixteenth. Any guesses as to the relationship between distance and force? ——— Time out, for guessing. We found precisely the same relationship between force and distance when we studied gravitational force. The electric force is proportional to the inverse of the square of the distance, just as is the gravitational force. I’ll summarize all this as

200

Electromagnetism

Coulomb’s Law of the Electric Force Between any two small charged objects there is a force that is repulsive if both objects have positive charge or if both have negative charge, and is attractive if one has positive and the other has negative charge. This force is proportional to the amount of charge on each object, and proportional to the inverse of the square of the distance between them: electric force r

(charge on 1st object) * (charge on 2nd object) square of the distance between them

Using abbreviated symbols, F r

q1 * q2 d2

If electric charge is measured in “coulombs” (see below), distance in meters, and force in newtons, then this proportionality becomes1 F = 9 * 109

q1 * q2 d2

The definition of the measurement unit for electric charge, the coulomb (abbreviated C), was chosen for convenience of use in electrical circuits. It turns out that it’s the total charge of 6.25 billion billion electrons. That sounds like a lot of charge. CONCEPT CHECK 1 Two objects each carry a charge of 1 C. How much force do they exert on each other at a distance of 1 meter? (a) 1 N. (b) 9 N. (c) 9 million N. (d) 9 billion N. (e) about 10–10 N.

Concept Check 1 reveals one way of defining the “coulomb”: It’s the amount of charge that causes an electric force of 9 * 109 N on an identical charge at a distance of 1 m. This is about the weight of 25,000 fully loaded highway trucks! Clearly, 1 C is a lot of charge. Nevertheless, 1 C is only the amount of charge that passes through an ordinary (incandescent) 100-watt lightbulb in a little over a second. This seems paradoxical: Lightbulbs don’t exert forces equal to the weight of all those trucks! The resolution (see Section 3 for more on this) is that the electrons that flow through wires and bulbs are always surrounded by an equal number of stationary protons, so that the wire and the bulb are actually electrically neutral (no net charge) and exert no electrical force on surrounding objects.2 Coulomb’s law refers to two small objects (much smaller than the distance between them) having net charges q1 and q2, unbalanced by other charges of the opposite sign. The charges on most such objects are usually measured in millionths of a coulomb (10 –6 C). Although Coulomb’s law looks a whole lot like Newton’s law of gravity, there are differences. For one thing, the electric force can be either attractive or repulsive, but the gravitational force can be only attractive. “Negative mass”—mass that repels ordinary mass—has never been observed, but both positive and negative electric charge is all around us. 1 2

More precisely, the number on the right-hand side is 8.988 * 109. For reasons described in Section 6, they can, however, exert magnetic forces on surrounding objects.

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Electromagnetism

For another thing, the electric force between any two charged particles is generally vastly larger than the gravitational force between them. For example, the electric force between the electron and the proton in a hydrogen atom is more than 1000 trillion trillion trillion (or 1039) times larger than the gravitational force between them. This is why scientists can ignore the effects of gravity within an atom. Comparison of the huge “proportionality constant” 9 * 109 in Coulomb’s law with the tiny proportionality constant 6.7 * 10 - 11 in Newton’s law of gravity also indicates that, at the macroscopic level in ordinary units, electric forces tend to be much larger. This is why it’s so easy to demonstrate electrical forces between ordinary charged objects (Figures 1 and 2) but challenging to demonstrate gravitational forces between ordinary objects. Finally, the electric and gravitational forces arise from different sources: one from charge, the other from mass. You can, for example, add additional charge to a charged object without appreciably changing its mass, perhaps greatly increasing the electric force on the object but without noticeably changing the gravitational force. So these are fundamentally different forces. Nevertheless, their similarities encouraged Albert Einstein to spend the latter part of his life searching unsuccessfully for a “unified field theory” that would unify these two forces into a single force. More recently we’ve begun to realize Einstein’s dream, but at the microscopic level. We’ve learned that there seem to be four fundamental forces—gravitational, electromagnetic, strong, and weak. Using quantum physics, the weak and electromagnetic forces have been unified and the unification of these two forces with the strong force appears near at hand, but the ultimate unification of these three forces with gravity looks far more challenging.

2 THE ELECTRIC ATOM

The energy produced by the breaking down of the atom is a very poor kind of thing. Anyone who expects a source of power from the transformation of these atoms is talking moonshine. Ernest Rutherford, discoverer of the atomic nucleus, made this famous wrong prediction in 1933.

Although thermal energy, chemical reactions, and much more are comprehensible in terms of the Greek model of the atom as a single tiny rigid object, it’s difficult to fit electromagnetic phenomena into this indivisible-particle picture of the atom. Where does electric charge come from? How can rubbing produce it? Why are there two kinds of charge? These and other questions led, early in the twentieth century, to the planetary model of the atom.3 According to this theory, the atom is not a solid, indivisible object. To the contrary, the planetary model of the atom is almost entirely empty, is divisible, and is made of many parts. As its name implies, the planetary model of the atom resembles a miniature solar system. Figure 3 portrays single atoms of two different elements, helium and carbon. The defining feature of the planetary model of the atom is the tiny nucleus (greatly enlarged in the figure) at the center, surrounded by a number of even tinier electrons (“electrified ones”) that orbit the nucleus at a distance much greater than the size of the nucleus itself. The overall size of an atom, the distance across its electron orbits, is typically 10–10 m. This is roughly the distance between the centers of adjoining atoms in solid materials. But a typical nucleus is 10,000 times smaller, on the order of 10–14 m in size. To put this into perspective, if we built a scaled-up atom in which the nucleus were represented by a soccer ball, the orbiting electrons would be dust specks several kilometers away! An atom is nearly totally empty. 3

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It’s sad but true that the widely used term “planetary model of the atom” is self-contradictory. “A-tom” means indivisible, but “planetary” refers to the parts into which the atom can be divided! The old Greek name, atom, has stuck, but not its essence.

Electromagnetism

You can add or remove charge from macroscopic objects such as transparencies and tissues, but you can’t remove the charge from an electron—it’s permanently negatively charged. Furthermore, to as high a precision as we can measure, all electrons have precisely the same amount of charge and the same mass. The electron’s mass is about 2000 times smaller than the mass of even the least massive atom. Nobody has any idea of why any of this is so. The nucleus is itself made of several subatomic particles of two types, called protons and neutrons. A proton (“positive one”) is another permanently charged particle, charged precisely as strongly as the electron, but positively instead of negatively. When we say that electrons and protons are charged “equally strongly,” we mean that when they are placed near some other charged object, both exert the same amount of force (but in opposite directions) at the same distance away. But they don’t have equal masses. The proton is about 2000 times more massive than an electron. The neutron (“neutral one”) is an uncharged, or neutral, particle whose mass is nearly the same as the proton’s mass. Between one and a few hundred protons and neutrons form the nucleus of any atom. The “glue” that holds electrons into their orbits around the nucleus is the electric attraction between electrons and protons. The glue that holds the nucleus together, however, must be some nonelectric force, because the electric force between the positively charged protons is repulsive and neutrons do not exert an electric force. Since an atom’s electrons are relatively distant from the nucleus, it’s not surprising to learn that it’s easy to remove electrons from atoms. But on Earth, most atoms have just as many electrons as protons, because any atom having a deficiency of electrons carries a net positive charge and quickly attracts electrons from its environment, while any atom having an excess of electrons quickly loses its outermost electrons to its environment. That’s why it took so long to discover many electrical phenomena: Individual atoms are normally electrically neutral and exhibit no obvious electrical effects. An atom that does have an excess or deficiency of electrons is called an ion.

How do we know that electrons exist? In 1897, English physicist J. J. Thomson (Figure 4) was investigating a type of invisible beam known as a cathode ray. Cathode rays were produced in a nearly evacuated (emptied of air) glass tube whose two ends were attached by metal wires to a source of electric power. When the power was switched on, rays of unknown composition streamed along the length of the tube, as could be observed by the flashes of light where they hit one end of the tube. Suspecting that these rays were electrically charged, Thomson placed electric charges and magnets around them. The flashes of light shifted in position, which meant that the rays had to be charged. The only charged microscopic objects then known were ions, observed in certain chemical experiments. So Thomson hypothesized that the electrically charged cathode rays were streams of such ions. He then measured the deflections of the rays. Using the known electric and magnetic force laws, he deduced from these measurements that these rays were streams of charged particles whose charge was the same as the charge of typical ions but whose mass was far smaller, only about 1/2000th of an ion’s mass.4 These, then, were not ions. This was revolutionary. It established that atoms had parts. According to Thomson, “At first there were very few who believed in the existence of these bodies smaller than atoms.... It was only after I was convinced that the experiments left no escape from it that

Helium

Carbon

Figure 3

Two examples of the planetary model of the atom. Protons are green, neutrons are white, and electrons are black. The diagrams are not drawn to scale! If they were drawn to scale, the nuclei and the electrons would be too small to be seen.

American Institute of Physics/ Emilio Segre Visual Archives Figure 4

4

More precisely, he found that the ratio of the mass to the charge was 1/2000th of the ratio of mass to charge for any known ion.

J.J. Thomson, discoverer of the electron. He was the first to show that atoms are made of smaller, electrically charged parts.

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Electromagnetism I published my belief in the existence of bodies smaller than atoms.” Thomson had discovered the electron. Today, Thomson’s rays, now known as electron beams, are used in TV tubes, fluorescent bulbs, computer screens, and much more.

CONCEPT CHECK 2 Which one of the following has the smallest mass, and which one has the largest mass? (a) Proton. (b) Electron. (c) Helium nucleus. (d) Neutron. (e) Water molecule. (f) Oxygen atom.

C.E.Wynn-Williams/American Institute of Physics/Emilio Segre Visual Archives Figure 5

Ernest Rutherford, codiscoverer of the nucleus and one of the greatest experimental physicists of the twentieth century, talks with a colleague. Because Rutherford’s resounding voice could upset delicate experimental apparatus, the sign overhead was playfully aimed at him. His research on radioactivity and nuclear physics influenced a generation of experimental physicists early in the twentieth century. Not exist—not exist! Why I can see the little beggars there in front of me as plainly as I can see that spoon! Rutherford, When Asked over a Dinner Table Whether He Believed That Atomic Nuclei Really Existed

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How do we know that every atom has a nucleus? Like others around 1910, New Zealander Ernest Rutherford (Figure 5) was trying to determine the atom’s internal structure. He knew that atoms contained electrons, so a positive charge must be present too. It was known that atoms are pressed right up against one another in solid materials and that huge forces are required to compress solids into smaller volumes. This suggested to scientists that atoms must be filled with matter throughout most of their volume. To test this hypothesis, Rutherford used what has become a traditional physics technique: He threw tiny things at other tiny things in order to see what would happen. He “threw” a recently discovered ray known as an alpha ray at the atoms residing in a thin metal foil, similar to aluminum foil. The alpha ray was a high-energy stream of positively charged and fairly massive “alpha particles” (helium nuclei, made of two protons and two neutrons) that emerged from certain “radioactive” substances. The idea was to observe how far the foil deflected the fast-moving alpha particles from their original directions and, from this, to deduce how matter must be distributed within the foil’s atoms (Figure 6). The deflection was measured by observing flashes of light where the alpha particles hit a screen placed partially around the foil, as shown in the figure. Similar experiments had been done before, and it had been found that the foil had surprisingly little effect on the motion of the alpha particles. Most deflections were less than 1 angular degree, as shown in the two “magnified” portions of the diagram. Since the foils were about 500 atoms thick, alpha particles apparently passed straight through most atoms without deflection. Apparently, atoms were fairly porous, open structures. Then in 1911, Rutherford decided to see whether any alpha particles were deflected through very large angles, greater than 90 degrees. He studied this by nearly surrounding the foil with the detection screen. He expected to see no large deflections, because a fast-moving and massive alpha particle was thought to pass through an atom like a cannonball through pudding and would have to experience an enormous force to be deflected by very much. His coworkers came to Rutherford a few days later with the news that a few alpha particles had been deflected backward. In Rutherford’s words, “It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch [artillery] shell at a piece of tissue paper and it came back and hit you.” Apparently, nearly all of the atom’s material was concentrated at its center. The cannonball had struck another cannonball and bounced back. Rutherford and his colleagues had discovered the atomic nucleus.

CONCEPT CHECK 3 Which of the following objects has an overall, or “net,” charge? (a) Proton. (b) Electron. (c) Helium nucleus. (d) Neutron. (e) Ionized oxygen atom. (f) Nonionized hydrogen atom.

3 ELECTRIC CURRENT AND OTHER APPLICATIONS OF THE ELECTRIC ATOM Like all good theories, the planetary model of the atom explains many things. It explains our experiments with charged objects. When you rub a transparency with tissue, some of the loosely attached outermost electrons in the transparency’s

Electromagnetism

Metal foil (target) Part of metal foil, magnified Alpha ray

Source of alpha ray Receiving screen to detect alpha particles

An occasional alpha particle is bounced back by a close encounter with a nucleus

Single atom, magnified

Most alpha particles pass straight through

Figure 6

Rutherford’s alpha-scattering experiment.

atoms are rubbed off and transferred to the tissue’s atoms, charging the transparency positively and the tissue negatively. That’s why the two attract each other after rubbing. The two rubbed transparencies repel each other because both have a deficiency of electrons. The planetary model provides a microscopic explanation of many chemical phenomena, including why all atoms of a single element have identical chemical behaviors and why different elements are chemically different. An element’s chemical behavior results from the behavior of its orbital electrons. Table salt, NaCl, makes a good example. Sodium (Na) combines readily with chlorine (Cl) because the Cl atom has a stronger attraction for electrons than does Na, so one electron from Na is attracted to Cl, leaving an overall positive charge on the Na and a negative charge on the Cl. The electric force between opposite charges then attracts and holds the Na and the Cl atoms together. The property that really defines an element is the number of protons in the atomic nuclei of that element, because the number of electrons in the neutral (nonionized) atom must equal the number of protons, and the number of electrons in turn determines the chemical properties of the atom. Two neutral atoms with the same number of electrons have the same chemical properties because their electron orbits have the same shapes. So it makes sense to number the different elements according to the number of protons in an atom of the element. This number is called the element’s atomic number.

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Since an atom is mostly empty space, what keeps solid matter solid? The answer is that the repulsion of electrons by other electrons keeps atoms from penetrating one another. All contact forces, such as your hand slapping a table, can be interpreted microscopically as forces by the orbital electrons in atoms (in your hand, for instance) on the orbital electrons in other atoms (in the table). Every time you touch something, you experience the force between orbiting electrons! In fact, all the forces in your daily environment come down to just two kinds: The gravitational force explains weight, while the electromagnetic force explains all the contact forces and also the forces between charged objects and, as we’ll see, magnetized objects. That’s some unification! When you electrically charge an object, electrons (or it could be protons, which are also moveable in some situations) are merely moved onto or off of the object. No electrons are created or destroyed. Charge is “conserved.” In fact, careful measurements have verified to high precision that the net (positives minus negatives) amount of charge is conserved in every process. Even in high-energy physics processes where charged particles such as electrons are actually created rather than merely transferred, equal numbers of oppositely-charged particles such as “antielectrons” are always created. This is another conservation law, similar to, and as fundamentally important as, conservation of momentum and conservation of energy: The Law of Conservation of Charge Although charge can be moved around and although charged particles can be created or destroyed, no net charge (positives minus negatives) can be created or destroyed.

Nobody has ever figured out how to split an electron into parts. So when you charge an object, it always gains or loses some whole number multiple of the electron’s charge. An object can have a charge of 1 electron, 2 electrons, 3 electrons, etc., but it cannot have a charge of 1.6 or 2.9 electrons. We call this quantization of charge, with the smallest quantity or “quantum” of charge being the electron’s (or proton’s) charge.5 Another way to say this is that charge is a discontinuous (computer buffs might say “digital”) quantity rather than a continuous (or “analogue”) quantity: It increases or decreases in steps that are whole number multiples of the electron’s charge. This is our first example of a quantized aspect of nature. The planetary atom underlies many electrical devices, such as batteries. Any battery has two “terminals,” one positively charged and the other negatively charged. Chemical processes within the battery maintain these charges by removing electrons from one terminal and depositing them on the other. If we attach the two ends of a single copper (or other metal) wire to the terminals, every charged particle in the wire instantly feels an electric force. These forces produce practically no disturbance of the positively charged copper nuclei, which are fixed in position. But in copper or any other metal, each atom’s outermost electrons are only loosely held to

5

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We’ve discovered that protons are made of smaller particles called “quarks,” and quarks carry charges having one-third and two-thirds the magnitude of the electron’s charge. However, quarks have never been found in isolation—they’re always combined into particles such as the proton carrying a whole number multiple of the electron’s charge.

Electromagnetism

their parent atom. As soon as these conduction electrons feel the electric forces established by a battery, they all simultaneously begin to move along the wire, through the body of the metal. As they move, they constantly bump into atoms, slowing the electrons and causing the atoms to vibrate, warming the wire. Such a flow of charged particles is called an electric current (Figure 7). Such a battery with a simple wire attached to the terminals forms an electric circuit: a closed loop around which electric current can flow. But try the experiment shown in Figure 7 carefully and only with a small flashlight battery. The problem is that electrons flow so easily through ordinary wires that the electric current is large, so the warming effect described above can burn out either the battery or a portion of the wire. Materials such as copper and other metals through which electric current can easily flow are called electrical conductors; they have atoms whose outermost electrons are only loosely attached. Materials such as rubber and dry wood whose outermost electrons are more firmly attached don’t permit the easy flow of electric current and are called insulators. If, as shown in Figure 8, we insert a lightbulb into the circuit of Figure 7, electrons will flow through the bulb’s thin filament and heat it by simply bumping into atoms, causing them to vibrate energetically. As compared with the circuit of Figure 7, the insertion of the narrow filament causes the flow of current in the circuit to decrease, for the same reason that the flow of water in a garden hose is reduced if you squeeze the hose at one point to make the cross-sectional area smaller: Conduction electrons can’t get through the narrow part (the filament) in such large numbers. We say that the filament creates electrical resistance in the circuit, by which we mean that the filament reduces the overall flow of current throughout the entire circuit. This is why you can easily burn out the battery of a circuit like Figure 7, while a circuit like Figure 8 doesn’t burn out quickly. The filament, on the other hand, is made of a heat-resistant material such as tungsten and is designed to heat up until it glows. It heats up because conduction electrons move faster through the thin filament than through the fatter wire, much as water “spouts” more rapidly out Figure 7 Short length of wire, magnified Electrons move through wire

–

+

Wire loop

A battery produces electrical forces that cause conduction electrons (black dots in the magnified view) to move through the volume of the metal wire (the green circles are the wire’s atoms). Electrons flow around the wire circuit, repelled by the battery’s negative terminal and attracted toward the positive terminal. Chemical forces within the battery then push the electrons through the battery from the positive electrode back to the negative electrode. Note that electrons within the battery feel electric forces toward the positive terminal, while leftward chemical forces push them toward the negative terminal.

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Electromagnetism Figure 8

Because electrons flow so easily through ordinary wires, the circuit of Figure 7 would soon burn out the wire or battery. Inserting an incandescent lightbulb or other “circuit element” reduces the flow to safe levels. This happens because the lightbulb’s glowing filament is very thin and thus restricts or “resists” the current, much as a squeezed garden hose restricts the flow of water.

– +

–

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of a garden hose when you narrow the nozzle. These faster electrons collide more violently with the atoms of the filament, heating the filament to high temperatures. The average forward speed of electrons in typical electrical circuits is surprisingly slow because they constantly run into atoms and bounce in all directions, enormously slowing their forward motion. A conduction electron’s average forward speed along a wire, called its drift velocity, is typically less than one millimeter per second! You might wonder, then, how the lightbulb in your room lights up so quickly after the switch is turned on, since it’s at least a few meters from the switch to the bulb and a conduction electron would take about an hour to cross this distance. The answer is that the electric force is not transmitted by the motions of the conduction electrons. Instead, the light switch connects a wire containing the bulb’s filament to a power source such as a battery, and electric forces and energy spreading outward from the power source at the speed of light exert forces on all the electrons in the circuit, causing them all to move at practically the same instant. All the standard electrical appliances—toasters, lightbulbs, electric motors, and so forth—are based on a similar flow of the electrons that ordinarily orbit within atoms, and they all create electrical resistance in the circuit that powers them. Figure 8 shows how the process works when a battery causes the electric force. Electrical outlets in your home work in a similar way, except that the current reverses its direction of flow many times every second (Figure 9). Such a current is called alternating current or AC, while current that always flows in the same direction is called direct current or DC. In any battery, one terminal remains negative and the other remains positive because the chemical reactions inside continue operating in the same direction, so batteries produce DC. Although devices exist that can transform DC into AC and vice-versa, AC typically arises from an entirely different process. This different process is the rotation of a loop of wire in a magnetic field. Commercial electric power plants in the United States use a rotational frequency of 60 cycles per second. So when you plug your lamp into a wall outlet, the conduction electrons in the electric cord and in the lamp instantly (well, with a slight delay determined by the speed of light) begin moving back and forth 60 times per second, moving forward and then backward at a drift speed of less than 1 mm per second. Think of electrons jiggling back and forth over tiny distances while bouncing rapidly in all directions.

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Electromagnetism Figure 9

Electrical appliances are based on the motion of unseen electrons that move back and forth in the appliances. A wall socket (more precisely, a generating station connected by wires to a wall socket) is the electrical equivalent of a water pump. It “pumps” (energizes) the electrons so that they can move back and forth in the circuit.

Wall socket

When the plug is plugged in, electrons flow first in the direction of the black arrows, and then in the Plug direction of the green arrows, reversing directions many times every second. Normally, the two wires shown stretching from the plug to the bulb are both placed inside a single electrical cord.

4 ELECTRIC CIRCUITS Electricity, in the form of electrons flowing through all sorts of devices, is so pervasive today that it’s a defining feature of modern life. Many poorer regions of the planet have little or no electricity and so live very different lives from you and me. Think of how your life—the past 24 hours for example—would be different if you’d lived two centuries ago when there was no electricity! To understand in more detail how electricity works, look again at Figure 8. This simple electric circuit has three essential electrical elements: a battery, a metal wire that conducts electric current, and a light bulb. The battery moves electrons from the positive to the negative terminal by means of chemical reactions that we won’t further delve into here. Even without the wire, an isolated battery has electrons piled on the negative terminal and excess protons (a deficiency of electrons, really) on the positive terminal. Since there’s nowhere for these electrons to go, nothing happens: The battery just sits there with charge piled on both terminals and no current flowing. But once you attach the wire and bulb, electrons have someplace to go: from the negative terminal onto the wire, and from the wire onto the positive terminal. As described in the previous section, conduction electrons immediately start flowing everywhere in the circuit. Let’s look at this in terms of energy. With the disconnected battery (no wire), excess electrons are more or less at rest on the negative terminal. Nevertheless, they have energy. The reason is that energy is the capacity to do work, and these electrons could do work if you connected the external circuit to the terminals and allowed the electrons on the terminal to flow through the circuit. For this circuit, the work would be done within the lightbulb filament as electrons heat the filament. What do you suppose we call the form of energy that the electrons have when they’re at rest on the negative terminal? ——— (Pause, for supposing)

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The answer is electrical (or electromagnetic) energy; it could be defined as “the energy that an electrically charged object has due to electrical (or electromagnetic) forces on it.” A useful analogy: The electrical energy of an electron on the negative terminal is analogous to the gravitational energy of a rock at the top of a hill. Just as the rock can give up its gravitational energy (while creating kinetic and thermal energy) by sliding down the hill, the electron can give up its electrical energy (while creating radiant and thermal energy) by flowing around the external circuit (the wire and bulb) to the positive terminal. Electrons arriving at the positive terminal are like the rock arriving at the bottom of the hill. They’ve lost their electrical energy. The purpose of the battery is then to “pick up” these electrons from the positive terminal and push them back “up” to the negative terminal, against the electrical attraction of the positive terminal and the repulsion of the negative terminal, so that they can again flow around the external circuit. The battery re-energizes the electrons. Quantitatively, a battery’s voltage is a measure of the amount of energy it gives to each electron. More precisely, a battery’s voltage is the number of joules of electric energy that the battery provides to each coulomb of electrons (some 1019 electrons) flowing through the battery. More generally, the voltage between (the official term is “across”) any two points A and B along an electrical circuit is the amount of electrical energy that a coulomb of charge would lose (or gain) in moving from A to B. For example, the voltage across a lightbulb is the number of joules of electrical energy lost by one coulomb of electrons in flowing from the negative side to the positive side of the bulb (Figure 8). Note that electrons lose energy in flowing “downhill” across a circuit element such as a lightbulb, and they gain energy when they are pushed “uphill” across a battery. From its definition, voltage is measured in joules/coulomb. But we have an abbreviation for this: the joule/coulomb is called—you guessed it (perhaps)—the volt. CONCEPT CHECK 4 As electrons move around the circuit of Figure 8, the energy transformations first in the battery and then in the bulb are (a) ChemE ¡ ElectE and then ThermE ¡ ElectE; (b) ElectE ¡ ChemE and then ThermE ¡ ElectE; (c) ChemE ¡ ElectE and then ElectE ¡ ThermE + RadE; (d) ElectE ¡ ChemE and then ElectE ¡ ThermE + RadE; (e) ThermE ¡ ElectE and then ElectE ¡ ThermE + RadE. CONCEPT CHECK 5 Suppose the battery of Figure 8 has a voltage of 2 volts, and that the battery sends 3 coulombs of electrons from the negative terminal and through the bulb to the positive terminal. The energy of these 3 coulombs of electrons as they leave the negative terminal, as compared with their energy upon arriving at the positive terminal, is (a) –6 joules; (b) +2 joules; (c) +6 volts; (d) +2 volts; (e) +6 joules.

Now let’s look more closely at the wire carrying current. I described this process microscopically in the previous section. The amount of current flowing in the wire could be quantitatively measured in electrons per second flowing into, out of, or across any cross-section of, the wire. This would be a huge number in most circuits, something like 1019 electrons per second. A more practical (because it’s a smaller, less cumbersome number) measure is the number of coulombs flowing, measured in coulombs/second. There’s an abbreviation for the coulomb/second: It’s called the ampere, or amp. The amount of current flowing out of the battery, into the bulb, out

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of the bulb, or in fact across any cross-section of the wire, filament, or battery, is the same everywhere along the circuit of either Figure 7 or 8. If this weren’t true, for instance if there were more electrons (per second) coming out of the small magnified segment of wire in Figure 7 than were going into it, then electrons would have to be created inside the small segment, which would violate conservation of charge. Finally, let’s look at the bulb. I explained in the previous section that the bulb’s filament slows or “resists” the flow of electrons mainly because the filament is narrow. The filament’s length and the material of which it’s made also help determine its resistance to the flow of electrons. Unsurprisingly, a battery of higher voltage causes a larger electric current to flow in the circuit of Figure 8 because the higher voltage provides more energy to each electron. But how much larger? Measurements show that, in most circuits, the current through any circuit element is proportional to the voltage across the element. For instance, if you double the voltage of the battery in Figure 8, you’ll double the current through the bulb. This proportionality is called Ohm’s law, although it’s just a practical rule that holds for most circuit elements in most circuits rather than a basic physical law comparable to Newton’s law or conservation of momentum. Maybe we should call it “Ohm’s rule of thumb.” Ohm’s law says that current is proportional to voltage. Equivalently, voltage must be proportional to current: voltage r current, or voltage = R * current where R is some fixed number for any particular circuit element. The standard symbols for voltage and current are V and I in which case V = RI or, equivalently, V = IR. Summing up: Ohm’s Law V = IR where V is the voltage across any circuit element and I is the current through it. Although V and I may vary, R is a fixed number for any particular circuit element. R is called the circuit element’s resistance.

“Resistance” is the right word for R because small R means that only a small voltage is needed get 1 amp (say) of current to flow in the element, while large R means a large voltage is needed to get 1 amp to flow. According to Ohm’s law, R is measured in volts/ampere. But we have an abbreviation for this combination, called the ohm. V = IR can be rearranged into two other useful forms: I = V>R,

and R = V>I

Example: Returning to the circuit of Figure 8, a resistance of 200 ohms is typical for an incandescent bulb filament. In this case, a 10-volt battery would create an electric current of I = V>R = 10 volt > 200 ohm = 0.05 amp. Every wire has a certain amount of electrical resistance. For example, the resistance of a 1-meter length of copper wire having a cross-sectional diameter of 1 millimeter is about 0.02 ohms. As we saw in the preceding section, a thinner wire (such

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as a lightbulb filament) would have a larger resistance. A longer wire should also have a larger resistance because the conduction electrons bump into more atoms as they more down a longer wire as compared with a shorter wire. So it’s not surprising that the resistance of 100 m of 1 mm diameter copper wire is 100 times the resistance of a 1 m length, or 2 ohms. From the preceding two paragraphs, you can see that in a circuit like Figure 8, each of the two strands of wire usually have a far smaller resistance than the circuit element. From an energy point of view, as electrons pass through the external circuit they lose a little energy in each strand of wire, but they lose most of their energy in the circuit element. CONCEPT CHECK 6 Suppose a 1 m strand of 1 mm diameter copper wire is attached directly from the negative to the positive terminal of a 6-volt battery, as in Figure 7. The current in the wire is then (a) 3000 amp; (b) 300 amp; (c) 30 amp; (d) 3 amp; (e) 0.3 amp.

Don’t try this at home. The current found in concept check 6 is large enough to raise the copper wire to its melting point within a few seconds! This is why you don’t want to attach a low resistance device such as a simple strand of wire across a battery, or worse yet (because the voltage is larger) a wall outlet. Such a situation is called a short circuit.

Fields are conditions of space itself, considered apart from any matter that may be in it. Fields can change from moment to moment and from point to point in space, in something like the way that temperature and wind velocity are conditions of the air that can vary with time and position in the atmosphere.... In the modern theory of elementary particles known as the Standard Model, a theory that has been well verified experimentally, the fundamental components of nature are a few dozen different kinds of fields. Steven Weinberg, Co-Inventor of the Standard Model of Elementary Particles

5 FORCE FIELDS: A DISTURBANCE OF SPACE To some people, it seems unbelievable that Earth could exert a force on the moon across 400,000 km of nearly empty space. After all, when you push a box across the floor, you exert a force on the box by actually touching or “contacting” it. How might you push a box without contacting it? Well, you could put another box between your hand and the box you want to push, and push on this other box (Figure 10). But it’s hard to see how you could push on box 2 in the figure without box 1 or something else to fill the space between your hands and box 2. By the same token, many nineteenth-century physicists felt that if Earth exerts a gravitational force on the moon, then something must fill the space between Earth and the moon to transmit the force. They called this “something” a gravitational field. It’s not really a “thing” in any ordinary sense—not material, not made of atoms. It’s one example of a “force field” (or simply “field”), a concept that lies at the heart of most new fundamental physics since 1900. Although physicists invented this idea in order to understand forces like gravity that act across empty space, we’ll discover that force fields are not imaginary but are physically quite real.

Figure 10

Could you push on box 2 from some distance away without having box 1 or some other object between you and box 2?

2 1

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Earth’s gravitational field fills the space around Earth, out to far beyond the moon. Think of this and other force fields as the effect that the source of a force (Earth, in this case) has on the surrounding space: not on the things in the space, but on the space itself. Think of a field as a distortion or disturbance of space. Earth’s gravity disturbs the space between Earth and the moon, and this disturbance—this gravitational field—pulls on the moon. So the gravitational field transmits Earth’s gravitational force to the moon. Earth’s gravitational field exists everywhere around Earth, even in places where there is nothing to feel any gravitational force. It would exist even if the moon weren’t there. The field fills the surrounding space in the way that smoke can fill a room. Earth’s gravitational field exists everywhere that a material object would feel Earth’s gravitational pull if such a material object were present. More generally: A gravitational field exists throughout any region of space where an object would feel a gravitational force if an object were placed there. Please ponder that sentence. A gravitational field is “the possibility of a gravitational force.” Every material object can exert gravitational forces and so is surrounded by its own gravitational field. We can speak of the gravitational field of the sun, the moon, a rock, your body, and so forth. Strictly speaking, each object’s gravitational field fills all space, but from a practical point of view, most fields can be neglected at large distances from the objects that create them. This force field concept applies to every force that can act across empty space, so it applies to the electric force. Just as a gravitational field surrounds every object possessing mass, an electric field surrounds every charged object. An electric field exists wherever any other charged object would (if it were present) feel an electric force; it is “the possibility of an electric force.” It fills the space between two or more separate charged objects and causes them to exert forces on each other. Force fields might seem abstract at first, but they are a simple and natural concept. Every mass, and every electric charge, is surrounded by a force field that you can’t see but that you can feel when the field exerts a force on other masses or other charges. For instance, an electrically charged transparency creates an electric field around itself, and this field exists even in the absence of other material objects around the transparency. Even if the transparency is isolated in outer space, it still creates an electric field throughout the space around it. You can demonstrate the existence of this electric field by holding a second charged transparency in the vicinity of the first transparency and finding that this second transparency feels an electric force. To visualize fields, we represent them by “field lines.” For an electric field, the field lines are in the direction of the force that would be exerted on a positive charge. For two examples, Figure 11(a) and (b) show some of the electric field lines surrounding a small positive charge and a small negative charge, respectively. In Figure 11(b), the lines point inward toward the negative charge since this is the direction of the force that would be exerted on a positive charge placed anywhere near the negative charge. For two more examples, Figure 12(a) shows some of the electric field lines surrounding two small equal (the same number of coulombs) but opposite (one positive and one negative) charges, while Figure 12(b) shows the electric fields lines surrounding two small equal positive charges. CONCEPT CHECK 7 The direction of the force on a positive charge placed at point A in Figure 11 would be (a) upward (toward the top of the page); (b) downward; (c) leftward; (d) rightward. What would be the direction of the force on a negative charge placed at point A?

⫹

A

(a) B ⫺

(b)

Figure 11

Electric field lines (a) near a small positive charge and (b) near a small negative charge.

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Electromagnetism Figure 12

C

Electric field lines surrounding (a) two small equal but opposite charges and (b) two small equal positive charges.

D ⫹

⫺

(a)

⫹

⫹

(b)

CONCEPT CHECK 8 The direction of the force on a negative charge placed at point B in Figure 11 would be (a) upward; (b) downward; (c) leftward; (d) rightward. CONCEPT CHECK 9 The direction of the force on a positive charge placed at point C in Figure 12 would be be (a) upward; (b) downward; (c) leftward; (d) rightward. What if the positive charge were instead placed at point D?

During the nineteenth century, scientists began using electric fields and also magnetic fields to help them understand and visualize electric and magnetic forces. In fact, it’s possible to state the basic laws of electricity and magnetism entirely in terms of force fields. Written in terms of the electric field, but leaving out the quantitative details, we can restate the electric force law (Coulomb’s law) this way: The Electric Force Law, Stated in Terms of Fields6 An electric field surrounds every charged object. Furthermore, any charged object that happens to be located at a point in space where an electric field exists will feel an electric force due to that field. Briefly, charged objects create electric fields and feel forces due to the electric fields of other objects.

6 ELECTROMAGNETISM Have you ever played with magnets? Everyone should have the opportunity to experience these intriguing toys. You can buy some at a toy store. If you bring two bar 6

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If you really want to know the quantitative details, here they are: The “electric field strength” at a distance d from a small charge q1 is E = (9 × 109) q1/d2, and the force on any charge q 2 placed at any point in an electric field E is F = q2 E. Putting the two formulas together, we recover Coulomb’s law.

Electromagnetism

magnets near each other, you will discover that their ends either attract or repel each other even when they are not touching. The ends of a magnet are called its north and south magnetic poles. Experiment shows that similar poles repel each other and dissimilar poles attract each other. This reminds us of the forces between electric charges: Likes repel, and unlikes attract. It’s plausible that the force acting between magnets actually is the electric force. This hypothesis is easy to check, for it predicts that magnets should exert forces on electrically charged objects such as a rubbed transparency or tissue. If you try this, you’ll find that the magnets do not exert forces on a rubbed transparency or tissue.7 So our hypothesis is false. The force between bar magnets is not the electric force, and the two ends of a magnet are not electrically charged. There are other big differences between magnetism and electricity. First, a bar magnet’s magnetism is permanent and has nothing to do with rubbing. Second, every magnet has both a north and a south pole. Nobody has found an object that possessed either kind of pole without the other kind, despite serious searches for such “monopoles.” On the other hand, it’s easy to find objects such as a rubbed transparency that possess only one kind of electric charge. We call this new type of force the magnetic force. The similarities between the electric and magnetic force suggest that they might be related. One of the great triumphs of nineteenth-century physics was the demonstration that this is in fact the case. The most concrete evidence was an experiment, first conducted in 1820, in which electrically charged particles that were in motion exerted a measurable force on a small magnet. Note that it is only moving charged objects that can exert forces on magnets. As we have seen, stationary charged objects do not exert forces on magnets. Further experiments during the nineteenth century showed that all magnetic forces can be traced to the motion of charged objects. Moving charged objects exert and feel magnetic forces over and above whatever purely electric forces they would feel if they were at rest. This additional force, due to the motion, is the magnetic force. This means that the separate concept of magnetic poles is not needed, so we can drop the idea of magnetic poles and just think of moving charges instead. For example, Earth’s magnetic effects are due to electrically charged material flowing within Earth. This is a good thing, because it permits the operation of magnetic compasses so that Boy Scouts and Girl Scouts can find each other. But if all magnetic forces can be traced to the motion of charged objects, where are the moving charges responsible for permanent magnets? The answer is that the moving charges are found at the subatomic level, in both the orbiting and spinning motions of electrons in atoms. Because of these motions, each electron in a bar magnet exerts its own tiny magnetic force on each electron in another bar magnet. In most materials, these tiny magnetic forces cancel one another because all the electron motions have different orientations. But the treatment of the iron when a magnet is manufactured locks many electron orbits into similar orientations, causing the many microscopic magnetic forces to add up to a large macroscopic effect. Permanent magnets can temporarily magnetize objects such as nails by forcing many of the nail’s electrons to orient themselves in similar directions. This is why non-magnetized nails are attracted to permanent magnets. 7

A small attractive force is sometimes obtained with the transparency, due to an electric effect called “electrical polarization.” This force occurs equally strongly even if an unmagnetized piece of metal is used in place of the magnet, so it is not caused by the magnetic poles. Rather, it is caused by redistribution of the highly mobile electrons within a metal when a charged object is brought near it. A similar electrical polarization occurs quite dramatically when a charged transparency is brought near an empty aluminum can that is free to roll. Try this!

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Electromagnetism

Summarizing: The Magnetic Force Law Charged objects that are moving exert and feel an additional force beyond the electric force that exists when they are at rest. This additional force is called the magnetic force. All magnetic forces are caused by the motion of charged objects.

How do we know that moving charges act like magnets? With the help of a D cell, a short length of insulated copper wire, and a sensitive magnetic compass with a well-balanced needle, you can demonstrate the magnetic force law. When the compass is placed on a level surface and the wire connected to the poles of the battery in such a way that the wire runs north-south (parallel to the needle) and is above or below the compass, the needle should rotate to a new position. Explanation: The battery causes electrons to flow along the wire, and the flowing electrons exert a magnetic force on the magnet. Caution: Connect the wire for only a few seconds to prevent the battery from quickly burning out. Use insulated wire because the wire might get hot.

CONCEPT CHECK 10 The force between two bar magnets cannot be due to gravity because (a) it’s far too strong to be caused by gravity; (b) it’s far too weak to be caused by gravity; (c) it can be attractive, while gravity is always repulsive; (d) it can be repulsive, while gravity is always attractive; (e) gravity acts even over large distances, while magnetism acts only over short distances. CONCEPT CHECK 11 If you rub a transparency with a tissue to charge both objects and then hold them at rest several meters apart, the forces they exert on each other will be (a) zero; (b) electrical repulsion only; (c) electrical attraction only; (d) electrical repulsion, plus a magnetic force; (e) electrical attraction, plus a magnetic force; (f ) a magnetic force only. CONCEPT CHECK 12 If you saw off one end of a magnet, you will have (a) two nonmagnetized pieces of metal; (b) two magnets; (c) two magnets that have magnetic poles on only one end; (d) none of the above.

N

S

Figure 13

A representation of the magnetic field of a bar magnet.

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Just as charged objects create electric fields in their vicinity, moving charged objects create magnetic fields; they also feel forces arising from the magnetic fields created by other moving charged objects. Like electric fields, magnetic fields can exist in a vacuum—in a region of space that contains no material particles. Just as an electric field exists wherever any charged object would (if it were present) feel an electric force, a magnetic field exists wherever any moving charged object would (if it were present) feel a magnetic force. Like electric fields, we can visually represent magnetic fields by “magnetic field lines.” For example, Figure 13 shows the magnetic field lines in the vicinity of a bar magnet. They point in the direction along which a small compass needle would orient itself. Just as electric field lines point outward from positive charges and inward toward negative charges (Figure 11), magnetic field lines point outward from north poles and inward toward south poles (Figure 13). Figure 14 is an experimental demonstration that a bar magnet’s field actually does have the shape drawn in Figure 13. Figure 14 is made by sprinkling small iron filings in the vicinity of a bar magnet. The long slender filings are temporarily

Electromagnetism

magnetized by the magnetic field, and the magnetic forces on their north and south poles then cause the filings to line up parallel to the field of the bar magnet. Summarizing: The Magnetic Force Law, Stated in Terms of Fields A magnetic field surrounds every moving charged object. Furthermore, any moving charged object that happens to be located at a point in space where a magnetic field exists will feel a magnetic force due to that field. Briefly, moving charged objects create magnetic fields and feel forces due to the magnetic fields of other moving charged objects.

As you’ve seen, electric and magnetic forces both arise from electric charge, so we think of them as different aspects of a single electromagnetic force. Similarly, both electric and magnetic fields arise from electric charge; we think of them as different aspects of a single electromagnetic field. English physicist Michael Faraday (Figure 15) was the first scientist to take electromagnetic fields seriously. Partly because of his ability to visualize electromagnetic phenomena in terms of fields, he was one of history’s greatest experimental scientists. During the mid-nineteenth century, he carried out a wide variety of experiments involving electromagnetic forces and fields. In the course of these experiments, he investigated what happens when a magnet is brought close to a simple circular loop of wire. He found that nothing at all happens so long as both the magnet and the wire loop are stationary. But when he moved either the wire loop or the magnet, an electric current was created in the wire. As soon as the motion of the wire loop or the magnet ceased, the electric current ceased. Apparently, the moving magnet exerted a force on the electrons within the wire loop, even in the case when the wire loop was stationary. Visualizing this in terms of fields, Faraday realized that moving either the loop or the magnet caused the magnetic field in the vicinity of the wire loop to change and that this changing magnetic field must in turn create an electric field in the vicinity of the loop because it takes an electric field to cause electrons to begin flowing in a stationary metal wire. This was something new. Today it is called

Figure 14

An experimental demonstration, using iron filings, that the field of a bar magnet actually does have the shape drawn in Figure 13.

Faraday’s Law When a wire loop is placed in the vicinity of a magnet and when either the loop or the magnet is moved, an electric current is created within the loop for as long as the motion continues. Stated in terms of fields: A changing magnetic field creates an electric field.

Like the electric and magnetic force laws, Faraday’s law has enormous social consequences. It is behind large-scale electric power generation in steam–electric (fossil-fueled or nuclear-fueled) power plants, hydroelectric power plants, and wind turbines (windmills that generate electricity). These power plants are based on machinery that is caused to rotate by either high-pressure steam, high-pressure water, or wind. Once you have rotating machinery, you can use Faraday’s law to generate electric current by wrapping many loops of wire around the machinery and allowing it to rotate in the presence of magnetic fields created by powerful magnets.

American Institue of Physics Emilio Segre Archives Figure 15

Michael Faraday was reared in a nineteenth-century British working class family, and had little formal schooling. He entered science at age 21 when he applied for a job as technical assistant to a well-known chemist, Humphry Davy, whose public lectures he had attended. Faraday’s enthusiasm and talent for science soon established him as an independent researcher who, at age 34, succeeded Davy as director of the Royal Institution of Great Britain.

217

Electromagnetism

Each loop generates additional electricity, so large amounts of electric energy can be supplied in this way. The basic principle, showing only a single loop, is illustrated in Figure 16. CONCEPT CHECK 13 A proton is placed at rest in the middle of a “vacuum chamber,” an enclosure that has been emptied of all matter. Consider some point X near a particular corner of the chamber. Neglect all influences other than the proton. Then at point X there is (a) an electric field; (b) an electric force; (c) a magnetic field; (d) a magnetic force; (e) none of the above, because there is nothing at point X. CONCEPT CHECK 14 In the preceding question, suppose that the proton is made to shake back and forth. Then at point X there is (a) an electric field; (b) an electric force; (c) a magnetic field; (d) a magnetic force; (e) none of the above, because there is nothing at point X. CONCEPT CHECK 15 Suppose that, as in Concept Check 13, a proton is placed at rest in the middle of a vacuum chamber, and also that an electron is placed at rest at point X. Then the electron (a) feels an electric force due to the electric field created by the proton; (b) feels a magnetic force due to the magnetic field created by the proton; (c) feels no electromagnetic force.

Shaft from power source

N

Moving (slip) rings

S Stationary conducting brushes

Figure 16

The principle of electric power generation, showing only a single rectangular loop of wire. A power source such as wind or steam causes the shaft to rotate, which rotates the loop. Stationary magnetic north and south poles are placed above and below the loop, so that their magnetic field passes through the loop as the loop is rotated. As Faraday’s law predicts, this generates an electric current that flows around the loop and onto the moving metal “slip rings” that are rigidly attached to the loop. This current then flows onto the stationary metallic brushes that are in electrical contact with the moving rings. The current from the brushes can then flow to an external consumer of electricity such as the lightbulb shown.

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© Sidney Harris, used with permission.

Electromagnetism

219

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Electromagnetism Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions

22. State Ohm’s law. 23. What is a short circuit?

ELECTRIC FORCE

FORCE FIELDS AND ELECTROMAGNETISM

1. When you rub two transparencies with tissue and hold them close together, they stand apart. Give two reasons that the force causing this cannot be gravity. 2. Cite the evidence supporting the claim that there are two, and only two, types of electric charge. 3. Suppose that the electric force between two objects is 2 N and that you then double the distance between the objects. What is the new force? 4. Suppose that the electric force between two objects is 2 N and that you then double the electric charge on each object. What is the new force? 5. Describe at least two ways in which the gravitational force and the electric force differ. 6. In what ways are the electric and gravitational force laws similar?

THE ELECTRIC ATOM 7. List the types of particles that are found within the atom. 8. Which types of particles within the atom are electrically charged? 9. What holds, or binds, an atom’s orbiting electrons to the nucleus? 10. What is an ion? 11. What is an electric current? 12. List several phenomena that require the planetary model of the atom, rather than the Greek atom, for their explanation. 13. Name and briefly describe the three kinds of subatomic particles found in atoms. 14. Give evidence supporting the claim that most of an atom’s mass is concentrated in a tiny nucleus at the center. 15. Explain what happens at the microscopic level in a wire when a battery creates an electric current in the wire. 16. What is an atomic number, and how is it related to the chemical elements?

ELECTRIC CIRCUITS 17. 18. 19. 20. 21.

What is an electric circuit? Give an example of electric energy. What is the meaning of a battery’s “voltage”? What is an ampere? How does AC differ from DC?

24. Name two kinds of force fields. 25. Which force fields would be felt by a stationary charged object: gravitational, electric, or magnetic? 26. Which force fields would be felt by a moving uncharged object: gravitational, electric, or magnetic? 27. Which force fields would be felt by a moving charged object: gravitational, electric, or magnetic? 28. What does it mean to say that there is an electric field throughout this room? 29. What does Faraday’s law say about magnets and wires? 30. What does Faraday’s law say about magnetic fields and electric fields? 31. Give two reasons that the force between bar magnets cannot be the electric force. 32. Magnetic forces are always caused by what types of objects?

Conceptual Exercises ELECTRIC FORCE 1. Suppose that you rub a transparency with a tissue to charge both objects and then hold them several meters apart and shake both of them back and forth. Name the forces that they exert on each other. 2. Since matter is made of electrically charged particles, why don’t we and the objects around us feel electric forces all the time? 3. When you remove a wool dress from a garment bag, the sides of the bag might tend to stick to the dress. Explain. 4. Figure 17 shows an electroscope. The leaves (made of metal foil) normally hang down, but they spread apart when the metal sphere on top touches a charged object. Explain. 5. How does the operation of the electroscope (previous exercise) demonstrate electric current? 6. When tiny scraps of paper are placed between two flat metal plates that have been oppositely charged (one plate charged positively and the other charged negatively), they bounce back and forth between the plates. Explain this phenomenon. 7. Highway trucks can become electrically charged as they travel. How can this happen? This can be dangerous, especially for gasoline tank trucks. How can it be prevented? 8. What happens to the electric force between two charged objects if the charge on one of them is reversed in sign?

From Chapter 8 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Electromagnetism: Problem Set Metal sphere

Helium

Wire

Leaves of thin metal foil

Figure 17

Why do the leaves stand apart? 9. What happens to the electric force between two charged objects if the charges on both of them are reversed in sign? 10. Ping pong ball A has an electric charge that is 10 times larger than the charge on ping pong ball B. When placed sufficiently close together to exert measurable electric forces on each other, how does the force by A on B compare with the force by B on A? 11. Objects A and B are both electrically charged. If the distance between them is halved while the charge on A is also halved, what happens to the force between them? 12. If the distance between two charged objects is reduced to one-fourth of its original value, what happens to the electric force between them? 13. If the distance between two charged objects and the charge on each of them are all doubled, what happens to the electric force between them?

THE ELECTRIC ATOM 14. While brushing your hair, you find that the hairs tend to stand apart from one another and that they are attracted toward the brush. Explain this in microscopic terms. 15. A covered mystery shoebox is placed on a table. What are a few ways that you could learn something about its contents without directly touching it or having it lifted? 16. After you walk across a rug and scuff electrons off the rug, are you positively or negatively charged? 17. According to Figure 3, what are the atomic numbers of carbon and helium? Roughly how much more massive is the carbon atom than the helium atom? 18. Some science fiction stories portray atoms as true miniature solar systems populated by tiny creatures. What are some differences, other than size, between our solar system and the planetary model of an atom? 19. An atom loses its two outermost electrons. How does the resulting ion behave when it is near a positively charged transparency? A negatively charged tissue?

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Carbon

Figure 3

Two examples of the planetary model of the atom. Protons are green, neutrons are white, and electrons are black. The diagrams are not drawn to scale! If they were drawn to scale, the nuclei and the electrons would be too small to be seen.

20. In the preceding question, would anything be different if it lost only one electron? 21. MAKING ESTIMATES About how many atoms thick is a sheet of paper? 22. MAKING ESTIMATES Which is bigger, an atom or a wavelength of light? Roughly how much bigger?

ELECTRIC CIRCUITS 23. Is a wire that carries current electrically charged? 24. At what point or points in the circuit of Figure 8 do electrons have the least energy? What kind of energy? 25. Does more current flow out of a battery than into it? 26. Only a small fraction of the electric energy that is used up in an incandescent lightbulb is transformed into light. What happens to the remaining energy? 27. In the circuit of Figure 8, would a thicker lightbulb filament produce a larger current, smaller current, or neither? Explain. 28. Do the headlights of an automobile carry AC or DC? What about a toaster in your kitchen? 29. The filament of a lightbulb glows, while the connecting wires do not. Why?

Electromagnetism: Problem Set Figure 8

Because electrons flow so easily through ordinary wires, the circuit of Figure 7 would soon burn out the wire or battery. Inserting an incandescent lightbulb or other “circuit element” reduces the flow to safe levels. This happens because the lightbulb’s glowing filament is very thin and thus restricts or “resists” the current, much as a squeezed garden hose restricts the flow of water.

– +

–

+

FORCE FIELDS 30. Do the electric circuits in your home produce magnetic fields? Suggest a measurement that might check your answer. 31. Is an electric field a form of matter? Explain. What about a gravitational field? 32. A proton is placed, at rest, at some point A within a room that is otherwise devoid of all matter. At some other point B within the room is there an electric field? An electric force? A magnetic field? A magnetic force? Is there energy at point B? 33. Suppose that, in the preceding exercise, the proton is oscillating back and forth. Is there an electric field at point B? An electric force? A magnetic field? A magnetic force? Energy? 34. Suppose you have a piece of metal wire and a bar magnet. Describe two ways in which you could create an electric current. What law of physics is involved here?

ELECTROMAGNETISM 35. You have three iron bars, only two of which are permanent magnets. Because of temporary magnetization, all three bars at first appear to be magnetized. How can you determine which one is not magnetized, without using any other objects? 36. Suppose you have two iron bars (see the previous exercise), one magnetized and one not magnetized. Can you then determine which one is magnetized, without using any other objects? 37. If you place a proton at some point in an electric field and then release it, what will happen? 38. How would a proton’s motion differ from the motion of an electron placed at the same point in the same electric field? 39. How would a proton’s motion differ from the motion of an electron placed at the same point in the same gravitational field?

Problems THE ELECTRIC FORCE 1. Two small electrically charged objects are placed 8 cm apart, where they exert an electric force F on each other. How far apart must they be in order to exert an electric force of (1/4) F on each other?

2. Referring to Problem 1, how far apart must the objects be in order to exert an electric force of 4F on each other? 3. Two small electrically charged objects are placed a certain distance apart, where they exert an electric force of 4 N on each other. Suppose the charge on object #1 is doubled. What happens to the force? What if the charge on both objects is doubled? 4. Two small electrically charged objects are placed a certain distance apart, where they exert an electric force of 4 N on each other. Suppose the charge on object #1 is halved. What happens to the force? What if the charge on both objects is halved? 5. Two small objects, each containing an electric charge q, are placed a certain distance apart, producing an electric force F by each object on the other. Suppose half of the first object’s charge is transferred to the second object. What happens to the force? 6. How does the electric force between two helium nuclei placed a certain distance apart compare with the force between two hydrogen nuclei placed the same distance apart? 7. How does the electric force between a helium nucleus and a lithium nucleus placed a certain distance apart compare with the force between two hydrogen nuclei placed the same distance apart? 8. How does the electric force between two hydrogen nuclei placed a certain distance apart compare with the force between two helium nuclei placed twice as far apart? 9. What is the electric force between two pellets that each have a charge of 10–6 C placed 1 cm apart? 10. A dust particle carrying a charge of -3 * 10 - 10 C is 2 mm to the left of another dust particle carrying a charge of +4 * 10 - 10 C. Find the magnitude and direction of the electric force on the first particle. 11. Two objects carrying equal amounts of charge are placed 10 cm apart, where the force between them is found to be 0.9 N. Find the charge on each of the objects. 12. How far should an object carrying a charge of 2 * 10 - 6 C be from another object carrying a charge of 4 * 10 - 6 C in order for them to exert a force of 6 N on each other?

ELECTRIC CIRCUITS 13. What should be the resistance of a lightbulb in order for it to draw a current of 2 amp when plugged into a 120-volt outlet?

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Electromagnetism: Problem Set 14. A blow dryer with a resistance of 6 ohms is plugged into a 120-volt outlet. How much current does it draw from the outlet? 15. If the coils of a heater have a resistance of 12 ohm, how much current does it draw when plugged into a 120-volt outlet? 16. An 0.5 amp current runs through a lamp whose resistance is 150 ohm. What is the voltage across the lamp? 17. If a lightbulb has a resistance of 40 ohm and a current of 2 amp, at what voltage is it operating? 18. A 1.5-volt battery is short-circuited by a 1-meter length of wire having a resistance of only 0.02 ohm. How large is the current flowing through the wire (before the wire or the battery burn out)?

13. The force is unchanged. 15. Roll a ball so that it collides with the box to see whether the

17. 19.

21.

Answers to Concept Checks 1. If you set q1 = 1 C, q2 = 1 C, and d = 1 m, Coulomb’s law 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

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tells you that F = 9 * 109 N, (d). Smallest mass (b), largest mass (e). (a), (b), (c), and (e) (c) (e) I = V>R = 6 volts>0.02 ohm = 300 amps, (b). (c). The force on a negative charge placed at A would be in the opposite direction, (d). The field points downward at point B, so the force on a small negative charge placed at point B would be upward, (a). The direction of the field at point C is rightward, so the force on a positive charge placed at point C would be rightward, (d). If the charge were instead placed at point D, the force on it would be downward, (b). (a) and (d) (c) (b) (a). Note that (b) is incorrect because there is no material object at point X to feel a force; (c) is incorrect because the proton is stationary, so it doesn’t create a magnetic field. (a) and (c) (a)

23. 25.

27. 29. 31. 33.

35.

contents have high mass or low mass. Tie a string around the box and pull it; the box’s mass can be determined by measuring the pulling and the acceleration. Fire bullets into the box from all directions. Hit the box with a hammer to see if anything inside rolls around. 6 and 2. The ratio is about 12 to 4, which is the same as 3 to 1. The ion will carry a positive charge, so it will be repelled by a positively charged transparency, and attracted to a negatively charged tissue. A 500-sheet stack of typing paper is about 5 cm thick, so the thickness of one sheet of paper is about 5 cm>500 = 0.01 cm = 10 - 4 m. An atom is about 10 - 10 m across, so a piece of paper is about 10 - 4>10 - 10 = 106 times bigger. No, the number of protons in any segment equals the number of electrons. But the protons remain at rest while the electrons move along the wire. The amount of current flowing out always equals the amount flowing in. For instance, if more electrons were flowing out than in, electrons would have to pile up someplace along the circuit. A thicker filament would allow electrons to flow through more easily, so it would produce a larger current. The filament gets a lot hotter because, due to the thinness of the filament, the moving electrons bump into so many of the filament’s atoms. An electric field is not made of atoms or of other material particles, so it is not a form of matter. The same goes for a gravitational field. At B there is an electric field (because of the charge at A), there is no electric force (because there is no charge at B), there is a magnetic field (because the charge at A is moving), there is no magnetic force (because there is no charge at B), and there is energy (because of the presence of an electromagnetic field). The two permanent magnets will be able to both attract and repel each other, depending on which ends are put together. The non-magnetized bar will only be attracted to (and not repelled by) the other two bars. It will start to move in the direction of the field. They wouldn’t differ. They would both move at the same acceleration and the same speed, just like two falling objects having different masses (Galileo’s principle of falling).

Answers to Odd-Numbered Conceptual Exercises and Problems

37. 39.

Conceptual Exercises 1. Both an electric force and a magnetic force will be exerted. (There will also be a tiny gravitational force.) 3. The bag and the dress become oppositely charged, due to friction, causing them to attract each other and cling together. 5. Charge must have flowed, or moved, from the metal sphere down to the leaves; this motion of charge is electrical current. 7. The truck picks up charge from the road by friction (the tires rubbing against the road). It can be prevented by providing an easy pathway for charge back to the road—a chain hanging from the truck to the road, for example. 9. The forces on each charge are unchanged. 11. The force is doubled.

Problems 1. Because of the inverse-square nature of the force, placing them twice as far apart (16 cm) will cause the force to be 1/4 as strong. 3. With doubled charge on #1, the force will double to 8 N. If both charges are doubled, the force will be 16 N. 5. #1 now contains a charge of q/2, while #2 contains (3/2)q. Thus q1q2 = (1>2)q * (3>2)q = 3>4q2. So the force is reduced to 3/4 of its previous value. 7. Lithium has three protons, helium has two protons, and hydrogen has one proton. So the force between a lithium and helium nucleus is 3 * 2 = 6 times larger than that between two hydrogen nuclei.

Electromagnetism: Problem Set 9. From Coulomb’s law,

F = 9 * 109 *

(10 - 6) * (10 - 6)

= 90 N. (0.01)2 11. Coulomb’s law tells us that F = 9 * 109 q2>d2, from which 0.9 = 9 * 109 q2>(0.01)2. q2 = 0.9 * (0.01)2> Thus 9 - 14 9 * 10 = 10 . Taking the square root, q = 10 - 7 C.

13. R = V>I = 120>2 = 60 ohm. 15. I = V>R = 120>12 = 10 amp. 17. V = IR = 2 * 40 = 80 volt.

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Waves, Light, and Climate Change Quite simply, I think it is no exaggeration to say that climate change is the biggest problem our civilization has ever had to face up to in its 12,000 years, because it requires a collective response. David King, Chief Science Adviser to the British Government, 2001–2007

T

his chapter continues our quest to grasp the nature of light, and looks at the planetary consequences of recent human interference with the radiations (including light) that arrive from the sun. In the first two sections, you’ll learn about waves, a topic that we’ll also need for studying quantum physics. In Section 3, you’ll see what waves have to do with light. Section 4 explains light in terms of electromagnetic fields. Section 5 makes the important, even revolutionary, point that electromagnetic fields are physically real and looks ahead to the modern view that, at the most fundamental level, the universe is made of fields. The electromagnetic field theory of light leads to an understanding of several other lightlike “radiations” (Section 6) and of sunlight (Section 7). In line with my goal of presenting important physics-related social implications as soon as the physics background is prepared, I’ll describe in Sections 8 and 9 two ways in which humans have significantly altered the interaction of our planet with the sun: (1) alteration in the planetary impact of the sun’s ultraviolet radiation, caused by human depletion of atmospheric ozone, and (2) alteration of the planetary impact of the sun’s infrared radiation, caused by human emissions of carbon-dioxide and other gases. The two problems have a lot in common. Encouragingly, humans have solved the first of these, but the second looms ever larger.

1 WAVES: SOMETHING ELSE THAT TRAVELS You’re probably familiar with some kinds of waves (Figure 1). Stretch a few meters of flexible rope along the floor, fix one end (perhaps under a friend’s foot), and give the free end a single shake. Something travels down the rope. Figure 2 is a series of pictures taken with a movie camera, showing a similar “something” traveling down a long spring that has been given a single up-and-down shake at the right-hand end. As another example, imagine (better yet, try it!) stretching a Slinky™ toy between your two hands along a tabletop. Quickly move the left end a short distance toward the right and then back to the left, holding the right end fixed. Something travels along the Slinky from your left to your right hand (Figure 3). This “something”

From Chapter 9 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Jeff Greenberg/Omni-Photo Communications, Inc. Figure 1

Water waves.

that travels across the water, along the rope, and along the Slinky is called a wave. As another example, the continued shaking of one end of a rope causes a long continuous wave to travel along the rope (Figure 4). But wave is just a word that names the behavior without telling us what it really is. What actually happens here? In Figure 2, how do the individual parts of the spring actually move? As you can see from the motion of the small ribbon tied to the spring, each loop just moves up and then back down. How does a particular part of the water move in Figure 1? Fill a bowl with water, float a small cork in it, and drop a small pebble in, several centimeters from the cork, to create ripples. Observe the cork as ripples pass by. If the ripples are small, the cork will move up and down, not outward along with the ripples. Each portion of the water just shakes or “vibrates” up and down, but does not travel along the water surface. The Slinky wave is similar, except that the vibrations are parallel instead of perpendicular to the Slinky. One thing that is traveling with each of these waves is energy. You can verify this for yourself by holding the fixed end of a rope while a friend shakes the other end. Your hand vibrates as the pulse arrives. It takes work to force your hand back and forth this way, and we know that work requires energy. So waves transfer energy. On the other hand, no material substance is transferred by waves: No water is transferred outward in Figure 1, no part of the spring is transferred from right to left in Figure 2, and no part of the Slinky is transferred from left to right in Figure 3. This type of motion is unlike any motion we have examined before. Previously we studied balls, books, molecules, and other material objects actually moving from one place to another. What do we see traveling along the spring in Figure 2? Well, we see a bump traveling along the otherwise straight spring! In Figure 3 we see a compression, a squeezed region, traveling along the Slinky. We could describe both as “disturbances” that travel along the otherwise undisturbed spring or Slinky. The situation Compression

Compression moves toward the right

Figure 3

With the right-hand end of the Slinky held fixed, a quick motion of the left-hand end to the right and back again to the left creates a pulse that travels down the Slinky. Uri Haber-Schaim Figure 2

A series of pictures taken with a movie camera, showing a wave moving along a spring. A ribbon is tied to the spring at the point marked by the arrow. The ribbon moves up and down as the wave goes by, but it does not move in the direction of the wave.

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Direction of wave motion

Figure 4

Continued shaking of the end of a rope creates a continuous wave that travels down the rope.

Waves, Light, and Climate Change

is similar for water waves. The material through which the disturbance travels—the spring or Slinky or water—is called the medium for the wave. So a wave is a disturbance that travels through a medium in such a way that energy travels through the medium but matter does not. A “sports wave” in a large stadium filled with people is an instructive example. It begins when all the people at one end of the stadium stand up briefly with their hands in the air. As they sit down, the people in the adjoining part of the stadium stand up briefly with their hands in the air, and so forth all around the arena. Although this gives the appearance of something traveling around the stadium, this “something” is not any single thing. The people who are momentarily standing constitute a disturbance of the otherwise-seated crowd, and it is this disturbance that travels through the crowd. This is precisely the sort of situation we have in mind when we use the word wave. We need some quantitative terms. The wavelength of a continuous, repeated wave is the distance from any point along the wave to the next similar point, for example, from crest to crest or from trough to trough in Figure 5. A wave’s frequency is the number of vibrations that any particular part of the medium completes in each second. Waves are usually sent out by a vibrating source of some kind, in which case the wave’s frequency must be the same as the source’s frequency. The frequency could also be defined as the number of waves that the source sends out during each second. The unit for measuring frequency is the vibration per second, also called a hertz (Hz). A wave’s amplitude is its maximum height or depth (Figure 5), in other words, its maximum disturbance from the “neutral” or undisturbed situation. The wavespeed is the speed at which the disturbance moves through the medium.1 In most waves, disturbances are able to travel through a medium because of the connections between the parts of the medium. (In the sports wave, this connection is mental rather than directly physical.) For example, when you shake one end of a rope, this disturbance moves down the rope because the different parts of the rope are connected, so that when one part is lifted, its neighbor is soon lifted also. So it’s reasonable to suppose that the wavespeed is determined mainly by the medium and is roughly the same for differently shaped disturbances in the same medium. Experiments confirm this notion that differently shaped disturbances all travel through any particular medium at roughly the same wavespeed. Wavelength Amplitude

Wavespeed

Figure 5

The meaning of wavelength and amplitude. The wavespeed is the speed at which a crest or a trough moves down the rope.

1

Quantitatively, a wave’s wavelength l, frequency f, and wavespeed s are related by s = f l. For example, if three waves are sent out by the wave source every second ( f = 3 vib>s) and each wave has a length of 2 meters (l = 2 m), then it seems reasonable that the wavespeed should be 3 * 2 = 6 m>s. Extending this argument, you can see that the general relation is s = f l.

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CONCEPT CHECK 1 Which of the following is a true wave? (a) A row of falling dominoes. (b) Ripples on the surface of a pond, extending outward from a pebble dropped in the water. (c) A large water wave coming into the beach, one that surfers can ride on. (d) Water rushing downstream.

Figure 6

Which wave has the higher (larger) frequency, assuming that both have the same wavespeed?

CONCEPT CHECK 2 In Figure 6, which wave has the larger or “higher” frequency, and which carries more energy? (a) The top wave has higher frequency and carries more energy. (b) The top wave has higher frequency but the bottom wave carries more energy. (c) The bottom wave has higher frequency and carries more energy. (d) The bottom wave has higher frequency but the top wave carries more energy.

From Concept Check 2, note these useful rules: Shorter wavelength means higher frequency and, if the amplitude remains unchanged, then higher frequency means higher energy.

2 INTERFERENCE: A BEHAVIOR UNIQUE TO WAVES Before meeting

Figure 7

Two waves travel in opposite directions along a rope. What happens when they meet?

Before

During

After

Figure 8

Two waves meeting: interference.

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How do different waves traveling through the same medium interact with one another? For example, what happens when the large upward wave in Figure 7 meets the small downward wave? Experiment shows that they just pass through each other without distortion (Figure 8). You might have expected this, because each wave simply lifts or lowers the rope as it travels, so when the two waves meet, the rope is raised a lot by the large wave and simultaneously lowered a little by the small wave. Effects such as this, occurring when two waves are present at the same time and place, are called wave interference, or just “interference.” CONCEPT CHECK 3 Suppose that the two waves in Figure 7 had the same size and shape, with the wave on the right being inverted as shown, and that the wave on the left is a 2 cm crest (high point). The resulting interference would be (a) a 4 cm crest; (b) a 2 cm crest; (c) flat; (d) a 4 cm trough (low point); (e) a 2 cm trough.

Two equal waves of opposite orientation interfere by canceling each other (Figure 9), and two equal waves of the same orientation interfere by reinforcing each other (Figure 10). These two cases, cancellation and reinforcement, are called destructive interference and constructive interference. Wave interference shows, once again, the stark difference between the motion of a material object and wave motion. Two moving material objects—say, two freight trains—don’t pass through each other undisturbed! This distinction will be significant later in this text. Interference gets more interesting when it happens in two or three dimensions. An undisturbed rope has only one significant dimension, length. The surface of a lake is “two-dimensional” because it has length and width; the space in a room is “three-dimensional” because it has length, width, and height. In a two- or threedimensional medium, the waves created by a small source are outward-spreading circles (Figure 1) or spheres, respectively. As a two-dimensional example, suppose you fill a rectangular pan with water and tap your fingers at the same steady rate on the surface at two points along one side of the pan. Continuous waves will spread out from each of the two sources and soon cover the water surface. With the help of Figure 11, you can predict the interference effects. The figure shows the water as viewed from above. The two

Waves, Light, and Climate Change

sources are marked A and B. The solid circles represent crests from source A acting alone, and the dashed circles represent crests from source B acting alone. The troughs, not drawn, lie midway between the crests. To predict the interference pattern, work through Concept Check 4. CONCEPT CHECK 4 In Figure 11, draw an “x” (a color enhances the effect) at every point of constructive interference. (Hint: These are places where crest meets crest, and where trough meets trough.) Can you see a pattern? Next, draw an “o” (in a different color) at every point of destructive interference. (Hint: These are places where crest meets trough.) Now can you see a pattern?

Before Before

During During After After

Figure 9

Figure 10

Two waves meet and interfere destructively.

Two waves meet and interfere constructively.

Figure 11

Continuous surface waves spreading out from two sources. What will the interference effects look like?

A

B

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Uri Haber-Schaim Figure 12

Interference between continuous surface waves spreading out from two sources: experimental results. Crests (constructive interference) are bright, troughs (also constructive interference) are dark, and flat places (destructive interference) are gray.

Figure 12 is a photograph of this experiment, looking down onto the water’s surface. The photographic technique causes crests to appear bright, troughs to appear dark, and flat places to appear gray. As you can see, the interference pattern has lines of undisturbed water radiating outward as though they came from a point somewhere between the two sources. The interference is destructive along these lines. Between these undisturbed lines are other lines of constructive interference, with large crests and troughs. Just what Concept Check 4 predicted, right? Our analysis so far has been at one instant in time. Now “turn on time” by imagining a moving picture that begins with the snapshot in Figure 12. Since the individual circular waves move outward from A and B, the entire pattern must move outward also; in other words, the “rays” of destructive and constructive interference remain fixed in place, while the large waves within the constructive rays move outward, as shown by the arrows in Figure 13. Finally, imagine that the water is a rectangular swimming pool and that you examine the waves as they slosh against the right-hand wall of the pool. What would you observe? You can predict the answer using Figure 13. The observer should find some points where large waves pound against the wall, marked with large Xs in Figure 14, and other points, marked with large Os, where no waves roll in. The difference between the pattern from a single source and from two sources is striking. Waves from a single source, say source A operating alone, roll into all parts of the bordering wall (Figure 15). If we also turn on the second source, B, the pattern along the wall shifts to an interference pattern (Figure 14). The most dra-

X x

A

Uri Haber-Schaim Figure 13

Lines of constructive and destructive interference remain fixed in place, and constructive (large) waves move outward within the constructive regions in the directions indicated by the arrows.

B

x

x

x

x x o o x x o o o x x oo x x o o o x x x x xo o x x x x o x x o x x x o o o o ox x x o o o o o o x o o o o o x x x x x x x x x x x x x x o o o x x xo o o o o o o o o o o o x x o o o x x x x x x o x x x x o o x x o o x x x o o x x o o x x o o x o x x x x x

O X O X O

Observer stands on this side and looks down at pool, observing large waves rolling into points marked X at side of pool, and no waves rolling into points marked O. Small x’s and o’s are places on surface of pool where interference is constructive (x’s) and destructive (o’s).

X O

X

Figure 14

An observer scanning a wall at the far border of the pool finds points where large waves come into the wall, interspersed with points where no waves come in.

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Waves, Light, and Climate Change Figure 15

If one of the two wave sources shuts off, an observer scanning the wall will find that waves arrive at all points. Since waves now spread out from only a single source, there is no longer any interference. A

matic change is that now no waves come into the points marked O, even though they did come into these points when only one source was operating. It seems paradoxical: When you add a second source you get a reduced (in fact, zero) effect at the points marked O. Now let’s look at light.

3 LIGHT: PARTICLES OR WAVE? When you turn off the lightbulb at night, it gets dark. So the light in your room must have come from the lightbulb and not, for example, from your eyes. When you look at a luminous (light-emitting) object like a lightbulb, light goes from the bulb to your eyes. In order for you to see a nonluminous object, such as the wall of your room, light from the lightbulb must bounce (reflect) off the wall and then into your eyes. The light reflecting from the wall does not give you a nice mirror reflection, however, because the rough surface of most walls scatters the incoming light in many different directions. But what enters your eyes when you see light? This has been debated for centuries, with most of the suggested answers falling into either the “particle” or “wave” category. Experiment is the ultimate judge. We need an experimental test that distinguishes between the particle and wave models of light. The preceding section suggests a good candidate: wave interference. Particles might interact in various ways, but they do not interfere in the way that waves do. What does light do? To answer this, we need an experiment like the water–wave interference experiment, but with light. The first experiment you might think of is to simply shine two flashlights on a flat surface. But if you try this (do it!), you’ll find that it gives no observable interference effects—no alternating regions of constructive (brightly lit) and destructive (darker) regions. So maybe light is a stream of particles.

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Waves, Light, and Climate Change Figure 16

The double-slit experiment with light. What will we see on the screen? A B

Partition with two very small thin slits (shown here greatly enlarged) to let light through

?

Screen

But in our discussion of water–wave interference, we assumed that the two wave sources had identical vibrations. For example, if the two sources in Figure 12 were changing their frequency all the time and in different ways, we would not expect to see a recognizable interference pattern. A flashlight bulb’s light is produced by heating up the bulb’s thin wire, or filament, until it glows. The microscopic thermal motions that make the filament hot enough to glow are highly random, so we wouldn’t expect two different bulbs to have identical vibrations, so they wouldn’t show interference even if light were a wave. How do we know whether light is a wave, or particles? In 1801, Thomas Young solved the problem of finding two sources with identical vibrations by using a single light source that he split into two parts. He then recombined these parts to see whether they interfered. Figure 16 shows how to do this. A single light source sends light through two very small and narrow parallel slits (shown greatly enlarged) in a partition that blocks all the light except that going through the slits. The two slits act as two new sources of light. If light is a wave, these two sources should have synchronized vibrations, because the light from each slit originates in the same filament.2 Young found an experimental result like that shown in Figure 17. This photograph was made by placing photographic film at the position of the receiving screen as shown in Figure 18. To interpret Figure 17, let’s compare it with water–wave interference. The receiving screen of Figure 16 is similar to the right-hand wall in Figure 14. But water waves occur on the two-dimensional surface of water, while light fills three-dimensional space. The two sources of light are not tiny points like A and B in Figure 14 but instead are slits that extend into the third dimension. If these slits send out light waves, we would expect the interference pattern on the receiving screen to be alternating lines of constructive and destructive interference running parallel to the slits, not small points like those marked X and O in Figure 14. In other words, we would expect alternating bright (lit) and dark lines—precisely the outcome in Figure 17. Conclusion of this double-slit interference experiment with light: light is a wave.

Figure 17

The double-slit experiment with light: experimental result.

A B

Very narrow slits, shown here greatly enlarged.

What happens if we close one of the slits, leaving only one slit open? If light is a wave, we would expect waves to spread out from the open slit, without interference, just like the water waves in Figure 15. A broad band of light should then cover a large area of the receiving screen (Figure 19). This is, in fact, what happens. The clearest evidence that the flashlight is sending out waves and not particles can be found at the positions of the dark lines in the double-slit experiment, the points 2

Figure 18

The double-slit experiment with light: the experimental setup and result.

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More precisely, the light must first be filtered so that it is all of one color (one frequency), and it must first come through a single narrow slit so the vibrations at slit A are synchronized with those at slit B. This makes Young’s experiment exactly analogous to Figures 13 and 14. In those figures, the waves from A and B have the same frequency and they are synchronized.

Waves, Light, and Climate Change

where no light arrives. With only one slit open, light spreads out over the entire receiving screen. How is it, then, that if we simply open a second slit, no light arrives at these particular positions? It’s difficult to see how particles coming through the two slits could cancel one another in this way, but it’s just what we expect of waves. By measuring the distance from one bright constructive-interference line to the next such line in a pattern such as Figure 17 and using a little geometry, it’s possible to calculate the wavelength of the light that created the pattern. Measurements of interference patterns like this are the usual method of measuring the wavelength of light. This wavelength turns out to be very small. Light sources have wavelengths ranging from about 0.4 * 10-6 m to 0.7 * 10-6 m (less than a millionth of a meter, or less than one-thousandth of a millimeter). How do we know that light is a wave? You can easily demonstrate light-wave interference yourself, using a single-slit wave-interference effect that occurs when the slit is hundreds of times larger than the wavelength of the light. With such a wide slit, the light coming through the slit acts like hundreds of tiny sources. If light is a wave, then all the individual waves from these hundreds of sources should interfere with one another to form an interference pattern. Here’s the experiment: Focus your eyes on a well-lit wall or other surface. Make a slit by holding your thumb and forefinger about a millimeter apart and several centimeters in front of your eye. Focus on the light source (the wall), not on your fingers, so that your fingers look blurred. Where the blurs overlap, you should see narrow bright and dark lines running parallel to your fingers. These lines are constructive and destructive interference regions, formed at the position of your eye.

A

Extremely narrow single slit, shown here greatly enlarged. To get the noninterference pattern shown, the slit’s width must be less than the wavelength of the light!

Figure 19

If one of the two slits is closed, light will arrive at all points along the receiving screen. The white screen indicates that there is no interference pattern—that light arrives everywhere on the screen, in contrast to the pattern that emerged in Figure 18.

CONCEPT CHECK 5 If water waves of longer wavelength were used in the experiment shown in Figure 14, the Xs and Os along the right-hand side of the figure would be (a) farther apart; (b) closer together; (c) unchanged. CONCEPT CHECK 6 If shorter-wavelength light were used in the experiment whose result is shown in Figure 17, the alternating bright and dark lines in the figure would be (a) unchanged; (b) greater in number, but unchanged in length or width; (c) wider; (d) narrower; (e) longer; (f) shorter.

MAKI NG ESTI MATES Roughly, how does a typical light wavelength compare with the thickness of a piece of paper?

4 THE ELECTROMAGNETIC WAVE THEORY OF LIGHT Water waves, rope waves, and Slinky waves are waves in water, ropes, and Slinkies. But what medium vibrates when light waves travel? Hmm... It’s not an easy question. We don’t see light beams directly the way we see water waves (Figure 20). It is as though we could see the impact of water waves against the edge of a swimming pool, without being able to see the water. We can see light beams in dusty air

Choose a typical wavelength of light, say 5 * 10-7 m (in making estimates, choose simple but reasonable numbers). To estimate the thickness of a sheet of paper, estimate the thickness of, say, 500 sheets (about 5 cm) and divide by 500 (5 cm>500 = 0.1 cm = 10-4 m). The number of wavelengths in this thickness is 10-4>(5 * 10-7) = 10-4 + 7>5 = 103>5 = 1000>5 = 200. SO LUTION TO MAKI NG ESTI MATES

Figure 20

Light beams cannot be seen from the side. What invisible medium is carrying these waves?

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Waves, Light, and Climate Change

Uri Haber-Schaim Figure 21

You can see a light beam by allowing it to reflect off dust particles in the air.

Figure 22

If you shake a charged object such as a charged comb, other charged objects will shake in response.

(Figure 21), but only because the light is reflected off the dust particles. The medium for light waves is itself invisible. Could the medium be air? Sounds plausible. But what about light traveling here from the sun, moon, and stars? Air extends, in any appreciable amounts, only a few miles above Earth’s surface, so air cannot be the medium for light waves.3 One odd thing about light is that it moves through outer space where there is essentially no matter at all. But something must be out there in so-called empty space, because without a medium to do the waving, you can’t have a wave. The medium for light, then, must be nonmaterial—not made of atoms or other forms of matter. Nineteenth-century scientists devoted lots of effort to learning what kind of wave light is. It turns out, as we’ll see, that the answer is bound up with electricity. Suppose you pull a rubber comb through your hair, scuffing electrons from your hair onto the comb. The charged comb then creates an electric field in the surrounding space, a field that can be detected by a rubbed (and hence electrically charged) transparency held near the comb. If you quickly shake the comb once, up and back down, the electric field in the surrounding space will shake too. This can be detected by the transparency, which will shake in response to the comb’s motion. The comb also creates a magnetic field during the brief time that it’s moving. This temporary magnetic field could in principle (the force would be very small) be detected as a brief shake of a magnet placed near the comb. In summary, the comb’s motion causes changes in the electromagnetic field around the comb, changes that can be detected by other charged objects and magnets (Figure 22). There is an interesting question about this experiment, a question that many nineteenth-century scientists asked: When will a distant detector feel the changes in the field? Is the effect instantaneous? Suppose you shake the comb precisely at noon. Does the transparency shake precisely at noon, too, or a little later? During the 1860s, British physicist James Clerk Maxwell (Figure 23) did some hard thinking about electromagnetism, putting all that was then known about the subject together into a single theory. Maxwell’s theory emphasized fields and described how electrically charged objects create electromagnetic fields. Three basic principles of the electromagnetic force were known at that time. Stating these three principles in terms of fields, the electric force law says that charged objects create electric fields, the magnetic force law says that moving charged objects create magnetic fields, and Faraday’s law says that any change in a magnetic field must create an electric field. Maxwell, like most theoretical physicists, felt that a correct and fundamental description of the natural world should also be fitting, balanced, symmetric, or, in a word, beautiful. It seemed to him that the laws of electricity and magnetism should treat electricity on the one hand, and magnetism on the other, symmetrically. The three basic laws (electric force law, magnetic force law, Faraday’s law) seemed to be missing something in this regard. The first two state that electric fields and magnetic fields arise from charged objects and from moving charged objects, respectively. Faraday’s law then states that electric fields can be created in a second way, namely by a changing magnetic field. It seemed to Maxwell that there should then be a fourth law, one that would balance Faraday’s law by providing a second way to create magnetic fields. Such a fourth law should be symmetric to Faraday’s law; in other words, it should state that magnetic fields can be created by a changing electric field. 3

236

Air is the medium for sound waves rather than for light waves. Since sound does not bear directly on the major purposes of this text, we won’t discuss it further here.

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Then the theory would treat electric and magnetic fields symmetrically: Changes in one field always create the other field. Maxwell’s invention, when combined with the other three laws, led him to predict the existence of so-called “electromagnetic waves” (see below) and to predict that light is a wave of this sort—predictions that are now amply verified and that represent a stunning success for theoretical reasoning. Once again we see the importance of beauty and symmetry in science.4 Maxwell’s theory, which he formulated in precise mathematical language, predicted a time delay for electromagnetic forces. The key ingredient was the way that changes in one field created the other field. This meant that electric and magnetic fields can create and re-create each other. Once the fields at one point in space are changed—for instance, by giving a charged comb a single shake—Maxwell’s theory implied that the change is transmitted outward as a change in the nearby fields a short time later, and these changing fields in turn transmit the change farther outward, and so forth. It follows from this that electric and magnetic forces are not transmitted instantaneously. Maxwell’s analysis showed that if you disturb an electromagnetic field at one point, the disturbance will move outward through the field. This is exactly the kind of behavior that we called “wave motion” in Section 1. But this new type of wave is not a wave in a material (made of matter) medium such as water. Rather, the medium is the electromagnetic field itself. Any such disturbance that moves through an electromagnetic field is called an electromagnetic wave. You can’t directly see electromagnetic waves in the way that you can see water waves, because the medium for electromagnetic waves is a nonmaterial electromagnetic field instead of a material substance such as water. Nevertheless, electromagnetic waves can be detected by other charged or magnetized objects at some distance from the source of the wave and at some later time after the wave was sent out. Figure 24 pictures these invisible waves. This was all worked out quantitatively in Maxwell’s theory. The theory predicted not only a delay in the transmission of electromagnetic forces but also the speed of transmission. The predicted speed was about 300,000 km/s or 3 * 108 m>s. This particular speed had come up before, in experiments performed nearly two centuries before Maxwell invented his theory. But these previous experiments seemed entirely unrelated to the electromagnetic effects that Maxwell was studying. This speed, 300,000 km/s, was the known speed at which light travels!

American Institute of Physics/ Emilio Segre Visual Archives Figure 23

James Clerk Maxwell, the “Isaac Newton of electromagnetism.” He cast the principles of electricity and magnetism into the form of four equations involving the electric and magnetic fields created by electric charges and electric currents. The theory led to a unification of the electric with the magnetic force and to understanding electromagnetic radiation, which underlies much modern technology including radio, television, and lasers.

How do we know the speed of light? People once thought that light requires no travel time—that its speed was infinite. Light certainly travels much faster than sound, as you verify whenever you see lightning before you hear thunder. Galileo was one of the first to try to measure the speed of light, or “lightspeed” as I will call it, by measuring the total round-trip time for light to travel to a distant mountain and back. His experiment didn’t work because the time turned out to be far too short to measure using Galileo’s timing methods. Either greater timing accuracy or a greater travel distance was needed. The first evidence for a finite, and not infinite, speed of light came from astronomical observations several decades later. Pointing telescopes at a moon of Jupiter, astronomers

4

In 1894, physicist Pierre Curie suggested that the symmetry between electricity and magnetism exhibited by Maxwell’s four laws would be complete if, in addition to electric charge, there existed in nature a pure “magnetic charge” called a “monopole.” Every magnet ever observed has two poles, north and south. You can’t isolate one from the other. A monopole would be a pure north, or pure south, pole, and would create a magnetic field even when it was not moving. Although monopoles have never been observed, theorists have never discarded the idea. Today, theorists searching for a “grand unified theory” of the fundamental particles suggest that monopoles exist or at least that they once existed, during the early stages of the big bang.

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Figure 24

When you shake a charged object, it sends out an electromagnetic wave in all directions. This invisible wave is a disturbance in the charged object’s electromagnetic field.

found that the time they measured for this moon to orbit Jupiter didn’t remain constant. This was weird. Why should a moon take longer for some orbits than for others? Earth’s moon takes 27.3 days, every time. The astronomers found that these variations were not caused by Jupiter’s moon at all but were instead related to Earth’s motion around the sun. The variations were just what would be expected if the light from Jupiter’s moon traveled at a finite, and not infinite, speed (see Figure 25 for the explanation). From the measured variations in orbital time, lightspeed could be estimated.

Maxwell hypothesized that light might actually be an electromagnetic wave. But scientists knew of no way to verify this tantalizing suggestion until two decades later, when it became possible to check Maxwell’s theory directly by causing charged objects to oscillate and observing the effects some distance away. How do we know that electromagnetic waves exist? One way to verify Maxwell’s theory would be to shake a charged object at the frequency of visible light, about a thousand trillion Hz, to see whether the shaking created light. You’d have a hard time shaking your hand that fast! Such high frequencies are hard to achieve in the laboratory, even today. German physicist Heinrich Hertz (Figure 26)—the hertz is named for him—figured out how to do an experiment of this sort, but at a frequency of “only” about a billion Hz. He constructed an electric circuit that contained a small open gap. Ordinarily, such a gap stops the flow of electric charge, but Hertz built the endpoints of the gap in such a way that large amounts of charge (excess electrons on one side, excess protons on the other) could build up on them. After enough buildup, electrons were forced to jump across the gap. We observe such charge jumping as a spark. Lightning is a spark of this sort. In Hertz’s circuit, a jump of charge triggered a brief series of such jumps back and forth across the gap, at a rate of a billion back-and-forths per second. If Maxwell was correct, these oscillations should create electromagnetic waves with a frequency of a billion Hz.

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Waves, Light, and Climate Change Earth, at two different points in its orbit B Jupiter’s moon, on two successive orbits A Sun

Jupiter

Figure 25

It takes light from Jupiter’s moon about 35 minutes to reach point A, and 43 minutes to reach point B. Thus, if Earth moves from point A to point B while Jupiter’s moon is orbiting Jupiter one time, the orbital time as measured on Earth will be 8 minutes longer than the true orbital time, because of the extra 8 minutes that it takes light to reach point B. This effect creates a variation in the measured orbital time that can be explained by assuming that light has a finite, not infinite, speed. At some distance from this predicted source of electromagnetic waves, Hertz placed a second circuit. This circuit was entirely passive, with no battery or other internal source of electric current. If Maxwell’s theory was correct, electromagnetic waves from the first circuit should cause an electric current to oscillate in the second circuit, also at a billion hertz. The transmission from one circuit to the other should occur at lightspeed. Hertz’s results entirely confirmed these predictions. Although Hertz’s waves were not light waves, his work convinced scientists that electromagnetic waves really existed and that light is actually an electromagnetic wave. As a by-product, Hertz’s work came to the attention of an ingenious Italian inventor named Guglielmo Marconi, launching the radio and television revolution. Today, we know Hertz’s waves as radio waves.

Hulton-Deutsch Collection/ CORBIS Figure 26

Two decades after Maxwell predicted the existence of electromagnetic waves, German physicist Heinrich Hertz discovered them experimentally. Hertz’s waves were in the radio region of the electromagnetic spectrum, and provided the scientific basis for the radio and television revolution.

It’s a stunning unification: Maxwell’s theory correctly describes electricity, magnetism, light, and radio. All these are different manifestations of one underlying reality: electric charge. Summarizing: Electromagnetic Wave Theory of Light Every vibrating charged object creates a disturbance (wave) in its own electromagnetic field. This disturbance spreads outward through the field at lightspeed, 300,000 km/s, or 3 * 108 m>s. Light is just such an electromagnetic wave.5

CONCEPT CHECK 7 Suppose you electrically charge a comb by running it through your hair and then shake it back and forth at a frequency of 1 Hz. This will produce (a) a sound wave having a frequency of 1 Hz and a speed of 300,000 km/s; (b) a sound wave having a wavelength of 300,000 km and a speed of 300,000 km/s; (c) an electromagnetic field that vibrates at a frequency of 1 Hz, but no electromagnetic wave; (d) an electromagnetic wave having a frequency of 1 Hz and a speed of 300,000 km/s. 5

More precisely, the speed of light, and all electromagnetic waves, is 299,792.458 km/s in a vacuum. When traveling through matter, however, light is slower than this because it is continually absorbed and re-emitted by atoms.

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Hertz’s waves had a frequency of 109 hertz. Could a normal home radio receive waves of this type and frequency? (a) No, because radios receive only sound waves, and Hertz created only electromagnetic waves. (b) No, because these waves are neither in the AM radio frequency range nor the FM radio frequency range. (c) Yes, these waves could be received by an AM radio. (d) Yes, these waves could be received by an FM radio. CONCEPT CHECK 8

MAKI NG ESTI MATES About how long does it take light to get to your eyes from a lightbulb in your room?

5 FIELDS ARE REAL

We now think of electric and magnetic and gravitational fields as being as real as rocks and people. Normally we are not aware of those fields, but you can confirm that they are all around you by turning on a radio. Gordon Kane, Particle Physicist, in His Book Supersymmetry

Since ancient Greek times, scientists have generally thought that the universe was made of tiny material particles, atoms. As Democritus put it, there are only atoms and empty space. It’s a view that is remarkably compatible with Newton’s physics, which was once thought of as the rules according to which atoms move. This worldview— atomic materialism coupled with Newtonian physics—dominated science during the eighteenth and nineteenth centuries. When Faraday first proposed electromagnetic fields around 1830, most scientists thought of them as only a useful way to picture electromagnetic forces and not as real physical objects. Then Maxwell and Hertz showed that waves can travel in electromagnetic fields and that light is one example of these waves. So electromagnetic fields were not just a useful fiction; they were physically real, as real as light. The most convincing argument for the reality of electromagnetic fields comes from conservation of energy. Suppose a radio transmitter sends a message (an electromagnetic wave) to a receiver on Mars and that the message’s travel time is 20 minutes. Energy must travel from the sender to the receiver because it takes energy to cause the receiver to respond. Where is this energy during the 20 minutes between sending and receiving? Not in the sender. Not in the receiver. And energy never just vanishes. So it must be in the space between sender and receiver, in the electromagnetic field. So electromagnetic fields contain energy. Philosophers might disagree about what is “real,” but to a physicist nothing is more real than energy. Today we know that the universe is filled with gravitational, electromagnetic, and other kinds of invisible, but physically real, fields that are smoothly spread out in space and not made of tiny particles. This represents a real break with the Newtonian worldview. As Einstein put it, We may say that, before Maxwell, Physical Reality, in so far as it was to represent the processes of nature, was thought of as consisting in material particles. . . . Since Maxwell’s time, Physical Reality has been thought of as represented by continuous fields, . . . and not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and the most fruitful that physics has experienced since the time of Newton.6 6

A. Einstein, in James Clerk Maxwell, A Commemoration Volume. (The Macmillan Company, New York, 1931).

SO LUTION TO MAKI NG ESTI MATES The travel time for light is the distance to the lightbulb divided by lightspeed, 3 * 108 m>s. To make the arithmetic easy (remember that in estimates you want to choose approximate numbers that make the arithmetic easy), suppose the distance is 3 m: 3 m>(3 * 108 m>s) = 10-8 s, or a hundredth of a millionth of a second.

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This is quite a far-reaching statement. Einstein is not saying that the universe is made of some combination of material particles and fields, but rather that it’s made only of fields. Einstein’s view is beautifully confirmed by the development of physics since about 1950, when a theory based on quantum physics and known as “quantum field theory” came to dominate the way that physicists think about the structure of matter and energy. According to this extremely well-confirmed theory, the universe is made entirely of fields. Atoms, for example, are made of several kinds of fields known collectively as “matter fields,” and also of electromagnetic fields spread out smoothly over distances the size of an atom or smaller. Nineteenth-century physicists resisted giving up the Newtonian clockwork universe. So ingrained was the Newtonian worldview that scientists could not imagine that energy might exist apart from tiny material particles. They developed the idea that an extremely light gaslike material substance called ether filled all space. Ether was assumed to be a form of matter but made of some unknown substance rather than of the atoms that are familiar to us. Electromagnetic forces and other forces that act at a distance were assumed to be transmitted by ether. Light and other electromagnetic waves could then be explained mechanically, in terms of the motions of the material ether and in keeping with the Newtonian tradition. Maxwell, Hertz, and others accepted this ether theory of the electromagnetic force. After two centuries of the Newtonian worldview, it was difficult to think in any other way. Albert Einstein, early in the twentieth century, was one of the first to break out of this mold. He showed that the ether theory had to be rejected. But surprisingly, this had no effect on Maxwell’s theory or on the interpretation of light and radio as electromagnetic waves. It only affected the mechanistic interpretation of the electromagnetic field. After Einstein’s work, electromagnetic fields could no longer be interpreted as properties of a material substance. So the electromagnetic field turned out to be philosophically revolutionary, the first of many post-Newtonian physical ideas. Apparently the universe is not made entirely of atoms, not made like a mechanical clock. There is something else: fields. Although the Newtonian worldview still dominates much popular culture and even lies behind many scientists’ intuitive view of nature, the mechanical universe began to unravel around 1900 and is by now seriously out of tune with much of contemporary physics. The two major modern theories, relativity theory and quantum theory, contradict both the specific predictions and the conceptual underpinnings of Newtonian physics. Physics is still in the middle of the post-Newtonian revolution, and it is not clear what new scientific worldview will emerge.

6 THE COMPLETE SPECTRUM The possible frequencies of light lie in a narrow range near 1015 Hz. We have seen that Heinrich Hertz produced electromagnetic waves, now called radio waves, having a frequency around 109 Hz. These are only two examples of the huge range or “spectrum” of electromagnetic waves now known. We call this range the electromagnetic spectrum (Figure 27). In the figure, wavelengths are arranged from the longest at the bottom to the shortest at the top, with typical objects having the size of these wavelengths listed for comparison. Frequencies are shown from the lowest (smallest) at the bottom to the highest at the top, with typical sources of these frequencies listed. In order to display a large range, Figure 27 shows wavelengths and

241

Waves, Light, and Climate Change Typical Sources That Send out Waves at This Frequency:

Typical Object Whose Size Is the Same as This Wavelength:

Frequency, Hz

1022 Processes by protons and neutrons in atomic nuclei

Electrons in atoms, high-energy processes Electrons in atoms, low-energy processes

Gamma ray

X-ray

1018

Ultraviolet

1016 1014

Microwave oven

1012

Radar antenna

violet green yellow red

Cell phone

Infrared

108

TV, FM radio

AM radio antenna

106

AM radio

Atom

108

DNA molecule Amoeba

106

Fine dust particle

104 Millimeter

Radar

FM radio, TV antenna

1010

Visible

Microwave 1010

102

Centimeter

1

Meter

102

Soccer field Kilometer

Radio

60 Hz power-line radiation

Nucleus

1012

1020

Thermal vibrations of molecules

1014

104

104

102

106

1

108

Earth

Wavelength, m

Figure 27

The electromagnetic spectrum. There are no definite ends to the spectrum and no sharp boundaries between the regions.

frequencies on a so-called “logarithmic scale” in which each increment is a factor of 10: 1, 10, 100, 1000, and so forth. A “linear scale”—such as 10, 20, 30, 40, and so forth—could not do justice to the entire range. All these waves are the same phenomenon, namely, an electromagnetic field disturbance that is created by a vibrating charged object and that travels at lightspeed outward through the field. Energy from all these waves can be received by other charged objects that the wave moves as the wave passes by. All can travel through empty space. And all carry energy, known as radiant energy. Electromagnetic waves are often called electromagnetic radiation because they “radiate” out in all directions from charged objects. Since higher frequency means higher energy (Section 1), energies increase as we move up the scale from bottom to top. Let’s tour the electromagnetic spectrum. It’s useful to arrange it into the six regions shown in Figure 27, regions that correspond to different ways of either sending or receiving electromagnetic radiation. The longest wavelengths, down to about a millimeter, form the radio waves, comprising AM and FM radio, TV, and microwaves.

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Humans can create and control these waves electronically by causing electrons to vibrate in human-made electric circuits. Hertz’s waves fall into this category, and so does a lot of modern technology. AM radio waves at around 1000 kilohertz (106 Hz), FM radio and TV waves at around 100 megahertz (108 Hz), and cell phone transmission waves at around 1 gigahertz (109 Hz) are created by electrons moving back and forth along a metal antenna that is part of an electric circuit. Radar and microwaves, with frequencies up to a trillion (1012 ) hertz, also are created electronically. Many natural processes also create radio waves. Radio astronomers use radio receivers or “radio telescopes” pointed at stars or other astronomical objects to learn about the universe. In fact, astronomical objects produce electromagnetic radiation in all parts of the spectrum. During the past few decades, many new sorts of receivers, stationed on or above Earth, have produced an explosion of astronomical knowledge. Infrared radiation has wavelengths ranging from 1 millimeter to below 1/1000 of a millimeter—the size of a fine particle of baby powder. Infrared is typically created by the random thermal motion of molecules due to their thermal energy. Since all objects have thermal energy, all objects produce infrared and hotter objects produce more of it. Infrared detectors, such as certain infrared-sensitive chemicals, can detect warmer objects against a cooler background, which is the basis for night-vision devices and infrared photography. You cannot see infrared radiation but you can feel it. Since it’s created by thermal motion, it’s not surprising that it has the proper frequency to shake molecules into thermal motion—so it warms the objects it hits. When you feel the warmth of a fire or a hot plate at some distance away, you are using your skin as an infrared detector, “seeing” with your skin. Some animals have evolved highly developed infrared sensors for nocturnal vision. Many animals, including humans, have sensors that detect a narrow range of frequencies just above infrared. This range of visible radiation or “light” has wavelengths centering on 5 * 107 m. This is smaller than the finest dust particles and 5000 times larger than an atom. Light is typically created by electrons moving within individual atoms. The visible region’s defining characteristic is simply that the human eye is sensitive to it. Light waves entering the pupil of the eye strike the retina at the back (Figure 28). The retina is covered with light-sensitive cells that act like tiny antennae to receive electromagnetic waves in the visible range. Some cells respond differently to different wavelengths, and the brain interprets these as different colors. Suppose that you have a variable-frequency source of electromagnetic waves and that you set it to 6 * 1014 Hz—the frequency of green light. If you gradually decrease the frequency, this green light will change to yellow, then orange, and finally red. As you continue decreasing the frequency, the red becomes deeper and darker until, at about 4 * 1014 Hz, the frequency is so low that your retina can no longer respond to it. The source no longer emits visible light. The waves have crossed the boundary into infra- (below) red. The source still radiates, but your eye cannot detect it. Now go in the other direction. Beginning with 6 * 1014 Hz, increase the frequency. The color changes from green to blue to violet. The violet light darkens until, around 8 * 1014 Hz, your eye can no longer detect it. The waves have crossed into the ultra- (above) violet region. Ultraviolet radiation is created in the same way that light is created, by electrons moving within individual atoms. Although similar to light, ultraviolet’s higher energy has important consequences. Ultraviolet radiation has the proper frequency to shake many biological molecules, so it is readily absorbed by living things. And it has enough energy to split molecules, which can disrupt or kill living cells. If absorbed by a cell’s DNA, this can lead to cancerous growth.

Light beam focused on retina

Lens Light beam entering eye

Retina Optic nerve

Figure 28

The human eye.

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Waves, Light, and Climate Change Heater X-rays

Electrons   High voltage

Collision with metal plate creates X-rays

Figure 29

The operation of an X-ray tube. Electrons are boiled off a thin, heated wire filament and are accelerated toward a positively charged metal plate at the other end of the vacuum tube. When the electrons smash into this plate, their rapid deceleration causes them to emit X-rays, and the collision also causes the plate’s atoms to emit X-rays.

X-ray radiation also comes from electrons in individual atoms, but only from the highest-energy electron activities within atoms. X-ray wavelengths span a range around 10–10 m—about the size of an individual atom. Humans make X-rays in highenergy X-ray tubes, as described in the caption of Figure 29. X-rays have important interactions with biological matter. They have enough energy to ionize molecules within biological cells, that is, to knock the electrons right out of some molecules. Like ultraviolet, this radiation can cause cancers. Radiation energetic enough to ionize biological matter is called ionizing radiation. X-rays and gamma rays are ionizing radiations, and so is the higher energy (higher frequency) portion of the ultraviolet region. Because X-rays are able to penetrate deeply into biological matter, they can be put to the useful cause of examining the interior of the human body without surgery. There is a certain logic to our tour through the electromagnetic spectrum. As we move to smaller wavelengths, we move toward higher frequencies and hence higherenergy radiation, which in turn implies higher-energy processes to create the radiation. We also have progressed toward processes that occur in smaller and smaller regions of space: Radio waves are created in macroscopic electric circuits, infrared is created in molecules, and the next three (visible, ultraviolet, and X-ray) are created in atoms. It should come as no surprise then that the shortest-wavelength radiation, gamma radiation, carries the highest frequency and highest energy and comes from the highest-energy processes in the smallest regions of space. Gamma rays are created within atomic nuclei by high-energy nuclear processes involving the strong forces that hold the nucleus together. Gamma rays are created in radioactive materials and in the nuclear reactions known as “fission” and “fusion.” Like X-rays, gamma rays are a form of ionizing radiation and can damage biological matter. But this very feature is often put to use to destroy diseased cells and so cure some cancers. Since gamma ray wavelengths are much smaller than individual atoms, atoms cannot readily respond to them, and so they penetrate deeply into matter. The room you are in is full of electromagnetic waves. Hundreds of television and radio broadcasts, radio pulsations from neutron stars, radio noise from millions of normal stars, the faint background radiation from the big bang, possibly communications from extraterrestrial life, radiations from the sun and the center of our galaxy, and much more are passing through your room right now. Your body is equipped to receive directly only the tiny visible portion of the complete spectrum of these waves. With the proper receiver, you could sense any of the other frequencies. The universe would appear far different in other wavelength ranges and would appear complex indeed if you could directly receive the entire spectrum. The reality that meets your eye is only a tiny fraction of nature’s reality. CONCEPT CHECK 9 When your radio is tuned to 100 on the FM dial, it is receiving (a) a 100 Hz sound wave; (b) a 108 Hz sound wave; (c) a 100 Hz electromagnetic wave with a wavelength about the size of Earth; (d) a 106 Hz electromagnetic wave with a wavelength of around 100 m; (e) a 108 Hz electromagnetic wave with a wavelength of around 100 m; (f) a 108 Hz electromagnetic wave with a wavelength of around 1 m. CONCEPT CHECK 10 In the preceding question, the sound coming from the radio is (a) an electromagnetic wave traveling at 300,000 km/s; (b) an electromagnetic wave that travels far more slowly than 300,000 km/s; (c) not an electromagnetic wave of any kind, and travels far more slowly than 300,000 km/s.

244

Waves, Light, and Climate Change MAKI NG ESTI MATES In the following list, which of these waves have wavelengths much bigger than your room (a few meters), which have wavelengths between a millimeter and a few meters, and which have wavelengths of less than a millimeter: AM radio, light, electromagnetic waves from the alternating current that oscillates 60 times each second in your house circuits, warming rays from a fire, rays from a microwave oven, radar, electromagnetic radiation from shaking an electrically charged blouse that you remove from the dryer?

7 SOLAR RADIATION: THE LIGHT FROM OUR STAR The sun, “Sol,” transmits electromagnetic waves in every region of the spectrum. Most of this solar radiation is in the visible, infrared, and ultraviolet parts of the spectrum and is created at the sun’s visible surface. Other solar radiation is created in the rarefied, very hot gas that surrounds the sun in the same way that Earth’s atmosphere surrounds Earth. Processes within the sun’s atmosphere create high-energy X-rays and some gamma rays, along with radio waves. The intense radiation created by high-energy processes deep within the sun is absorbed and altered within the sun, and little of it escapes directly. Figure 30 graphs the relative amounts of radiant energy emitted by the sun at different wavelengths. When you sit in the sunlight, your eyes detect the sun’s visible radiation and your skin detects its infrared as warmth. Your skin also detects ultraviolet but you don’t notice it until a little later, as the cellular damage known as “sunburning.” The amount of solar energy reaching Earth is different at different locations, in different seasons, in different weather conditions, and at different times of the day. In the United States, an average 200 watts (200 joules every second) strikes every square meter of the ground.

Relative amount of energy

Figure 30

Ultraviolet Visible

0

500

The relative amounts of energy at different wavelengths in the solar spectrum. Most of the sun’s radiant energy is in the ultraviolet, visible, and infrared portions of the electromagnetic spectrum.

Infrared

1000 1500 2000 2500 Wavelength, in nanometers (109m)

3000

SO LUTION TO MAKI NG ESTI MATES Use Figure 27. Much bigger than a few meters: AM radio, waves from alternating current, waves from the blouse. One millimeter up to a few meters: microwaves, radar. Less than 1 mm: warming rays, light.

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Waves, Light, and Climate Change MAKI NG ESTI MATES Photovoltaic cells are devices that transform solar energy into electric current. If such devices were 100% efficient, about how much area would need to be covered by these cells in order to provide the average 1.3 kilowatts of electric power that a typical family home uses? Actual photovoltaic cells are only about 15% (one-seventh) efficient. At this efficiency, how much area must be covered? Could you put this on your roof ?

CONCEPT CHECK 11 When energy from the sun is absorbed by your skin (a) it remains there as electromagnetic energy; (b) it remains there as radiant energy; (c) it transforms into nuclear energy; (d) it transforms into kinetic energy; (e) it transforms into thermal energy; (f ) it gives you the heebie-jeebies.

8 GLOBAL OZONE DEPLETION: A VULNERABLE PLANET7

High-temperature object such as your kitchen

ThermE out Cooling system

Work ThermE in

From electric company

Low-temperature object such as the inside of your refrigerator

This and the next section apply our knowledge of molecules, energy, and electromagnetic radiation to discuss two environmental issues: ozone depletion and global warming. These represent a new kind of social issue: environmental effects that cannot be contained locally and are truly global. As part of the dues that we all must pay if Earth is to pull through its current experiment with powerful new technologies, we’d better think seriously about such issues. An invisible trace of gas drifting 10 to 50 kilometers overhead protects life on Earth from the sun’s ultraviolet radiation. This wispy “ozone layer” would be only 2 millimeters thick if compressed to normal atmospheric pressure, but without it most life on Earth would soon cease. During the past century, humans using common household chemicals destroyed a large portion of this ozone. Belatedly realizing what we had wrought, we all came together in the nick of time to agree to ban the offending chemicals. The story of ozone depletion demonstrates the threat of global environmental destruction, shows that strong collective counter-action is possible, and offers an encouraging lesson for these times. The story begins in 1928 when the General Motors Corporation first synthesized chlorofluorocarbons (CFCs), molecules made from atoms of chlorine, fluorine, and carbon, for its Frigidaire refrigerators. CFCs are chemically inert, meaning that they do not readily react with other substances. They normally form a gas but become liquid when put under high pressure. Being inert, they are nontoxic to humans, noncorrosive in mechanical devices, and nonflammable. Such a chemical can have many uses, one of which is as a coolant. Refrigerators and air conditioners operate like heat engines in reverse (Figure 31). Just as heat engines use hot gases as the “working fluid” that does work while exhausting thermal energy, so refrigerators and air conditioners use a “coolant” that has work done on it while extracting thermal energy from a refrigerator or house.

Figure 31

Energy flow diagram for a refrigerator. Refrigerators operate like heat engines in reverse. An outside energy source does work to push thermal energy “uphill,” from a low to a high temperature.

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7

The U.S. government maintains an informative Web page on this topic, at http://www.epa.gov. Click on “index” and find “ozone.”

SO LUTION TO MAKI NG ESTI MATES 1.3 kilowatts (1300 watts) of solar energy falls on 6.5 square meters (1300/200) of surface. At an efficiency of one-seventh, it would take seven times this much area: 45 square meters. If square-shaped, this would be about 7 m on a side and might fit on your roof.

Waves, Light, and Climate Change

CFCs soon became a universal coolant. Production soared. In the 1940s, CFCs were found to be useful as pressurized gases to propel aerosol sprays. In the 1950s, they created the air-conditioning revolution that facilitated America’s shopping malls, summer automobiling odysseys, and population shifts to Southwestern cities. CFCs created lots of business and little fuss until 1974 when scientists began to ask where all these inert gas molecules might be drifting. After all, being inert they were nearly indestructible, so essentially all the CFCs manufactured since 1930 should still be in the atmosphere. But where? And what became of them there? During the previous four decades of profitable production, nobody had bothered to ask. In 1974, two university chemists suggested an alarming possibility. Mario Molina and Sherwood Rowland (Figure 32) discovered that because CFC molecules are inert and gaseous, they are not chemically broken down or rained out in the lower atmosphere. Instead, they drift slowly into the upper atmosphere or stratosphere, 10 to 50 kilometers overhead, where they may remain intact for decades or centuries. Molina and Rowland theorized that high-energy solar ultraviolet radiation should eventually split CFC molecules apart, releasing large quantities of chlorine. This was alarming because chlorine reacts strongly with O3, known as ozone. Ozone is one of a long list of trace gases in the atmosphere—gases that, all together, make up far less than 1% of the atmosphere. Table 1 shows only a few of these trace gases, namely those whose concentrations are one part per million (that is, one molecule per million atmospheric molecules) or larger. The list would be far longer if it were extended down to one part per billion! Here’s how chlorine destroys atmospheric ozone: Ozone is produced naturally in the stratosphere from O2 when high-energy radiation from the sun breaks up O2 molecules and the resulting oxygen atoms then combine with O2 to create O3. But ozone can be easily broken down by this reaction with chlorine:

Nobelstiftelsen/The Nobel Foundation Figure 32

Paul Crutzen, Mario Molina, F. Sherwood Rowland. The theory of stratospheric ozone depletion by means of human-made compounds was recognized by the awarding of the 1995 Nobel Prize in Chemistry to the three scientists who discovered it. Crutzen’s work involved nitrogen compounds, while Molina and Rowland’s work involved chlorine compounds. The Nobel Committee commended them for having “contributed to our salvation from a global environmental problem that could have catastrophic consequences.” Table 1 Composition of the atmosphere Molecule

Major constituents

Cl + O3 : ClO + O2 Under bombardment by the ultraviolet radiation at that altitude, a second reaction then occurs:

78.084 %

O2

20.946 % 00.934 % Total

99.964 %

Trace gasesa (in ppmb)

This second reaction releases the chlorine, which is then free to destroy more ozone. Scientists found that a single Cl atom destroyed about 100,000 ozone molecules. A little chlorine goes a long way. But stratospheric ozone is essential to most life on Earth.8 Because ozone molecules vibrate naturally at ultraviolet frequencies and so absorb much of the sun’s ultraviolet radiation, they protect us from this biologically harmful radiation. Unfortunately, ozone is easily altered by human activities because there is so little of it in the atmosphere. Out of every 10 million atmospheric molecules, only 3 are ozone! To get a feel for this, imagine a huge city of 10 million, in which just 3 people are labeled “ozone.” We see in ozone the potential importance of each of the many atmospheric trace gases (Table 1). In 1974, most people regarded it as absurd to think that coolants and spray cans could cause a catastrophe, because it violated the intuitive notion that human activities were far too puny to alter the global environment. But some were alarmed. A debate, of Ground-level ozone, on the other hand, is a toxic pollutant. It is a consequence of automobile exhaust and is the primary component of urban smog.

N2 Ar

ClO + ClO + sunlight : Cl + Cl + O2

8

Concentration

CO2

377

H2O

20–20,000

Ne

12

He

5

NO2

2

CH4

2

Xe

2

Kr

1

O3

3 waves>s = 1>3 Hz. 7. Solving s = fl for l, l = s>f = 2 m>s > 3 Hz = 0.67 m. 9. Solving s = fl for l, l = s>f = 3 * 108 m>s > 108 Hz = 3 m. 11. (5000 km)>(300,000 km>s) = 0.017 s. 13. 3 * 105 km>s * 8.3 min * 60 s>min = 1.5 * 108 km = 150 million km. 15. Solving s = fl for l, we get l = s>f = (3 * 108 m>s)>60 = 5 * 106 m = 5000 km. 17. Solving s = lf for f, we get f = s>l = (3 * 108 m>s)> 589 * 10 - 9 m = 5.09 * 1014 Hz. 19. 2.7 * 1019>106 = 2.7 * 1013 = 27 trillion. Even at “only” 1 ppm, there are still a lot of Krypton atoms in a cubic centimeter of air!

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The Special Theory of Relativity

From Chapter 10 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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The Special Theory of Relativity

Nature and Nature’s laws lay hid in night: God said, “Let Newton be” and all was light. Alexander Pope

It did not last: the Devil, shouting “Ho Let Einstein be” restored the status quo. Sir John Collings Squire

P

hysics changed around 1900. Physicists began investigating phenomena farremoved from the normal range of human experience, things like the structure of atoms and the precise speed of a light beam. They found that Newtonian physics and nineteenth-century electricity and magnetism were far off the mark in phenomena involving very high speeds, very strong gravitational forces, large astronomical regions, and the microscopic world. To deal with these new realms, they invented new theories called special relativity (this chapter), general relativity, and quantum physics. All of these new theories reproduce, nearly exactly, the standard Newtonian predictions within the normal range of human perceptions. For example, special relativity correctly predicts new, non-Newtonian results for objects moving at speeds comparable to lightspeed, but also correctly predicts the normal Newtonian results for slower-moving objects such as cars and speeding bullets. But despite this similarity within the normal range of human perception, the concepts behind these new theories are quite unlike the concepts behind Newtonian physics. For example, Newtonian physics describes the universe as a kind of giant predictable clockwork mechanism. But you’ll find that, according to quantum physics, the universe is nothing like a clock, quite non-mechanical, and far from predictable. The new theories represent the most accurate knowledge known about the real physical universe, and they describe a different universe from what you would have expected on the basis of Newtonian or pre-Newtonian concepts. So expect your preconceptions about space, time, motion, gravity, matter, energy, and physical reality to be assaulted. Each of these new theories has a non-intuitive oddness about it, as might be expected since they deal with phenomena beyond your normal range of perception. In this chapter you’ll learn about some unexpected effects that happen when objects move at high speeds, speeds comparable to lightspeed. You’ll also learn that space and time aren’t quite what you thought they were, and you’ll learn something new and, for most people, amazing about energy. Einstein’s “special theory of relativity” is based on

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two simple ideas and all of its odd conclusions are off-shoots of these. This theory has a reputation for being difficult, but this comes really from its strangeness rather than any inherent difficulty. Its conclusions violate common sense. The main requirement for understanding this theory is not intelligence but mental flexibility. Einstein created two related theories of relativity. The “special” theory of relativity, discussed in this chapter, revolutionizes the way we think about space and time, and this leads to a further revolution in our concepts of mass and energy. The “general” theory of relativity revolutionizes our concepts of space and time even further, and radically reformulates the way we look at gravity. Following some historical context in Section 1, Section 2 discusses the older “Galilean” way of viewing the phenomena with which Einstein was concerned. Sections 3 and 4 cover the theory’s two key laws: the principle of relativity and the principle of the constancy of lightspeed. Sections 5 and 6 present Einstein’s prediction of the relativity of time. Section 7 presents two more predictions: the relativity of space and the relativity of mass. Section 8 presents Einstein’s famous prediction of the equivalence of energy and mass, the aspect of special relativity that Einstein himself thought was most important, and discusses its profound significance.

1 EINSTEIN: REBEL WITH A CAUSE The Scottish mathematician and physicist Lord William Thomson Kelvin stated in an address to physicists at the British Association for the Advancement of Science in 1900 that “There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.” Many scientists1 of that day shared Kelvin’s confidence that the known “grand unifying principles”—Newton’s laws and the laws of thermodynamics and electromagnetism—were complete and permanent. But the world soon changed. In 1900, Max Planck introduced a revolutionary new principle, the quantum of energy. And a scant five years later, in 1905, a quite different but equally revolutionary theory was hatched in the brain of an obscure patent clerk in Bern, Switzerland: Albert Einstein (Figures 1 and 2). Einstein was a rebel in more ways than one. In his midteens he got fed up with high school and dropped out. This surprised no one, for he had been a mediocre student and a daydreamer since beginning elementary school. Before that he had been a slow child, learning to speak only at 3 years of age. His high school teachers were glad to see him go, one of them informing Einstein that he would “never amount to anything” and another suggesting that he leave school because his presence destroyed student discipline. Einstein was delighted to comply. He spent the next few months as a model dropout, hiking and loafing around the Italian Alps. After deciding to study engineering, he applied for admission to the Swiss Federal Polytechnic University in Zurich, but he failed his entrance exams. It seems he had problems with biology and French. To prepare for another try, he spent a year at a Swiss high school, where he flourished in this particular school’s progressive and democratic atmosphere. He recalled later that it was here that he had his first ideas leading to the theory of relativity. The university now admitted Einstein. He was known as a charming but indifferent university student who attended cafes regularly (where he enjoyed discussing philosophy and science) and lectures sporadically (because he preferred to spend time in physics laboratories). He managed to 1

I thought of that while riding my bicycle. Einstein, on the Theory of Relativity, in the Quotable Cyclist.

Common sense is nothing more than a deposit of prejudices laid down by the mind before you reach eighteen. Einstein

My intellectual development was retarded, as a result of which I began to wonder about space and time (things which a normal adult has thought of as a child) only when I had grown up. Einstein

If I were a young man again and had to decide how to make a living, I would not try to become a scientist or scholar or teacher. I would rather choose to be a plumber or a peddler, in the hope of finding that modest degree of independence still available under present circumstances. Einstein, in a remark made near the end of his life.

But perhaps not most scientists. Many physicists were dissatisfied with the theoretical foundations of physics and rejected Newtonian mechanics as the basis for physics in favor of electromagnetism.

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Were I wrong, one professor would have been quite enough. Einstein, when asked about a book in which 100 Nazi professors charged him with scientific error.

Figure 1

Figure 2

Never one to take himself too seriously, Einstein stuck his tongue out when asked to smile on his seventysecond birthday.

Albert Einstein during his student days in Zurich, a few years before he created his special theory of relativity.

pass the necessary exams and eventually graduate with the help of friends who shared their systematic class notes with the nonconforming Einstein. Following his graduation in 1900, Einstein applied for an assistantship to do graduate study, but it went to someone else. After looking unsuccessfully for a teaching position, in 1902 a friend helped him land a job as a patent examiner. Einstein often referred to his seven years at this job as “a kind of salvation” that paid the rent and occupied only 8 hours a day, leaving him the rest of the day to ponder nature. And ponder he did. One of the many remarkable aspects of the theory of relativity is that it was invented nearly single-handedly.

2 GALILEAN RELATIVITY: RELATIVITY ACCORDING TO NEWTONIAN PHYSICS Here is a typical relativity question: Suppose that a train passenger, call her Velma, throws a baseball toward the front of the train. Both she and Mortimer, who is standing on the ground watching the passing train, measure the baseball’s speed (Figure 3). Will they get the same answer? If not, how will their answers differ? Think about it. This question concerns two observers who are moving differently. We say that Velma and Mort are in relative motion whenever they are moving at different speeds or in different directions. A theory of relativity is any theory that works out answers to questions concerning observers who are in relative motion. You can think of the train as being Velma’s laboratory, or her reference frame, within which Velma measures things like the speed of the ball. You can think of the ground beside the tracks as a second reference frame, Mort’s reference frame, for his measurements. The standard question that any theory of relativity asks is how measurements made in one reference frame compare with those made in another. Scientists have thought about questions like this since at least the time of Galileo.

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Figure 3

Velma throwing a ball, observed by Mort.

To be more specific, suppose that the train moves at 70 meters per second (150 miles per hour, a typical modern train speed). Suppose that Velma throws the baseball toward the front of the train at 20 m/s “relative to Velma” (as measured on the train, using meter sticks and clocks that are on the train). How fast does the baseball move “relative to Mort” (as measured on the ground)? …Think about that. Well, during each second, the baseball moves 20 meters toward the front of the train as measured by Velma. But as observed by Mort, the baseball moves an additional 70 meters during that same second, because the train itself moves 70 meters. So the ball must move at 90 m/s relative to Mort. Right? Because Galileo would have given the same answer four centuries ago, this straightforward and fairly intuitive form of relativity is called Galilean relativity. CONCEPT CHECK 1 Velma’s normal ball-throwing speed is 20 m/s. She is in a train moving eastward at 70 m/s and throws a ball toward the rear of the train. The velocity of the ball relative to Velma is (a) 50 m/s eastward; (b) 50 m/s westward; (c) 20 m/s eastward; (d) 20 m/s westward; (e) 70 m/s eastward; (f) 70 m/s westward. CONCEPT CHECK 2 In the preceding question, the velocity of the ball relative to Mort, who is standing beside the tracks, is (a) 50 m/s eastward; (b) 50 m/s westward; (c) 20 m/s eastward; (d) 20 m/s westward; (e) 70 m/s eastward; (f) 70 m/s westward.

Let’s turn to a similar example involving light beams instead of baseballs. Light is an electromagnetic wave moving at 300,000 km/s, a speed that I will symbolize by the letter c. It’s difficult to imagine such a high speed. A light beam travels from New York to Los Angeles in a hundredth of a second. Trains, jet planes, and even Earth satellites moving at 8 km/s are slowpokes by comparison. Imagine that Velma pilots a really fast rocket ship past Earth at 75,000 km/s, or 0.25c (25% of lightspeed), and that she holds a source of light—a flashlight or a laser—pointed forward. Mort stands on Earth. What would be the speed of the light

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Velma’s spaceship moves past Mort at a speed of 0.25c

Velma’s light beam moves away from Velma at speed c

Figure 4

How fast is Velma’s light beam moving, as observed by Mort?

beam—the moving tip of the beam—relative to Velma and relative to Mort? It seems plausible that Velma measures the light beam to move at speed c, since she’s holding the light source. In fact, experiments with light beams emitted by moving sources show this to be true: Any light beam from a moving source moves at speed c relative to the source.2 What speed would Mort measure for the same light beam (Figure 4)? Following the logic of the baseball example, the sensible answer would seem to be 1.25c. After all, the light beam travels 300,000 km in each second as measured by Velma, and Velma travels 75,000 km in each second as measured by Mort, so it seems sensible that the light beam would travel 300,000 km + 75,000 km in each second, or 375,000 km/s, as measured by Mort. This is the answer Galileo would have given, the answer given by Galilean relativity. It is the answer that all scientists would have given up through the end of the nineteenth century. It is indeed a most sensible answer. Nevertheless, it’s experimentally wrong. Nature does not always comply with our notion of what is sensible! To see why there might be something wrong with this answer and to learn nature’s answer, let’s turn in the next two sections to Einstein’s thoughts.

3 THE PRINCIPLE OF RELATIVITY You ride in a smoothly moving unaccelerated jet airplane in level flight at unchanging velocity. The flight attendant pours you a cup of coffee. Where should you hold your cup: directly under the spout, or someplace else to take into account the motion of the airplane? In other words, does the coffee pour straight downward relative to the airplane? Try it sometime and see. Or try dropping a coin from one hand to the other in a moving vehicle (but not when you are driving): Is the catching hand directly beneath the dropping hand? The answer is that the coffee pours straight downward relative to the plane. You could experiment with many things, all within a smoothly moving reference frame: a falling ball, frictionless air coasters, electric currents, magnets, and more.

2

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Provided the source is not accelerating; see Section 4.

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Just as for the poured coffee, you would find that the results are the same as when the experiments are performed in a reference frame at rest on Earth. Suppose you are a passenger on an airplane with no windows in the passenger compartment. You fall asleep and awaken later to find yourself alone in the compartment. Can you tell, without receiving information from the outside world,3 whether your airplane is in level flight at unchanging velocity or parked on the ground? The answer is no. You could throw a ball, do handstands, pick up nails with magnets, and the like, and everything would be the same, regardless of whether your plane was in flight or parked. This is another example of a symmetry principle. It says that, no matter from what nonaccelerating reference frame you view the universe, the laws of physics are the same. I’ll summarize this as: The Principle of Relativity Every nonaccelerated observer observes the same laws of nature. In other words, no experiment performed within a sealed room moving at an unchanging velocity can tell you whether you are standing still or moving.

Unless you look outside, you can’t tell how fast you’re going. It’s a plausible idea and was the key to Einstein’s thinking about relativity. It’s called the “principle of relativity” because it says that all motion is just relative motion. When you say “the car moves at 25 km/hr westward,” you really mean that “the car moves westward at 25 km/hr relative to the ground” or that “the car and the ground are in relative motion at 25 km/hr.” You could just as well say that the car is standing still and the ground is moving eastward at 25 km/hr. You could even say that the ground is moving eastward at 1600 km/hr (which it is, relative to Earth’s center, due to Earth’s spin) and that the car is moving eastward at only 1575 km/hr. It is only the relative speed, the 25 km/hr, that really counts. CONCEPT CHECK 3 What about acceleration—can this be detected without looking outside? (a) Yes, you can do simple experiments to tell you whether you are accelerating. (b) Yes, but the experiments must involve light beams. (c) No.

4 THE CONSTANCY OF LIGHTSPEED: STRANGE BUT TRUE Have you ever asked yourself what it would be like if you could keep up with a light beam? Some people do. The 16-year-old Einstein did, and his reflections on this question helped lead him to his theory of relativity. To Einstein, the possibility of moving along with a light beam seemed paradoxical, contradictory. The reason is that, to an observer moving along with a light beam, the light beam itself would be at rest. To this observer, the light beam would appear as an electromagnetic “wave” that was standing still! To Einstein, this seemed absurd. Here’s why.

3

The Lord is subtle, but He is not malicious. Einstein

Information from the pilot would be from the outside world, because the pilot’s information enters through the cockpit window and through radio receivers.

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The Special Theory of Relativity You could see that Einstein was motivated not by logic in the narrow sense of the word but by a sense of beauty. He was always looking for beauty in his work. Equally he was moved by a profound religious sense fulfilled in finding wonderful laws, simple laws in the universe. It was really a religious experience for him, of the most profound sort, even though he did not believe in a personal god. Banesh Hoffmann, Mathematician and Author, in Some Strangeness in the Proportion

Our understanding of electromagnetic waves, such as light, is based on Maxwell’s theory of electromagnetic fields. Recall that Maxwell’s theory predicts that any disturbance in an electromagnetic field, such as a disturbance caused by the motion of an electrically charged object, must propagate as a wave moving outward through the field at speed c. This particular speed, 300,000 km/s, is built into Maxwell’s theory. Einstein believed that Maxwell’s theory should, like all other laws of nature, obey the principle of relativity. So Maxwell’s predictions should be correct within every moving reference frame. Since speed c is built into Maxwell’s theory, Einstein concluded that every observer ought to observe every light beam to move at speed c, regardless of the observer’s motion. No matter how fast you move, a light beam should always pass you at speed c, relative to you. If every observer sees every light beam move at speed c, then nobody can even begin to catch up with a light beam, much less move along with a light beam. It’s a simple idea. But it’s also pretty crazy, which is why it took Einstein to think of it. After all, if you run after a departing light beam, common sense tells you that from your perspective the speed of the departing light must be less than 300,000 km/s. And if you run toward an approaching light beam, common sense says that the speed of the approaching light must be greater than 300,000 km/s. Einstein’s idea is so odd that other turn-of-the-century physicists who might have discovered it did not. It’s the second important principle underlying Einstein’s theory. I’ll summarize it as: The Principle of the Constancy of Lightspeed The speed of light (and of other electromagnetic radiation) in empty space is the same for all nonaccelerated observers, regardless of the motion of the light source or of the observer.

I don’t try to imagine a personal God; it suffices to stand in awe at the structure of the world, insofar as it allows our inadequate senses to appreciate it. Einstein

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Like the principle of relativity, this principle is valid only for nonaccelerated observers. The reason is that Maxwell’s theory, like most laws of physics, is valid only for nonaccelerated observers. To get a feel for it, we’ll apply this principle to several “thought experiments,” impractical experiments that could in principle be performed. Each experiment involves a light beam, which we take to be a laser beam but which could just as well be a flashlight beam. Suppose Velma moves away from Mort at a quarter of lightspeed and holds a laser pointed forward, as in Figure 4. As noted in Section 2, she observes the beam to move away from her at speed c. What speed does Mort observe for the laser beam? Galilean relativity and our intuitions answer 1.25c, or 375,000 km/s. But Einstein’s relativity predicts that the answer is c, or 300,000 km/s! Another example: Mort has the laser and he shines it in the direction of Velma who is departing from him at a quarter of lightspeed (Figure 5). Mort observes the beam to move away from him at speed c, but what does Velma observe? Galileo, and common sense, now predict 0.75c, but Einstein predicts c. To dramatize the oddness of this, imagine that Velma is moving away from Mort at a speed of 0.999 999c, just a hair slower than lightspeed (Figure 6). Mort switches on his laser and sees the light beam depart from him at speed c. As

The Special Theory of Relativity Velma’s spaceship moves away from Mort at a speed of 0.25c

Mort’s light beam moves away from Mort at speed c

Figure 5

What is the speed of Mort’s light beam relative to Velma?

Velma’s spaceship moves past Mort at a speed of 0.999,999c

Mort’s light beam moves away from Mort at speed c. According to Mort, Velma is moving only 0.000,001c, or 300 m/s, slower than his light beam

Figure 6

Now how fast is Mort’s light beam moving, as observed by Velma?

observed by Mort, Velma moves only slightly slower than the light beam—he says that she nearly keeps up with the light beam. Galilean relativity predicts that Velma observes the light beam passing her at only 0.000 001c. This is just 300 m/s—the speed of fast jet airplanes. But Einstein’s relativity says that she sees the light beam pass her at precisely 300,000 km/s, despite the fact that she is moving away from the light source at nearly lightspeed! Maybe you’ve noticed that we don’t allow Velma to have precisely speed c. If we imagined that she moves right at speed c, we’d get into the difficulty that Einstein noted: She would observe the light beam to be at rest. So an observer can move at nearly, but not precisely, speed c relative to another observer. Later, we’ll see why. How do we know that light goes the same speed for all observers? Strange though the constancy of lightspeed may seem, it’s verified daily. However, most experiments involve fast-moving microscopic particles rather than spaceships. In one especially striking experiment in 1964, a subatomic particle moving at nearly lightspeed emitted

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The Special Theory of Relativity electromagnetic radiation both forward and backward. Galilean relativity predicts that the forward-moving radiation should move much faster than c while the backward-moving radiation should move much slower than c, as measured in the laboratory. But measurement showed that both radiation beams move at speed c relative to the laboratory.

Maxwell and other nineteenth-century scientists had a more conventional view of light beams. They believed that light was a wave in a material medium, just as water waves are waves in water. They called this medium ether. Nobody had observed ether. It couldn’t be made of ordinary atoms, because light waves travel through outer space where there are essentially no atoms. Instead, ether was thought to be a continuous material substance filling the entire universe and made of some unknown nonatomic form of matter. The ether theory assumes that the “natural” speed of light, 300,000 km/s, is light’s speed relative to the ether. Observers moving through the ether should then measure other speeds for light beams, speeds that should depend on the observer’s speed through the ether. But as the principle of the constancy of lightspeed states, and as experiment shows, all observers measure the same speed for all lightbeams, so the ether theory must be wrong. Since Einstein, electromagnetic waves have been viewed as the vibrations of an electromagnetic field, which itself is not made of any material substance. This contrasts sharply with the materialist worldview of Newtonian physics. The constancy of lightspeed is the key principle that gives the theory of relativity its odd quality. It’s natural to question this principle. How do we know it’s true? The answer is simple but profound: It’s true because nature says so. Numerous experiments show that every light beam moves at speed c, regardless of the motion of the source or observer. Although this odd notion violates our preconceived beliefs, it is observation of nature, rather than preconceived beliefs, that determines truth in science. Our preconceived beliefs about motion are based on observations of objects moving far slower than lightspeed and are very nearly correct at such speeds. But at higher speeds, our preconceptions are radically incorrect. The foundations of Einstein’s theory are the principle of relativity and the constancy of lightspeed. Their role in the theory of relativity is identical to the role of Newton’s laws in Newton’s theory of force and motion: They form the logical basis of the theory, from which everything else is derived and which are themselves justified directly by observation. Physicists call this theory the special theory of relativity. The word special distinguishes this theory from another, related theory of Einstein’s called the general theory of relativity. The distinguishing feature of the general theory of relativity is that it allows accelerated observers, while the special theory allows only nonaccelerated observers, so the general theory is a more general—broader—theory than the special theory. Strictly speaking, Earth itself is an accelerated reference frame, because it spins on its axis and because it rotates around the sun. But these accelerations are so small that the predictions of the special theory are excellent approximations for any Earth-based observer. The remainder of this chapter explores five of special relativity’s most important predictions: the relativity of time, the relativity of space, the relativity of mass, c as the speed limit, and E = mc2. CONCEPT CHECK 4 Velma moves away from Mort at 0.75c. She turns on two lasers, one pointed forward and the other backward. According to Galilean relativity, how fast should the forward and backward beams move, as observed by Mort? (a) 0.25c and 1.75c. (b) 1.75c and 0.25c. (c) 0.25c and 0.75c. (d) 0.75c and 0.25c. (e) c and c.

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CONCEPT CHECK 5 In the preceding question, Mort actually observes (a) 0.25c and 1.75c; (b) 1.75c and 0.25c; (c) 0.25c and 0.75c; (d) 0.75c and 0.25c; (e) c and c.

5 THE RELATIVITY OF TIME The constancy of lightspeed suggests something is amiss in our intuitive conceptions of space and time. After all, speed measures how far an object moves through space divided by the time to move, so speed is intimately tied to space and time. We feel that we understand what “time” is, but its meaning fades when we ponder it. Being enmeshed in time, we cannot study it from a distance, so our attempts to define it are usually circular, implicitly using the concept of time in order to define time. Einstein’s insight into time was that it’s physical—part of the physical universe. Just as one can measure the properties of a stone or of a light beam, one can measure the properties of time. And how should we measure the properties of time? With clocks! This reply is more profound than it appears. The only way we can measure time is with real, physical “clocks,” by which we mean any phenomenon—a swinging pendulum, Earth’s rotation around the sun—that goes through identical repetitions. Physically, the concept of a clock really defines time. So to investigate the properties of time, we must investigate clocks. How do clocks really behave? Einstein managed to predict the properties of clocks using as his starting point only the two principles of the special theory of relativity. An ordinary spring-wound or battery-driven clock would be hard to study based only on Einstein’s two principles because these clocks are so complex, involving springs, electric current, gears, and so forth. So Einstein invented a simple kind of clock, a simple thought experiment, really. His light clock (Figure 7) involves no mechanically moving parts; its only motion is the motion of a light beam. Two parallel mirrors face each other, one above the other, and a light beam bounces up and down (reflects) between them. Although it’s not terribly practical for the clock maker, it’s convenient to imagine that the mirrors are separated by 150,000 kilometers, because then the time for one complete round trip of the light beam is just 1 second. You know it’s 1 second because the constancy of lightspeed says all light beams travel 300,000 kilometers in 1 second. We’ll assume this light clock ticks at the end of each round-trip. We begin investigating the properties of time by installing one light clock in Velma’s spaceship moving eastward past Earth, and another in Mort’s laboratory on Earth. Let’s think about Velma’s light clock. She sees her light beam bouncing straight up and down, covering 300,000 km per tick [Figure 8(a)]. Simple enough. But from Mort’s point of view, the tip of Velma’s light beam is not only moving up and down, it’s also moving eastward because of Velma’s eastward motion. So the tip of Velma’s light beam, as seen by Mort, moves along diagonal paths. Figure 8b shows Mort’s observations of Velma’s spaceship at three instants: when the tip of her light beam is at the bottom mirror, when it has moved up to the top mirror, and when it is back at the bottom mirror. Since the distance between the mirrors is 150,000 km, you can see from the figure that the distance along one of the two diagonals is greater than 150,000 km. This means that the total round-trip distance traveled by Velma’s light beam, as measured by Mort, is greater than 300,000 km. There is nothing surprising or subtle about this; Galileo would have said the same thing. Now comes the part that Galileo (and our intuitions) wouldn’t agree with: The constancy of lightspeed says that Mort observes

Mirrored faces

Tip of light beam

150,000 kilometers

Figure 7

A light clock. A light beam bounces up and down between two mirrors. If the distance between mirrors is 150,000 km, then 1 second will elapse during one complete round-trip up and back down.

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(a)

(b)

Mort’s light clock

Figure 8

(a) Velma in her spaceship, observing her light clock. (b) Velma’s spaceship and the light beam on Velma’s light clock as observed by Mort using his own light clock. According to Mort’s observations, the tip of Velma’s light beam moves along the diagonal path shown by the dashed arrows.

Velma’s light beam to move at just 300,000 km/s (Galileo would say that Mort observes Velma’s light beam to move faster than 300,000 km/s, because of Velma’s motion). Since Mort observes the round-trip distance to be greater than 300,000 km, it follows that according to Mort it takes more than 1 second for Velma’s light beam to make the round-trip! So, as measured by Mort using his clock, more than 1 second elapses between Velma’s ticks. According to Mort, Velma’s clock runs slow. Velma’s second is different from Mort’s second. The two observers measure different time intervals for the same event (one round-trip of Velma’s light beam). Time is relative to the observer. It’s simple, but hard to believe. Let’s turn things around. How does Mort’s clock appear to the two observers? To Mort, his own clock’s light beam travels 300,000 km in one round-trip and requires 1 second to do so. But from Velma’s viewpoint, Mort’s clock is moving westward, so the tip of Mort’s light beam is moving along a diagonal and therefore the total round-trip distance traveled by Mort’s light beam as observed by Velma is greater than 300,000 km. But because Velma observes Mort’s light beam to move at 300,000 km/s, she must observe that more than 1 second elapses between Mort’s ticks. According to Velma, it’s Mort’s clock that runs slow. The rule is moving clocks run slow: Mort and Velma both observe that the other person’s clock runs slow. This is not your normal situation caused by an inaccurate clock, in which if my clock runs slow according to your clock, then your clock must run fast according to my clock. This raises an interesting question: Whose clock is really running slow, and whose is really accurate? The answer is that Velma and Mort are both right! Velma observes that Mort’s clock is slow, and Mort observes that Velma’s clock is slow, and both observations are correct. This situation is not caused by inaccurate clocks; it is instead a property of time itself. There is no single “real” time in the universe, no “universal time”; there is only Mort’s time and Velma’s time and all the other possible observers’ times.

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As you might expect, there is a formula that quantitatively describes the relativity of time.4 Table 1 gives a few of the numerical results that can be calculated from this formula, and Figure 9 is a graph based on the same formula. As you can see from the table, the effect is negligible even at orbiting satellite speeds (10 to 20 km/s). It’s not until speeds of 0.1c—a speed that would get you around the world in 1 second—that the effect amounts to even a half of 1%. But at large fractions of lightspeed, the effect becomes quite large: At 99.9% of lightspeed (not shown on the graph), Mort and Velma’s seconds will be more than 22 seconds long as measured by the other observer. The relativity of time is also called time dilation, because a time interval of 1 second on a moving clock is expanded, or dilated, to more than 1 second as measured by an observer past whom the clock is moving. Although we investigated the relativity of time by studying light clocks, the conclusion holds for every type of clock—every regularly repeating phenomenon. Einstein thought about light clocks only in order to learn what the two principles of his theory implied about time. Every clock must behave the way a light clock behaves because they all measure the same thing: time. And every phenomenon that occurs during an interval of time must also behave in this way. Think, for example, of an ice-cream cone melting. Suppose you can make ice-cream cones that melt in exactly 10 minutes and that both Velma and Mort have one of these cones. These cones are a kind of clock, a clock that ticks in 10 minutes.

It requires a very unusual mind to undertake the analysis of the obvious. Alfred North Whitehead, TwentiethCentury Philosopher

Time in seconds

CONCEPT CHECK 6 Mort and Velma have identical 10-minute ice-cream cones. Velma passes Mort at 75% of lightspeed. Use Table 1 to predict the times measured by Mort for his and Velma’s cone to melt. (a) 10 minutes for Mort’s cone, 10 minutes for Velma’s cone. (b) 10.5 minutes and 10 minutes. (c) 10 minutes and 10.5 minutes. (d) 15 minutes and 10 minutes. (e) 10 minutes and 15 minutes.

4

3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0

Figure 9

0

0.1c

0.2c

0.3c

0.4c

0.5c 0.6c Speed

0.7c

0.8c

0.9c

c

The relativity of time. The graph shows the duration of one clock tick (representing 1 second in the clock’s reference frame) on a moving clock, for various speeds of the clock relative to the observer.

This formula can be derived from Figure 8 by using the Pythagorean theorem, which states that a right triangle’s short side lengths a and b are related to its diagonal length c by c2 = a2 + b2. The formula is T = To> 2(1 - y2>c2), where y is the relative speed, To is the time between two of Velma’s ticks as observed by Velma (To = 1 second), and T is the time between two of Velma’s ticks as observed by Mort.

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The Special Theory of Relativity Table 1 The relativity of time: some quantitative predictions To give you a feel for these speeds: 0.3 km/s is a typical subsonic jet plane speed, 3 km/s is twice the speed of a high-powered rifle bullet, at 3000 km/s you could cross the United States in 1 second, and at 30,000 km/s you could circle the globe in 1 second. Clearly, relativistic effects are small until the speed becomes very large! Relative speed (km/s) 0.3 3 30

Relative speed as a fraction of lightspeed (c)

Duration of one “tick” on a moving clock, as measured by an observer past whom the clock is moving (s)

10–6

1.000 000 000 000 5

10

–5

10–4

1.000 000 000 5 1.000 000 005

300

0.001

1.000 000 5

3000

0.01

1.000 05

30,000

0.1

1.005

75,000

0.25

1.03

150,000

0.5

1.15

225,000

0.75

1.5

270,000

0.9

2.3

297,000

0.99

7.1

299,700

0.999

22.4

Instead of ice-cream cones, they could have frogs. Suppose your local biology department hatches guaranteed 10-day frogs, having a 10-day lifetime. Biological life occurs in time, too, so these frogs can be thought of as a kind of clock. So if Velma passes Mort at 75% of lightspeed, he says that her frog lives 15 days but that his frog lives only 10 days (see Concept Check 6). And she says that his frog lives 15 days but that her frog lives only 10 days. So each observes their own frog to die first. And both observations are correct! Fantastic. “But,” you may ask, “whose frog really dies first?” If you are tempted to ask this, your unspoken belief is that there is one single, universal, “real” time. But there isn’t. There is only Mort’s time, and Velma’s time, and every other individual observer’s time. How do we know that time flows differently for different observers? The relativity of time has been verified repeatedly in laboratories, by observing fast-moving subatomic particles. One experiment, similar to the frog example, involved a type of subatomic particle known as a “muon.” Muons, unlike most ordinary matter, are not permanent objects. Instead, they have a “lifetime” after which they disintegrate spontaneously into other particles. The lifetime of a muon is only 2.2 microseconds (2.2 millionths of a second), as measured by you if the muon is at rest relative to you. But a muon moving rapidly past you lives much longer as measured by you, because of time dilation. For example, at 99% of lightspeed (muons often move this fast in high-energy physics labs), Table 1 says that its lifetime will be lengthened by a factor of 7.1, so it will not disintegrate until 7.1 * 2.2 = 15.6 microseconds have passed. This experiment has been done, and the moving muons were observed to have lifetimes that were lengthened by just the predicted amount.

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6 TIME TRAVEL: YOU CAN’T GO HOME AGAIN As you might have suspected, the next step is to investigate the life spans of Velma and Mort themselves. Suppose they are born at the same time5 and that they have 80-year lifetimes. In other words, Velma observes her lifetime to be 80 years and Mort observes his lifetime to be 80 years. If Velma and Mort spend their lives moving at 75% of lightspeed relative to each other, then Table 1 informs us that Mort’s descendants observe that Velma lives for 120 years, as measured by Mort’s clocks. And Velma’s descendants observe that Mort lives for 120 years, as measured by Velma’s clocks. From Mort’s viewpoint, Velma ages slowly; she ages by just a year during each of Mort’s 1.5 years; he dies after 80 of his years; and she dies after 120 of Mort’s years but having the physical appearance of a person who is only 80. According to Velma, all of this is reversed. And both of them are correct. Incredible. CONCEPT CHECK 7 When Velma observes herself to be 60 years old, she will observe Mort to be (a) 30; (b) 40; (c) 60; (d) 80; (e) 90.

This suggests a perplexing question. Suppose that Velma and Mort are born at the same time on Earth, as twins perhaps, and Velma then boards a spaceship, takes a fast trip to a far star, and returns to Earth. This scenario is different from the scenario in the preceding paragraph, because now Velma and Mort begin and end in the same reference frame. Once they are back together they must agree on who is older, because there is only a single time in any single reference frame. Which twin will be older, or will they be the same age? Let’s think about that. Recall that the special theory of relativity applies only to nonaccelerated observers. But in the scenario for the two twins, Velma leaves Earth, speeds up enormously, turns around to get back to Earth, and then comes to rest on Earth. Since this trip necessarily involves three enormous accelerations, the special theory of relativity does not apply to Velma’s observations. But the special theory does apply to Mort’s observations, since he didn’t accelerate. As you have seen, the theory predicts that he observes Velma to age slowly during her entire trip, because she is moving relative to him. For example, if she moves at 0.75c, he should observe that 1.5 of his years elapse for every 1 of hers (Table 1). If Velma’s trip takes 60 years as measured by Mort, he observes that only 40 of her years elapse. So he observes that when they get back together on Earth, he is 60 and she is 40! Her observations must agree with this, since the two are now in the same reference frame. This is how you can get to be 20 years younger than your twin brother.

The testimony of our common sense is suspect at high velocities. Carl Sagan, Astronomer and Author

How do we know that time travel is possible? This conclusion has been experimentally verified, but in a less dramatic fashion. Atomic clocks were flown around the world on commercial jet flights and compared to clocks that remained at rest on Earth. Although the predicted time difference was only a fraction of a second, it was measurable using highaccuracy clocks. As predicted, the clock that went on the trip came back younger (it hadn’t ticked as many times) than the clock that stayed home. And the quantitative difference in elapsed time was precisely as predicted. As you will see in a moment, such experiments demonstrate that time travel is possible, but only into the future. 5

You might wonder what “at the same time” means, since we are assuming that Mort and Velma are in different reference frames. To simplify matters, suppose that Mort and Velma are just passing each other. Then “at the same time” means that as either one comes into the world, he or she observes that the other is coming into the world too.

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This suggests some astonishing possibilities. Suppose your mother leaves Earth for the star Vega, a sunlike star lying relatively close to our sun and a possible candidate for a planetary system. The distance to Vega is 26 light-years, meaning that it takes light 26 years to reach Vega from here. A light-year is the distance light travels in 1 year. Suppose mom’s spaceship averages a colossal 0.999c. She spends 3 years on a planet that is orbiting Vega and returns home. Since she travels at nearly lightspeed, each one-way trip takes slightly more than 26 years, as measured on Earth. So she is gone for slightly more than 26 + 3 + 26 = 55 years, as measured on Earth. If you were 5 and mom was 30 when she departed, you would be 60 when she returned. But mom would no longer be 25 years older than you! Table 1 informs us that during the 52 “Earth-years” of space travel at 0.999c, she aged by only 1 year for every 22.4 years of “Earth time.” So she aged by only 52>22.4 = 2.3 years during the 52 Earth-years. Including the 3 years spent on Vega, she aged by only 5.3 years during the entire trip. So mom is 35.3 years old when she returns, and you are 60! This is how you can get to be older than your mother. It’s a form of time travel. Your mother took a trip to Earth’s future. She could travel much further into the future, hundreds or thousands of years into the future, by moving faster, say at 0.9999c. But it’s a one-way trip. You can’t go home again to the past from which you departed. Time dilation suggests that humans might travel to distant stars within a human lifetime. Suppose you travel to a star 200 light-years away, at 0.999c relative to Earth. Even though the trip takes a little over 200 years as measured on Earth clocks, it takes you only 200>22.4 = 9 years as measured in your spaceship. When you arrive at the star, two centuries have elapsed on Earth. Even if you immediately hurry back to Earth, you time-travel four Earth centuries into the future during the round-trip but you age by only 18 years. On Earth, you will be a relic from four centuries earlier. CONCEPT CHECK 8 It is physically possible for your mother to leave Earth after you were born and return (a) before you were born; (b) before she was born; (c) younger than you; (d) older than you; (e) younger than she was when she left; (f) older than she was when she left.

7 THE RELATIVITY OF SPACE AND MASS What is space? Just as time means “what is measured by clocks” (Section 5), space means “what is measured by rulers.” What operations should Mort perform to measure, say, the width of a window? For a window at rest relative to Mort, the prescription is to place a measuring rod along the window and compare the ends of the window with the marks on the rod. If the window is moving past Mort, he should continue using a measuring rod that is fixed in his own reference frame, because he wants to know the width of the moving window as measured in his own reference frame. If the width being measured lies along the direction of motion, Mort must measure the positions of the two ends simultaneously because otherwise the window will shift positions during the lag between measurements and Mort won’t measure the true width. In order to ensure that the front-end and back-end measurements are simultaneous, Mort must use two clocks—one at each end. This means that the measurement of the width of a moving object is mixed up with the measurement of time; time and space are tangled up with each other! Since time is relative, it then comes as no surprise to

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learn that space is relative too. I won’t go through the argument that proves this result; it’s similar to the argument in Section 5 showing that moving clocks run slowly. More specifically, Einstein’s theory predicts that Mort observes the window’s width along its direction of motion to be shorter than does Velma who is traveling along with the window (Figure 10). This effect is called length contraction. There is no length contraction along directions perpendicular to the window’s direction of motion. As with time dilation, length contraction works both ways: Just as Mort finds that Velma’s window is contracted, Velma finds that Mort’s window is contracted. A quantitative analysis leads to a formula, graphed in Figure 11.6 The figure graphs the predicted length of a 1-meter-long object such as a meter stick, held parallel to its motion, for various speeds of the object. Like time dilation, length contraction is barely detectable for speeds below about 0.1c but becomes large at higher speeds. Length contraction is not simply something that happens to meter sticks. Since space is defined by meter sticks, it is space itself that is contracted. Just as Velma’s time flow is different from Mort’s time flow, we must speak of “Velma’s space” and “Mort’s space” rather than a single, universal space. Space is different for different observers. Space is relative. CONCEPT CHECK 9 Velma measures her spaceship to be 100 m long and 10 m high. Is it possible for her spaceship to move fast enough past Mort for its length to be equal to its height, as observed by Mort? (a) Yes, by moving at about 0.9c. (b) Yes, by moving at about 0.99c. (c) Yes, by moving at about 0.1c. (d) No, because she would have to move at precisely lightspeed to accomplish this. (e) No, because objects do not change their shapes.

1m

1m

(a) Less than 1 m

1m

(b)

Figure 10

The window in Velma’s spaceship as measured by (a) Velma and (b) Mort.

Einstein’s new principle, the constancy of lightspeed, affects nearly everything in physics: time, space, and more, including Newton’s law of motion. This law states that an object’s acceleration is equal to the net force exerted on the object divided by the object’s mass, or in symbols a = F>m

Length, m

This implies that if you exert an unchanging force on an object, the object maintains an unchanging acceleration. Eventually, the object will be going at lightspeed and still accelerating. An observer riding on such an object could catch up with and pass a light beam.

6

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 11

The relativity of space. The predicted length of a meter stick for various speeds of the meter stick relative to the observer. 0

0.1c

0.2c

0.3c

0.4c

0.5c 0.6c Speed

0.7c

0.8c

0.9c

c

The formula is L = L0 2(1 - y2>c2) where L0 is the object’s rest length (the length as measured by an observer for whom the object is at rest), and L is the length of the object when it is moving at speed y.

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So Newton’s law of motion is not consistent with the theory of relativity! Apparently, relativity alters Newton’s law in such a way as to prevent objects from accelerating up to lightspeed. To describe this alteration of Newton’s law of motion, let’s imagine that Mort and Velma (who is moving past Mort) have identical 1 kilogram objects, 1 kilogram melons perhaps. If Mort pushes on his melon with, say, a 1 newton force, he will find that it accelerates at 1 m/s2, just as Newton’s law of motion predicts. If he now pushes on Velma’s melon (which is moving past him) with a 1 newton force, Newton’s law of motion predicts that Velma’s melon accelerates at 1 m/s2, but relativity theory predicts7that Velma’s melon accelerates at less than 1 m/s2. As was the case for other relativistic effects, this effect is negligibly small at normal speeds but large at speeds comparable to lightspeed. From Mort’s point of view, a 1 newton force applied to both melons produces a smaller acceleration in Velma’s melon than in his own melon. From Mort’s point of view, Velma’s melon has more inertia than does his own melon (recall that a body’s inertia is its resistance to acceleration). But this is the same as saying that Velma’s melon has more mass, because the fundamental meaning of mass is “amount of inertia.” In other words, Mort measures Velma’s melon to have a larger mass than his own melon, even though they are identical melons. As usual, the effect works the other way around: Relative to Velma, her melon has a mass of 1 kg, but Mort’s melon has a mass of more than 1 kg. Thus, mass is relative: An object’s mass increases with its speed, so different observers measure different masses for the same object. A quantitative analysis leads to a formula that predicts an object’s mass for various speeds.8 Figure 12 is a graph of this formula, along with the previous graphs for time dilation and length contraction. The formulas for mass increase and time dilation have identical forms, so their graphs have identical shapes. In Newtonian physics, “mass” (or inertia) means the same thing as “quantity of matter.” But in relativity, an object’s mass increases with its speed while its quantity of matter does not increase because it still contains the same atoms. So mass no longer means “quantity of matter.” But we need a word for an object’s quantity of matter. That word is rest-mass, the mass of an object as measured by an observer in a frame of reference in which the object is at rest. For example, Velma’s and Mort’s melons both have rest-masses of 1 kg, regardless of who observes them. This number, 1 kg, is a measure of its quantity of matter. An object’s mass, however, is the amount of inertia it possesses and is different for different observers. The mass and rest-mass of a slow-moving object are essentially the same, but the mass of a high-speed object is significantly greater than its rest-mass. How do we know that mass increases with speed? Relativistic mass increase is an everyday fact of life in high-energy physics labs. A subatomic particle can be accelerated to speeds so close to lightspeed that its mass is thousands of times greater than its rest-mass. One way to check this prediction is to bend a high-speed particle’s path by applying electric or magnetic forces and measure the curvature of the resulting path. If high-speed particles really do have larger masses, their paths should curve less than they otherwise would, because their larger inertia tends to keep them moving straight ahead. Measurements show that such paths are less curved than they would be in the absence of relativistic mass increase and that the amount of curvature agrees with Einstein’s predictions. 7

8

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The reason is that accelerations of Velma’s melon, as viewed by Mort, are reduced because distances are contracted and time intervals are expanded. The formula is m = m0> 2(1 - y2>c2), where m0 is the object’s rest-mass (the mass as measured by an observer for whom the object is at rest), and m is the mass of the object when it is moving at speed y.

Time in seconds, length in meters, or mass in kilograms

The Special Theory of Relativity 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Time dilation and mass increase

Figure 12

Length contraction

0

0.1c

0.2c

0.3c

0.4c

0.5c 0.6c Speed

0.7c

0.8c

0.9c

c

Relativistic mass increase, length contraction, and time dilation. The graph shows the duration of one clock tick (representing 1 second in the clock’s reference frame) on a moving clock, the length of a moving meter stick, and the mass of a moving standard kilogram, for various speeds of the clock, meter stick, and kilogram relative to the observer.

Time, space, and mass are relative, but not everything is relative. In fact, the two basic principles of Einstein’s theory tell us that the speed of any light beam is the same for every observer, and the same goes for the laws of physics. Relativistic mass increase explains why you cannot accelerate objects up to lightspeed. At high speeds, an object’s mass becomes very large, increasing without limit as the speed approaches c (Figure 12). Eventually, the force needed for further acceleration becomes so large that the object’s surroundings cannot provide it. But there is something that moves as fast as lightspeed: light itself. In fact, light never moves slower than 300,000 km/s.9 When you turn on a lightbulb, the light does not accelerate from zero up to lightspeed; instead, it moves at precisely lightspeed from the instant it is created. Light is quite different from any material object. When you put a material object down in front of you, it has rest-mass. Light beams must not have rest-mass, because if they did, then relativistic mass increase would make their mass infinite while moving at lightspeed. Anything that has no rest-mass and always moves at lightspeed, such as light and other forms of electromagnetic radiation, is classified as radiation. It’s a useful distinction: Matter has rest-mass and always moves slower than lightspeed, while radiation has no rest-mass and always moves at lightspeed. 9

However, light travels through material substances such as water or glass at an average speed that is sometimes far less than lightspeed. When moving through matter, light momentarily vanishes when absorbed by an atom and is re-created when emitted by the atom. Whenever the light actually exists as light, it moves at 300,000 km/s.

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CONCEPT CHECK 10 Which is a form of matter? (a) Red light. (b) invisible waves drawn (c) The invisible carbon dioxide gas emitted by automobiles. (d) The electron beam that creates the picture on a TV tube. (e) A gamma ray.

8 E = mc2: ENERGY HAS MASS, AND MASS HAS ENERGY As an object speeds up, its kinetic energy increases and, as you have just learned, its mass increases. So, at least in the case of kinetic energy, energy increase and mass increase go hand in hand. Working from the theory of relativity and the law of conservation of energy, Einstein found that mass is connected to every form of energy in this fashion. You can increase a system’s mass by simply lifting it (giving it gravitational energy), warming it (giving it thermal energy), stretching it (giving it elastic energy), or giving it any other form of energy. Does that surprise you? It surprises me. If you stretch a rubber band, you don’t expect its mass to increase. It’s still the same rubber band, after all. This is a new and surprising prediction: Any increase in a system’s energy increases its mass, regardless of what form that energy increase might take. Einstein’s analysis yields a simple formula that quantitatively relates the change in mass to the change in energy. The formula states that the change in mass equals the change in energy divided by the square of lightspeed:

S

N S

N

(a) Less energy

change in mass =

S

N

S

N

(b) More energy

Figure 13

The separated magnets of (b) have more energy, and hence more mass, than do the two joined magnets of (a). The excess energy and mass in (b) reside in the invisible and nonmaterial magnetic field, indicated by dashed lines.

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change in energy square of lightspeed

In the standard metric units, mass and energy are in kilograms and joules, and c = 3 * 108 m>s, so c2 = 9 * 1016 m2>s2. Note that, since the standard metric unit for use in physics formulas is meters rather than kilometers, you need to use 3 * 108 m>s for “c” rather than 300,000 km/s. Here’s an example. Suppose you stretch a large, strong rubber band by exerting an average force of 300 N through a distance of 0.6 m. Since work equals force times distance, you’ve done 300 N * 0.6 m = 180 J of work on the band. So the work–energy principle says you’ve added 180 J of energy to the band. This increases the band’s mass by 180 > 9 * 1016 = 2 * 10 - 15 kg = 0.000 000 000 000 002 kg. Not much. The increase is small because c2 is so large. This is why relativistic mass increase wasn’t noticed before Einstein: In ordinary situations, it’s too small to notice. As a second example, suppose you have two bar magnets and that the north pole of one is joined to the south pole of the other so that they cling together [Figure 13(a)]. Since it takes work to pull them apart, the separated magnets of Figure 13b must have more energy than do the joined magnets. But more energy means more mass. So the total mass of the two combined magnets increases simply by pulling them apart! The separation process creates a magnetic field in the space between the two magnets [Figure 13(b)]. The excess energy in the separated magnets resides in this invisible and nonmaterial magnetic field. You encountered such “field energy” before, in the energy of electromagnetic radiation. But now you can see that fields also have mass. This mass is in the “empty” space between the magnets. The work done in separating the two magnets is only a few joules, so the mass difference is again tiny. Nevertheless, it’s extraordinary that nonmaterial fields in empty space have mass.

The Special Theory of Relativity

Turning to more dramatic examples, nuclear reactions entail nature’s strongest forces, the forces acting within the atomic nucleus. For now, all you need to know about nuclear reactions is that they are analogous to chemical reactions but they involve changes in nuclear structure rather than changes in electron orbits. For example, in nuclear power reactors and nuclear weapons, the element uranium undergoes a nuclear reaction known as nuclear fission in which the nucleus of each uranium atom is altered.10 Fission is a little like combustion, but the forces involved are so strong that the thermal energy created is far larger than in any chemical reaction. So the rest-mass loss, after removing the thermal energy, is far larger. If a kilogram of uranium is fissioned, the rest-mass loss is about 0.001 kg (1 g), which is a 0.1% mass decrease and easily detected. This can be checked experimentally, and the results agree with Einstein’s predictions. Nineteenth-century scientists believed matter was indestructible, in other words, that rest-mass was conserved in every physical process. This is certainly plausible. Since the days of the early Greek materialists, most scientists have felt that matter is indestructible—that although its form might change, its total amount cannot change. Nineteenth-century chemists performing high-precision mass measurements concluded that rest-mass is conserved even in highly energetic chemical reactions. But Einstein’s relativity contradicts the conservation of matter. Matter—that is, rest-mass— is not conserved in chemical reactions, in stretching a rubber band, and so forth. But these changes in rest-mass are so small that they are experimentally undetectable. In high-energy processes such as nuclear fission, however, the changes are easily detected, and the results show clearly that matter is not conserved. Now take this reasoning one step further: Einstein believed that this result extended not just to changes in mass but to all of the mass of any system. In other words,

total mass of any system =

When I think of matter, I like to think mostly of fields. We are fields rather than particles. Freeman Dyson, Physicist

total energy of that system c2

or, in symbols, m =

E c2

This implies Einstein’s famous formula, total energy of any system = (system’s total mass) * (c2) E = mc2 So all energy has mass, and all mass has energy. Since energy means the capacity to do work, and mass means inertia, the practical meaning of E = mc2 is that any system of mass m should be able to do mc2 units of work, and any system of energy E has an inertia E/c2. How do we know that E = mc2? If Einstein is right, there should be some physical process by which mc2 units of work can be obtained from any object of mass m. Such processes, known as matter–antimatter annihilation, have been discovered. Here’s how they work. In addition to the protons, neutrons, and electrons that form ordinary matter, physicists have discovered three other material particles, known as “antiprotons,” “antineutrons,” and 10

If matter turns out in the end to be altogether ephemeral, what difference can that make in the pain you feel when you kick a rock? John A. Wheeler, Physicist

Each uranium nucleus splits to form two nuclei of various lighter-weight elements.

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The visible world is neither matter nor spirit but the invisible organization of energy. Heinz Pagels, Physicist

Science has found no “things,” only events. The universe has no nouns, only verbs. R. Buckminster Fuller, Architect and Futurist

There are no things, only processes. David Bohm, Physicist

“antielectrons.” If one of these “antiparticles” is brought close to its corresponding particle, the two particles vanish entirely, and high-energy radiation is created. It’s an extreme example of the nonconservation of matter: Matter entirely vanishes, to be replaced by radiation. So any material object can be turned into radiation by annihilating all its protons, neutrons, and electrons—although it would be difficult to collect enough antiparticles to annihilate a macroscopic object. The energy of this radiation can then be used to do work. Furthermore, when the radiation’s energy is measured, it is found to equal the total mass of the particles times c2.

E = mc2 is simple but subtle, and easy to misinterpret. Most of the confusion arises from confusion between mass (inertia) and rest-mass (matter). Following are two common misconceptions about E = mc2. It is sometimes said, incorrectly, that Einstein’s relation means that “mass is not always conserved.” It is true that matter (rest-mass) is not always conserved. But mass (inertia) is always conserved, because mass equals energy divided by c2, and energy is always conserved. It is sometimes said, incorrectly, that Einstein’s relation means that “mass can be converted to energy.” It’s true that rest-mass—matter—can be converted to nonmaterial forms of energy such as radiation. But you just saw that mass is always conserved, so mass can never be converted to anything else! In proton-antiproton annihilation, for example, the mass of the pair is precisely equal to the mass of the created radiation. But rest-mass, or matter, is destroyed, and is converted to radiation. One must be careful with the word mass. To summarize: The Principle of Mass–Energy Equivalence Energy has mass; that is, energy has inertia. And mass has energy; that is, mass has the ability to do work. The quantitative relation between the energy of any system and the mass of that system is E = mc2.

Mass–energy equivalence represents another sharp break with the Newtonian worldview, which follows the Greek materialists in believing that interactions between indestructible atoms moving in empty space determine everything that happens in the physical universe. Let’s think about the mass–energy relationship at the atomic level. Since all energy has mass, some of an atom’s mass must be due simply to the kinetic energy of its parts (electrons, protons, and neutrons) and to the energy of its various electromagnetic and nuclear force fields. This suggests an intriguing question: Is that all there is? Are atoms made only of fields and motion? If so, atoms are not only mostly empty space, they are entirely empty space, made only of fields similar to the magnetic fields in Figure 13, and the motion of those fields! High-energy physics has already provided part of the answer. It is now known that protons and neutrons are made of three smaller particles called “quarks.” Because quarks exert enormous forces on each other, the energy in their force fields is enormous. In fact, calculations show that the energy of these fields is sufficient to explain 90% of the mass of the proton (or neutron)! Since essentially all of the mass of ordinary matter comes from protons and neutrons, this result implies that some 90% of the mass of ordinary matter comes from the nonmaterial energy of fields and motion! The remaining 10% might arise in a similar way, although this is not yet confirmed. Our most accurate theory of physics (the “standard model”) suggests the existence

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throughout the universe of a field called the “Higgs field.” If verified, the Higgs field will explain the still-unexplained 10%. The fundamental theories of contemporary physics known as “quantum field theories” also suggest that all mass arises solely from nonmaterial fields. For example, Steven Weinberg, a leading high-energy theorist, states the following: [According to the physical theories developed during the 1920s] there was supposed to be one field for each type of elementary particle. The inhabitants of the universe were conceived to be a set of fields—an electron field, a proton field, an electromagnetic field—and particles were reduced to mere epiphenomena. In its essentials, this point of view has survived to the present day, and forms the central dogma of quantum field theory: the essential reality is a set of fields [Weinberg’s emphasis] subject to the rules of special relativity and quantum mechanics; all else is derived as a consequence of the quantum dynamics of these fields.

In this field view of reality, there is no “there” there (to quote the poet Gertrude Stein), no “things” at all. Electrons and other material particles are only non-material fields in space, similar to the magnetic field in the space between the poles of a magnet. All mass is due only to the energy of fields. Since fields are “possible forces,” and forces are interactions, this view implies that every “thing,” everything, is interactions and motion. It’s the interactions and motion themselves that are fundamental rather than the material particles that we had always supposed were doing the interacting and the moving. It’s a view that stands Newtonian materialism on its head.

We are such stuff As dreams are made on Shakespeare, The Tempest

CONCEPT CHECK 11 In which of the following processes does the system’s mass change? (a) A bullet that speeds up while moving down a gun barrel. (b) A rubber band that is being stretched around a package. (c) Two positively charged objects that are moved closer to each other and placed at rest. (d) An electron and an antielectron, at rest, that spontaneously annihilate each other.

© Sidney Harris, used with permission.

CONCEPT CHECK 12 In the preceding question, in which processes does the system’s rest-mass change?

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The Special Theory of Relativity Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions GALILEAN RELATIVITY 1. What is meant by relative motion, reference frame, a theory of relativity? 2. A train moves at 70 m/s. A ball is thrown toward the front of the train at 20 m/s relative to the train. How fast does the ball move relative to the tracks? What if the ball had instead been thrown toward the rear of the train? 3. A spaceship moves at 0.25c relative to Earth. A light beam passes the spaceship, in the forward direction, at speed c relative to Earth. According to Galilean relativity, how fast does the light beam move relative to the spaceship? Is this answer experimentally correct? If not, then what answer is correct?

THE PRINCIPLES OF RELATIVITY AND CONSTANCY OF LIGHTSPEED 4. How does travel in a jet airplane illustrate the principle of relativity? How must the airplane be moving in order to illustrate this principle? 5. State the principle of relativity in your own words. Does it apply to every observer? Explain. 6. State the principle of the constancy of lightspeed in your own words. Does it apply to every observer? Explain. 7. Use the principle of the constancy of lightspeed to explain why no observer can move at precisely speed c relative to any other observer. 8. What is the ether theory, and why did physicists ultimately reject it? 9. In Galilean relativity, space and time are absolute and lightspeed is relative. What is the situation in Einstein’s relativity? 10. What distinguishes the special from the general theory of relativity? 11. List the basic “laws” of the special theory of relativity.

THE RELATIVITY OF TIME 12. How is time defined in physics? 13. Describe the light clock. 14. Velma passes Mort at a high speed. Both observers have clocks. What does each observer say about Velma’s clock? What do they each say about Mort’s clock?

15. One twin goes on a fast trip and returns. Does the special theory of relativity apply to the observations of both twins? Why, or why not? 16. One twin goes on a fast trip and returns. Have the two twins aged differently during the trip? If so, how do their ages differ? 17. Explain how you can travel to the future.

THE RELATIVITY OF SPACE AND MASS 18. What do we mean by “space” or “distance”? 19. What does “space is relative” mean? 20. Velma passes Mort at a high speed. Each of them holds a meter stick parallel to the direction of motion. What does each observer say about Velma’s meter stick? What does each say about Mort’s meter stick? 21. According to Einstein’s theory, which of these are relative: time, lightspeed, rest-mass, length, mass? 22. Velma passes Mort at a high speed. Both observers carry a standard kilogram. What does Mort say about the mass of each of the standard kilograms? What does Velma say? 23. Mort exerts a 1 newton force on his standard kilogram. What acceleration does this give to the kilogram? What will he find if he exerts the same force on Velma’s standard kilogram while Velma is passing him at a high speed? 24. What is the distinction, if any, between rest-mass, mass, and matter? Which ones increase with speed? 25. What is the distinction between matter and radiation? 26. Why can’t material objects be sped up to lightspeed? Does anything move at lightspeed?

E = mc 2 27. What does E = mc2 mean? Does it mean that mass can be converted to energy? Explain. 28. Is matter always conserved? Is mass always conserved? Is rest-mass always conserved? Is energy always conserved? 29. According to Einstein’s relativity, is rest-mass precisely conserved in chemical reactions? 30. Describe an experiment in which a system’s entire rest-mass vanishes. Is matter conserved here? Mass? Energy?

Conceptual Exercises GALILEAN RELATIVITY 1. Two bicyclers, on different streets in the same city, are both moving directly north at 15 km/hr. Are they in relative motion?

From Chapter 10 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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The Special Theory of Relativity: Problem Set 2. According to the Galilean theory of relativity, does every observer measure the same speed for a light beam? 3. Velma moves toward Mort at half of lightspeed. Mort shines a searchlight toward Velma. What does Galilean relativity predict about the speed of the searchlight beam as observed by Velma? 4. Velma bicycles northward at 4 m/s. Mort, standing by the side of the road, throws a ball northward at 10 m/s. What is the ball’s speed and direction of motion, relative to Velma? What if Mort had instead thrown the ball southward at 10 m/s? 5. A desperado riding on top of a train car fires a gun toward the front of the train. The gun’s muzzle speed (speed of the bullet relative to the gun) is 500 m/s, and the train’s speed is 40 m/s. What is the bullet’s speed and direction of motion as observed by the sheriff standing beside the tracks? What does a passenger on the train say about the bullet’s speed? What if the desperado had instead pointed his gun toward the rear of the train? 6. Velma is in a train moving eastward at 70 m/s. Mort, standing beside the tracks, throws a ball at 20 m/s eastward. What is the ball’s speed and direction relative to Velma? 7. Velma is in a train moving eastward at 70 m/s. Mort, standing beside the tracks, throws a ball at 20 m/s westward. What is the ball’s speed and direction relative to Velma?

THE PRINCIPLE OF RELATIVITY 8. Does the principle of relativity require that every observer observe the same laws of physics? Explain. 9. If you were riding on a train moving at constant speed along a straight track and you dropped a ball directly over a white dot on the floor, where would the ball land relative to the dot? 10. Suppose that you drop a ball while riding on a train moving at constant speed along a straight track. If you measure the ball’s acceleration, will your result be greater than, less than, or equal to, the usual acceleration due to gravity? 11. Think of several ways that you could determine from inside an airplane whether the plane was flying smoothly or parked on the runway. Do each of these ways involve some direct or indirect contact with the world outside the airplane? 12. How fast are you moving right now? What meaning does this question have? 13. If you drop a coin inside a car that is turning a corner to the right, where will the coin land? 14. If you drop a coin inside a car that is slowing down, where will the coin land?

THE CONSTANCY OF LIGHTSPEED 15. Does every observer measure the same speed for a light beam? Explain. 16. A star headed toward Earth at 20% of lightspeed suddenly explodes as a bright supernova. With what speed does the light from the explosion leave the star? With what speed (as measured on Earth) does it approach Earth? 17. Is it physically possible for a person to move past Earth at exactly lightspeed? Explain. 18. Velma’s spaceship approaches Earth at 0.75c. She turns on a laser and beams it toward Earth. How fast does she see the

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beam move away from her? How fast does an Earth-based observer see the beam approach Earth? 19. A desperado riding on top of a freight-train car fires a laser gun pointed forward. What is this gun’s “muzzle velocity”? Suppose the train is moving at 40 m/s (0.04 km/s). How fast does the tip of the laser beam move relative to the sheriff, who is standing on the ground beside the train? What answer would the Galilean theory of relativity have given to this question?

THE RELATIVITY OF TIME 20. Velma passes you at a high speed. According to you, she ages slowly. How does she age according to her own observations? How do you age according to her? 21. Suppose you have a twin brother. What could be done to make him older than you? 22. The center of our galaxy is about 26,000 light-years away. Could a person possibly travel there in less than 26,000 years as measured on Earth? Could a person possibly travel there in less than 26,000 years of his or her own time? Explain. 23. A woman conceives a child while on a fast-moving space colony moving toward a distant planetary system. How long should it take before the baby is born, as measured by the woman? Would an Earth observer measure the same amount of time? 24. A certain fast-moving particle is observed to have a lifetime of 2 seconds. If the same particle was at rest in the laboratory, would its lifetime still be 2 seconds, or would it be more, or less, than 2 seconds? 25. Does the special theory of relativity allow you to go on a trip and return older than your father? 26. Does the special theory of relativity allow your father to go on a trip and return younger than you? 27. Does the special theory of relativity allow you to go on a trip and return younger than you were when you left? 28. When you go on a very fast trip, must you always return older than you were when you left? 29. A satellite orbits Earth at 8 km/s. Find its speed as a fraction of lightspeed. Would an orbiting astronaut directly notice the effects of time dilation without using sophisticated measurement techniques? 30. Velma passes Earth at 50% of lightspeed. On her video player, she watches a taped video program that runs 1 hour. How long does the program run as measured by an Earthbased observer? 31. Your fantastic rocketship moves at 30,000 km/s. If you took off, moved at this speed for 24 hours as measured by you, and returned to Earth, by how much time would your clock differ from Earth-based clocks? Would you have aged more than, or less than, people on Earth? By how much? 32. Answer the preceding question assuming that your extraordinarily fantastic rocketship moves at 99% of lightspeed. 33. Mort and Velma have identical 10-minute ice-cream cones. Velma passes Mort at 75% of lightspeed. How long does Mort’s cone take to melt as measured by Velma? 34. How fast must Velma move in order for her 10-minute icecream cone to melt in 30 minutes as measured by Mort?

The Special Theory of Relativity: Problem Set

THE RELATIVITY OF SPACE AND MASS 35. How fast must Velma move past Mort if Mort is to observe her spaceship’s length to be reduced by 50%? If Velma is flying east to west across the United States (about 5000 km wide) at this speed, how wide will she observe the United States to be? 36. Mort’s swimming pool is 20 m long and 10 m wide. If Velma flies lengthwise over the pool at 60% of lightspeed, how long and how wide will she observe it to be? 37. Mort’s automobile is 4 m long as measured by Mort. What length does Velma measure for Mort’s auto, as she passes him at 90% of lightspeed? 38. Velma, who is carrying a clock and a meter stick, passes Mort. Is it possible that Mort could observe length contraction of Velma’s meter stick but observe no time dilation of her clock? If so, how? 39. Velma, who is carrying a clock and a meter stick, passes Mort. Is it possible that Mort could observe time dilation of Velma’s clock but observe no length contraction of her meter stick? If so, how? 40. Velma drives a really fast rocket train northward past Mort, who is standing beside the tracks. Two posts are driven into the ground along the tracks. How does Mort’s measurement of the distance between the posts compare with Velma’s: longer, shorter, or the same? 41. If Velma passes Mort at a high speed, Mort will find her mass to be larger than normal. Will he also find her to be larger in size? 42. Velma’s spaceship has a rest-mass of 10,000 kg, and she measures its length to be 100 m. She moves past Mort at 0.8c. According to Mort’s measurements, what are the mass and the length of her spaceship? 43. How fast must Velma move past Mort if Mort is to observe her spaceship’s mass to be increased by 50%? How fast must she move if Mort is to observe her spaceship’s length to be reduced by 50%? 44. A meter stick with a rest-mass of 1 kg moves past you. Your measurements show it to have a mass of 2 kg and a length of 1 m. What is the orientation of the stick, and how fast is it moving? 45. Use Figure 12 to estimate how fast Velma must move, relative to Mort, for Mort to observe that her body’s mass is 50% larger than normal.

E = mc 2 46. When you throw a stone, does its mass increase, decrease, or neither? Can this effect be detected? 47. A red-hot chunk of coal is placed in a large air-filled container where it completely burns up. The container is a perfect thermal insulator—in other words, thermal energy is unable to pass through the container’s walls. According to E = mc2, does the total mass of the container and its contents change during the burning process? If so, does the mass increase, or decrease? 48. Referring to the previous question: Suppose that the container is not a thermal insulator—in other words, thermal energy passes through the walls. In this case, does the total mass of the container and its contents change during the burning process? If so, does the mass increase, or decrease?

49. An electron and an antielectron annihilate each other. In this process, is energy conserved? Is mass conserved? Is restmass conserved? 50. Two mousetraps are identical except that one of them is set to spring shut when the trigger is released, and the other is not set. They are placed in identical vats of acid. After they are completely dissolved, what, if any, are the differences between the two vats? Will the masses differ? 51. In a physics laboratory, an electron is accelerated to nearly lightspeed. If you were riding on the electron, would you notice that the electron’s mass had increased? If you were standing in the laboratory, what would you notice concerning the electron’s mass and energy?

Problems Use the time-dilation formula T = T0 > 2(1 - y2>c2) (explained in footnote 4) to answer questions 1–6. 1. Time dilation depends on the quantity 2(1 - y2>c2), which in turn depends on the fraction y2>c2. Evaluate the fraction y2>c2 for each of the following speeds: 3 km/s (high-powered rifle bullet), 30 km/s (speed of Earth in its orbit around the sun), 3000 km/s (fast enough to cross the United States in about 1 second). Is time dilation a very significant, noticeable effect at these speeds? 2. Time dilation depends on the factor 2(1 - y2>c2), Evaluate this factor for each of the following speeds: 30,000 km/s (fast enough to circle the globe in 1 second), 150,000 km/s. 3. Velma passes Mort at 30,000 km/s. What fraction of lightspeed is this? What is the duration of one of Velma’s seconds (a time interval that Velma observes to be 1 second in duration) as observed by Mort? 4. Velma passes Mort at 150,000 km/s. What fraction of lightspeed is this? What is the duration of one of Mort’s seconds (a time interval that Mort observes to be 1 second in duration) as observed by Velma? 5. Velma passes Mort at a high speed. His clock, as observed by her, runs at half of its normal speed—for example, his clock advances by only 30 minutes during a time of 1 hour as recorded on her own clock. What must be the value of the quantity 2(1 - y2>c2)? Find Velma’s speed relative to Mort. 6. Velma passes Mort at a high speed. Her clock, as observed by him, runs at 25% of its normal speed—for example, her clock advances by only 15 minutes during a time of 1 hour as recorded on his own clock. What must be the value of the quantity 2(1 - y2>c2)? Find Velma’s speed relative to Mort. 7. You give 90 J of kinetic energy to a 1 kg stone when you throw it. By how much do you increase its mass? 8. A large nuclear power plant generates electric energy at the rate of 1000 MW. How many joules of electricity does the plant generate in one day? What is the mass of this much energy? 9. If you had two shoes, an ordinary shoe and an “antishoe” made of antiparticles, and you annihilated them together, by how far could you lift the U.S. population? Assume that each person weighs 600 N, that each shoe’s rest-mass is 0.5 kg, and that all the energy goes into lifting.

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The Special Theory of Relativity: Problem Set 10.

Show that, if all the energy released (transformed) in fissioning 1 kg of uranium were used to heat water, about 2 billion kg of water could be heated from freezing up to boiling. (Assume that the uranium’s rest-mass is reduced by about 0.1%. Roughly 4 J of thermal energy is needed to raise the temperature of 1 gram of water by 1°C.) How many tonnes of water is this (a tonne is 1000 kg)? How many large highway trucks, each loaded to about 30 tonnes, would be needed to carry this much water? 11. Solar radiation reaches Earth at the rate of 1400 watts for every square meter directly facing the sun. Using the formula pR2 for the area of a circle of radius R, find the amount of solar energy entering Earth’s atmosphere every second. Earth’s radius is 6400 km. 12. Use the answer to the preceding question to find how many kilograms of sunlight hit Earth every second. MAKING ESTIMATES

Answers to Concept Checks 1. (d) 2. During every second, the ball moves 20 m closer to the rear

3. 4. 5. 6. 7. 8.

9. 10. 11.

12.

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of the train, while the train moves 70 m eastward. The net result of these two motions is that the ball moves 50 m eastward (relative to the ground). So the velocity observed by Mort is 50 m/s eastward, (a). (a). For instance, you could pour a cup of coffee. If your vehicle is speeding up, the coffee won’t pour straight down. (b) (e) Mort observes his own cone to melt in 10 minutes. Table 1 says that when he observes Velma’s cone, the time dilation factor is 1.5, so he observes Velma’s cone to melt in 15 minutes, (e). She observes him to be younger, in the same ratio as in the other examples (1.5 to 1), so when she has aged by 60 years, she observes him to have aged only by 40 years, (b). She cannot go into the past, which rules out answers (a), (b), and (e). She can accomplish (c) by moving sufficiently fast. She can accomplish (d) by moving slowly. Finally, (f) will be true regardless of how fast or how slow she moves. (c), (d), and (f). In order for 100 m to contract to 10 m, 1 m must contract to 0.1 m. Figure 11 tells us that this happens when the relative speed is about 99% of lightspeed, (b). (c), (d) Mass increases whenever energy increases. The bullet’s energy increases as it moves faster. The rubber band’s energy increases (because work must be done to stretch it). The two charged objects have more electrical energy after the move (because work must be done to move them closer). However, the electron–antielectron pair do not gain or lose energy (even though they are annihilated), because of conservation of energy, (a), (b), (c). The bullet’s rest-mass is unchanged. The rubber band and the two charged objects are left at rest, with greater energy and hence greater mass, so the rest-mass of these two systems increases. The electron–antielectron pair lose all their rest mass when they annihilate to create gamma radiation (which has zero rest mass), (b), (c), (d).

Answers to Odd-Numbered Conceptual Exercises and Problems Conceptual Exercises 1. No. 3. According to Galilean relativity, she observes the light beam to move at 1.5c. 5. 540 m/s as observed by the sheriff. 500 m/s as observed by the passenger. With the gun pointed toward the rear, the bullet moves at 460 m/s as observed by the sheriff, 500 m/s as observed by the passenger. 7. 90 m/s west. 9. The ball would land on the dot. 11. Look out the window, radio to the outside, ask the pilot to look out of the window and tell you what he sees, stick your hand out of the airplane, etc. All of these involve contact with the outside world. 13. To your left. 15. No. Only non-accelerated observers measure the same speed for a light beam. 17. No. All material objects move slower than lightspeed. 19. 300,000 km/s. 300,000 km/s. 300,000.04 km/s. 21. If you go on a sufficiently long and fast trip and return, he will then be older than you. 23. Nine months. An Earth observer would measure a longer time. 25. No, because you will age less than your father. 27. No. Time always goes forward, never backward. 29. (8 km>s)>300,000 km>s = 2.67 * 10 - 5. No. 31. The speed is 0.1c, so Table 1 says that one of your seconds is observed on Earth as 1.005 s. So when you return you will have aged by 24 hours while people on Earth have aged by 1.005 * 24 = 24.12 hr = 24 hr and 7.2 min. You have aged less than people on Earth, by 7.2 minutes. 33. Table 1 says that the time-dilation factor is 1.5, so Velma observes Mort’s cone to melt in 1.5 * 10 min = 15 min. (Note that Mort moves at 75% of lightspeed relative to Velma.) 35. According to Figure 12, the required speed is about 0.86c. She will observe the United States to be 0.5 * 5000 km = 2500 km wide. 37. 0.43 * 4 m = 1.8 m. 39. Yes. Velma could be holding her meter stick perpendicular to the direction of motion. 41. No, in fact he observes her to be shorter as measured along her direction of motion. 43. Using Figure 12, Mort observes her spaceship’s mass to be increased by 50% when she moves past him at a speed of 0.75c. Mort observes the length to be reduced by 50% when she moves past him at 0.86c. 45. About 0.75 c. 47. No; since the total energy is unchanged, the total mass is also unchanged. 49. Yes. Yes. No; in fact, rest-mass is entirely destroyed. 51. No. If you were standing in the laboratory, you would notice that the electron’s mass and energy had both increased.

The Special Theory of Relativity: Problem Set Problems 1. y>c = (3 km>s)>300,000 km>s = 10 - 5, y2>c2 = 10 - 10. 2 2 -8 Similarly, y >c = 10 when s = 30 km>s. y2>c2 = 10 - 4 when y = 3000 km>s. Time dilation is not very significant at these speeds. 3. This is 0.1 (or 10%) of lightspeed. y2>c2 = 0.01. 2(1 - y2>c2) = 2(1 - 0.01) = 20.99 = 0.995 T = T0> 2(1 - y2>c2) = 1 s>0.995 = 1.005 s. 5. From the given information, the time-dilation factor is 2, i.e., T = 2T0. Thus, 1> 2(1 - y2>c2) = 2. 2(1 - y2>c2) = 1>2 1 - y2>c2 = 1>4 y2>c2 = 3>4, so y>c = 20.75 = 0.87. Velma moves at 87% of lightspeed. Note that this answer agrees with Figure 10. 7. 90 J>c2 = 90 J>9 * 1016 m2>s2 = 10 - 15 kg, or 0.000,000,000,000,001 kg.

9. mc2 = (1 kg) * (9 * 1016 m2>s2) = 9 * 1016 J. This much

energy goes into lifting. The energy needed to lift a weight through a height is weight * height. The weight of the U.S. population is about (300 * 106 people) * (600 N>person) = 1.8 * 1011 N. The height through which this mass could be lifted is 9 * 1016 J>1.8 * 1011 N = 5 * 105 m = 500 kilometers! 11. Since 1 watt = 1 joule/second, 1400 joules hit each square meter in each second. Earth’s area facing the sun (the cross-sectional area facing the sun) is pR2 = 3.14 * (6.4 * 106 m)2 = 1.3 * 1014 m2. The energy hitting Earth each second is thus 1400 J>m2 * 1.3 * 1014 m2 = 1.8 * 1017 J.

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Einstein’s Universe and the New Cosmology Einstein’s Pegasus There’s Einstein riding on a ray of light, In which he cannot see his face in flight Because his jesting image, I now understand, Won’t ever reach the mirror since its speed, Too, is the speed of light. He rides, this fleeting day, As if on Pegasus, immortal steed Of bridled meditation, past the Milky Way, Out to my mind’s Andromeda, where I, Also transported, staring at a windless pool, Watch his repaired reflection whizzing by. Though he can’t see himself, this self-effacing fool Who holds all motion steady in his head, I won’t forget his facing what he cannot see In thought that binds the living and the dead, And ride with him, outfacing fixed eternity. Robert Pack, Middlebury College, 1991

T

he special theory of relativity describes the observations of nonaccelerated observers. What about accelerated observers? Einstein found a surprising connection between acceleration and gravity, and between gravity and a feature best described as “warps in spacetime.” Section 1 presents these key ideas. I’ve devoted the remainder of the chapter to cosmology: the study of the origin, structure, and evolution of the large-scale universe. The general theory of relativity is science’s basic tool for such matters. You are living in the golden age of cosmology. It started in 1992 when microwave receivers on orbiting satellites gathered the first detailed image of the “cosmic microwave background” showing the earliest light from the creation of the universe. It continues today with the search for dark matter, dark energy, and an elusive microscopic particle known as the “Higgs boson.” Cosmology is inspired by some of the oldest questions ever asked, and is perhaps the oldest story ever told. For thousands and probably millions of years, humans have looked for answers to questions such as: How did all this come to be? Where did Earth come from? What is the layout of the universe? Where do humans fit in? There’s been plenty of speculation about all this, but now for the first time we are finding evidence-based answers, answers that have at least as much to do with physics as with astronomy. I hope you’ll share in the excitement by pondering the discoveries and concepts in this chapter.

From Chapter 11 of Physics: Concepts & Connections, Fifth Edition, Art Hobson. Copyright © 2010 by Pearson Education, Inc. Published by Pearson Addison-Wesley. All rights reserved.

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Einstein’s Universe and the New Cosmology

Section 2 presents the “big bang” that created our universe and evidence that it actually occurred. Einstein’s connection between gravity and warped spacetime leads to a new way of viewing, in Section 3, the overall structure and expansion of the universe. Section 4 presents the recent microwave image of the big bang shortly after it occurred and its implications for the overall shape of the universe. Section 5 presents a surprising and exciting development: The universe is filled with enormous amounts of “dark matter” that doesn’t interact with light and so has not yet been directly observed. Section 6 describes two additional completely unexpected developments: the accelerating universe and the mysterious “dark energy” pushing this acceleration. Section 7 presents a recent hypothesis on how, and perhaps even why, the big bang banged. Because these results aren’t easy to believe, I’ve included quite a few “How Do We Know” subsections.

A rocket ship in outer space accelerates at 9.8 m/s2 in this direction

Due to the acceleration, this occupant feels a force by the floor against his shoes.

(a)

Due to gravity, this occupant at rest on the ground feels a force by the floor against his shoes.

(b)

Figure 1

Acceleration is indistinguishable from gravity. The occupant cannot tell the difference.

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1 EINSTEIN’S GRAVITY: THE GENERAL THEORY OF RELATIVITY The special theory of relativity begins with the principle that the laws of physics are the same for all unaccelerated observers. What about accelerated observers? This is the starting point for the general theory of relativity. You’ve probably noticed, when riding in an elevator accelerating upward from a building’s ground floor, that you felt squashed down, heavy, as though there were more gravitational pull on you than usual. This connection between acceleration and “apparent gravity” runs deeper than you might think. For example, imagine being inside an accelerating rocket ship in outer space, far from all planets and stars so that there are no gravitational forces [Figure 1(a)]. If the rocket’s acceleration is 9.8 m/s2 or 1g (“one gee”), you will feel the same as you do when you are stationary on Earth [Figure 1(b)]. The reason is that, according to Newton’s law of motion, the rocket’s floor must push upward on the bottoms of your shoes in order to accelerate you at 1g. Because this push is quantitatively equal to your weight on Earth, you feel the same force against your feet as you do at rest on Earth. So accelerations mimic the effects of gravity. If you were in a rocket accelerating smoothly at 1g through outer space, could you tell, without communicating with the world outside the rocket, that you were actually in space and not at rest on Earth’s surface? Think about it. You might try dropping a stone to see how it falls (Figure 2). But your rocket is accelerating at 1g so the floor accelerates upward to meet the stone. From your point of view inside the rocket, the stone “falls” down to the floor with an acceleration of 1g. Furthermore, Galileo’s law of falling is valid: A large-mass stone and a small-mass stone, released together, reach the floor (or rather, the floor reaches them) at the same time. You also could throw a stone horizontally (Figure 3). Because of the rocket’s upward acceleration, the stone gets closer to the floor as it moves across the rocket. As you view it, the stone “falls” to the floor exactly as though it were thrown horizontally on Earth. It seems it’s not easy to perform an experiment inside your rocket ship that can tell you whether you’re at rest on Earth’s surface or moving through space with a 1g acceleration. Einstein made this reasoning into a fundamental principle that’s similar to the principle of relativity. The principle of relativity says that there is no way, from within your own laboratory, to distinguish a state of rest from motion at constant

Einstein’s Universe and the New Cosmology Figure 2

Rocket ship accelerates at 1g

Floor accelerates upward and hits the stone

If you release a stone inside an accelerating rocket in outer space, it will appear to you that the stone falls “down” to the floor, just as though you were on Earth and feeling the effects of gravity.

Observer releases stone

Stone’s path relative to you

Rocket ship accelerates at 1g

Figure 3

If you throw a stone inside an accelerating rocket in outer space, it will appear to you that the stone falls to the floor as though you were on Earth and feeling the effects of gravity.

Floor accelerates upward to meet the stone Observer throws stone horizontally

velocity. The new principle states that there is no way, from within your own laboratory, to distinguish the effects of gravity from the effects of acceleration. Because it says that gravity is equivalent to acceleration, we call this The Equivalence Principle No experiment performed inside a closed room can tell you whether you are at rest in the presence of gravity or accelerating in the absence of gravity.

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Einstein’s Universe and the New Cosmology

Rocket ship accelerates at 1g

Observer points flashlight horizontally

Light beam’s path relative to you

Floor accelerates upward toward the flashlight beam

Figure 4

If you turn on a flashlight inside an accelerating rocket in outer space, the light beam bends relative to you.

Distant star

Star’s apparent position as seen from Earth

Starlight bent by sun’s gravity

From Earth, the star appears to lie in this direction Sun

Earth

Figure 5

Because the sun bends light beams, we can (during a total eclipse) see stars that are behind the sun.

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Light beams play a central role in the general theory just as they do in the special theory. How do accelerations affect light beams? If you accelerate through outer space and you turn on a flashlight horizontally, the light beam must bend downward relative to you (Figure 4), just like the path of a horizontally thrown stone (Figure 3). The equivalence principle implies that this experiment must come out the same way if performed in a stationary room in the presence of gravity. So gravity must bend light. How do we know that gravity bends light? Earth’s gravity is too weak to bend light very much. But the sun is massive enough to measurably bend the light from distant stars as the light passes close to the sun. The first measurement of this effect was made during a total eclipse of the sun in 1919, when astronomers could photograph the stars that appear near the edge of the sun (Figure 5). Measurements of these stars’ positions showed that the starlight does bend as it passes the sun and that the amount of curvature agrees with Einstein’s predictions.

Recall that the constancy of lightspeed led Einstein to the surprising discovery that time is relative. Similarly, the gravitational bending of light implies a surprising property of space, related to the concept of straightness. Just as time is a physical property of the universe that can be measured by a light clock, straightness is a physical property that can be defined as the path followed by a light beam. In fact, surveyors often use laser beams to determine straightness, and you use light beams to determine straightness when you aim a gun by sighting along its barrel. But what can it mean to say that gravity bends light beams, when light beams themselves are the definition of straightness? Just as the slowed ticking of moving light clocks implies that time itself slows down, Einstein saw that the bending of light beams

means that space itself is bent, or curved, or warped, by gravity. The path of a light beam is best described as the “straightest possible” path. In a curved space, even the straightest possible path must be curved. Space is warped. It’s an odd concept. It took an Einstein to think of it, but it’s not something that Einstein, or you or anybody else, can visualize. As Stephen Hawking (Figure 6) remarked, “It is hard enough to visualize ordinary three-dimensional space, let alone warped three-dimensional space.” The difficulty is that space has only three dimensions (length, width, and height), so there is no higher dimensionality from which to view the curvature of our three-dimensional space the way you can view, from your three-dimensional perspective, the bending of a twodimensional sheet of paper. The best anybody can do is visualize analogies to this important idea of curved space. For example, a flat tabletop is two-dimensional (the surface has length and width only) and can be considered to be a “flat two-dimensional space” (Figure 7). If we put a warp in it, a depression perhaps (Figure 8), the surface becomes a warped two-dimensional space. For another example, the surface (not the inside) of a sphere is a curved two-dimensional space. The two standard dimensions on the surface of a globe, for example, are called longitude (angular distance east or west of a circle running through the two poles and through Greenwich, England) and latitude (angular distance north or south of the equator). In this curved two-dimensional space, the straightest possible lines (analogous to the paths of light beams in curved threedimensional space) are the “great circles,” such as the equator and the circles of longitude running through the poles. Suppose you were a two-dimensional creature inhabiting a two-dimensional spherical space, something like a flat ant crawling on the surface of a large globe. How could you tell that your space was curved? You couldn’t stand outside or inside the globe’s surface, in the third dimension, to see that you are on a spherical surface, because there is no such third dimension in this two-dimensional analogy. One way you could learn that your space is curved is by performing geometry experiments. For instance, two lines, beginning parallel and extending as straight lines (or straightest lines, as perceived from our three-dimensional vantage point), should eventually meet (Figure 9). Similarly, you cannot directly see the curvature of three-dimensional space, but you can perform experiments to determine whether our space is curved. The 1919 experiment that measured the curvature of light near the sun was just such a geometry experiment. It found that even the straightest path, the path of a light beam, bends near the sun. We conclude that three-dimensional space itself is curved.

Figure 7

A flat tabletop is a flat twodimensional space.

UPI/M. Manni/Corbis/Bettmann

Einstein’s Universe and the New Cosmology

Figure 6

Stephen Hawking has made remarkable contributions to astrophysics and cosmology. Then I would have felt sorry for the dear Lord, for the theory is correct. Einstein’s reply when asked how he would have felt if the 1919 Solar Eclipse observations had disagreed with his General Theory of Relativity.

Figure 9 Figure 8

If you warp a flat two-dimensional space, it becomes a curved twodimensional space.

In a two-dimensional spherical space, two lines that start out parallel and extend as “straight” (or straightest) lines will eventually meet.

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At this point, many students develop the misconception that there must be a fourth spatial dimension into which three-dimensional space is curving. This is wrong. Our two-dimensional analogy is meant to be imagined with no reference to any “embedding” of those two dimensions in a third dimension; the real threedimensional space is curved despite the absence of a fourth spatial dimension into which three-dimensional space is curving. How do we know that space is curved? But does the bending of light really show space to be curved or does it merely show that light beams bend in ordinary or “flat” threedimensional space? The latter possibility was ruled out by an experiment in 1972 in which a spacecraft orbiting Mars beamed back radar signals sent from Earth (Figure 10). The radar beam’s travel time was measured at a time of year when the line of sight from Earth to Mars passed near the sun. This travel-time measurement can tell whether the bent light beam travels through a flat space or through a warped space. Here’s how. It’s easy to use the observed curved path to predict the travel time in a flat space by making a scaled-down drawing of the curved path on a flat sheet of paper and seeing how much longer it is than a straight line. In the experiment, the answer was about 10 m, so if the radar beam was merely bending in a flat space, it should have been delayed by about 30 billionths of a second, the time taken by light (and radar) to travel 10 m. But you can’t use a flat sheet of paper to measure distances in a warped space, for the same reason that you can’t determine the distance from Los Angeles to London by making measurements on a flat map: The “scale” keeps changing because of the curvature. Einstein’s formulas predict a delay of 200 millionths of a second, 7000 times longer than the predicted delay in a flat space. The experiment confirmed Einstein’s prediction.

Spacecraft in orbit around Mars

Radar beam from Earth is reflected back by spacecraft

Sun

Earth

Figure 10

An experiment to measure the total travel time for a radar beam to get to Mars and back.

Figure 11

Masses such as the sun cause space to curve.

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I have so far ignored one fact that I now must mention. Space and time are tangled up with each other. For example, to measure the width of a moving window, you need at least two clocks to ensure that you measure the two sides of the window at precisely the same instant. So distance measurements involve time measurements. In general relativity, this tangling of space and time means that any warping of space must also distort time, causing clocks (in other words, time) to go slower in stronger gravitational fields. It’s really space and time together, or spacetime, that is distorted by masses. Spacetime is not an especially subtle or difficult idea. It’s not hard to imagine two or three of its dimensions, but impossible to imagine all four at once. For example, if you’ve ever graphed the position “x” of an object moving along a straight line versus the object’s time of travel “t,” you’ve graphed the motion of an object in spacetime. The general theory of relativity revolutionized our view of gravity and of space and time. Newtonian physics viewed space and time as a passive, unchanging background against which events unfolded, while modern physics views spacetime as an active and changing physical participant in events. Spacetime forms a kind of fabric that can be molded by masses (Figure 11), much as a hammer can bend a sheet of metal. Spacetime has a shape, a shape that is molded by matter and that affects the motion of matter and radiation through space. For familiar situations on Earth such as the fall of a stone, general relativity’s predictions are nearly identical to Newton’s.1 For exotic situations such as near a 1

Even on Earth, the small differences from Newtonian predictions are important for practical applications requiring extreme accuracy. For example, the global positioning system (GPS) depends on satellites to provide an accurate determination of the position of any GPS receiver on Earth. It’s crucial that all 24 GPS satellites use the same time to a high degree of accuracy. For this, scientists must take the effects of both special and general relativity into account.

Einstein’s Universe and the New Cosmology

black hole or during the creation of the universe, general relativity’s predictions differ enormously from Newton’s. Conceptually, the two theories differ radically. In Einstein’s theory, gravitational effects such as Earth’s circular motion around the sun are not caused by forces at all but are instead due entirely to the curvature of spacetime. Earth’s orbit is pulled into a circle not by the force of gravity, but rather because the sun warps spacetime and Earth simply “falls” freely (experiencing no force at all) along those warps. Earth must move along a curved path in spacetime, because spacetime is curved. To ward off a common misconception, I’m not saying here that space is curved into a circle around the sun and that Earth follows these circles. Instead, spacetime is curved in such a way that Earth moves in a circle in the spatial dimensions while moving toward increasing time in the time dimension, producing a spiral in spacetime. How do we know that general relativity is accurate? What with curved space, bending lightbeams, and spacetime, this theory introduces some unusual concepts. It would be natural for you to question its validity. So it’s reassuring to know that scientists have checked the theory frequently and carefully and it has not yet failed a single test. The most demanding test was reported in 2006, and involved a pair of pulsating neutron stars that orbit each other in our own galaxy at 2000 light-years from Earth. Imagine two stars, each more massive than the sun yet squeezed down by gravity to a diameter of only a few kilometers, each containing a billion tonnes of material in every cubic centimeter, one star spinning an incredible 44 times per second and the other at 3 seconds per revolution, each sending out with each revolution a radio beep similar to a rotating lighthouse beacon, and separated from each other by only about 3 times the distance from Earth to the moon—close enough that each star affects the pulses of the other. The stars are converging at 7 mm per day, and will merge in a galaxy-shaking collision in 85 million years. The enormous gravitational fields created near these tiny but massive stars, the regularity of the stars’ motions and their clocklike radio signals make this system a perfect “laboratory” for testing many quantitative details of general relativity in a situation where spacetime is predicted to be strongly bent by gravity, and where Newtonian gravity is far wrong. According to Ingrid Stairs, a member of the team who reported the first measurements, “general relativity does a perfect job of describing what we know of the system so far.” The results showed that, despite the extreme gravity, the theory of general relativity is accurate to within the team’s measurement uncertainty of 0.05%. A similar but even more mind-blowing observation was reported in 2008 when scientists discovered the largest known black hole at the center of a galaxy 3.5 billion light-years from Earth. This black hole has the mass of 18 billion suns. To make matters even more interesting, they found a smaller 100-million-sun black hole orbiting the larger black hole every 12 years. Again, this system is a perfect laboratory to observe general relativistic effects. These observations have been done, and they fully agree with general relativity while ruling out several competing theories that had been proposed as alternative theories of gravity. Like the neutron stars, the two black holes are converging and will merge in about 10,000 years, a collision that will literally shake spacetime in a manner that should be detectable here on Earth, across 25% of the observable universe.

CONCEPT CHECK 1 Since accelerations can mimic the effects of gravity, accelerations should be able to cancel gravity. Thus, a person could experience weightlessness by (a) blasting off from Earth, straight upward, at an acceleration of 1g; (b) falling from a high place such as a diving board or airplane (skydiving); (c) orbiting Earth; (d) standing on the surface of the moon.

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CONCEPT CHECK 2 The equator is a “straightest possible” line on the surface of a globe. Are the other east-west circles of latitude “straightest possible” lines? (a) Yes. (b) No, they curve more than the equator’s curvature. (c) No, they curve less than the equator’s curvature. (d) No, despite the fact that their curvature is the same as the equator’s curvature.

2 THE BIG BANG You don’t have to search far to locate where the big bang occurred, for it took place where you are now as well as everywhere else; in the beginning, all locations we now see as separate were the same location. String Theorist Brian Greene in The Elegant Universe

You are living in the golden age of cosmology: the study of the origin, structure, and evolution of the large-scale universe. I will take full advantage of that fact by presenting some of the mind-blowing recent cosmological discoveries. The golden age began in 1992 when an observing satellite charted the first detailed map of the early universe. The keys to the new cosmological discoveries are the wonderful new observing instruments such as the Hubble Space Telescope. The key to understanding these discoveries is the general theory of relativity. When applied to the universe as a whole, general relativity predicts the possible ways our three-dimensional universe could evolve throughout past and future time. When supplemented with certain astronomical observations (described in the following discussion), general relativity leads to a striking description of the origin and evolution of the universe: About 14 billion years ago,2 the universe began in a single event called the big bang that created the different forms of energy and matter, causing the “observable” universe (the portion that can be observed with telescopes) to expand from a much smaller initial size. The reality of the big bang is strongly confirmed by serval independent lines of observational evidence, but the theoretical understanding of how and why this event occurred is just beginning (Section 7). How do we know there was a big bang? Four independent lines of evidence support the big bang theory:

1. Astronomers first hypothesized the big bang in 1929 because they discovered evidence that all the galaxies throughout the universe are receding from one another just as if they had been driven apart by an explosion. Extrapolating backwards in time from the speeds and distances we see today, the galaxies should have all been together 14 billion years ago. 2. In 1964, radio astronomers first detected the cosmic microwave background, the faint afterglow that still fills the universe from the hot initial explosion. The radiation has now cooled all the way down to –270°C.3 This cold radiation has too little energy to be visible and is observable today only as faint radio static in the microwave and radio regions of the spectrum. Its observed characteristics, such as its temperature, agree with the big bang theory’s predictions. 3. In 1992 and again in 2003, observing satellites mapped the cosmic microwave background arriving at Earth from all directions in space (Figure 12). The results (Figure 13) showed that this radiation contains subtle and highly complex “ripples” of precisely the sort expected if the initial big bang did indeed develop into the structured universe of galaxies and clusters of galaxies that we see today. The existence of the radiation mapped in Figure 13, and the close relationship between that radiation and the universe we see around us today, are strong evidence for the big bang theory.

NASA Earth Observing System Figure 12

The Wilkinson Microwave Anisotropy Probe (WMAP) leaving the Earth/moon system, headed for a gravitational blance point in space known as “L2.” In 2003, this satellite looked nearly 14 billion years back in time to observe our universe when it was in its infancy—about 400,000 years old. This is comparable to viewing a baby picture of an 80-year-old man taken when he was less than one day old. WMAP used the moon to gain velocity for a slingshot to L2.

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2 3

More precisely, 13.73 billion years with a surprisingly small 1% margin of error. This is just 3 degrees above absolute zero, the lowest possible temperature, the temperature at which all microscopic motion is the least it can be without violating the quantum uncertainty principle.

The Wilkinson Microwave Anisotropy Probe/NASA

Einstein’s Universe and the New Cosmology

Figure 13

First light: A “fossil” of the creation of the universe. This map, a portrait of the 14-billion-year-old microwave whisper that now remains from the mighty flash of radiation that was released a mere 380,000 years after the big bang, shows the temperature differences in the universe as it existed at the time the radiation was released; darker regions are slightly cooler, lighter regions are warmer. The map shows every direction in space; think of horizontal and vertical axes going through the map’s center, with units in angular degrees extending 360° along the horizontal axis and 180° along the vertical axis. Although the expansion of the universe has by now stretched the fabric of space so much that the radiation has stretched into the microwave region of the spectrum, you are looking at a photograph (“microwave-graph” actually) of the first light that traveled through the universe. Before this time, the universe was so hot that its atoms were electrically charged, which prevented light from traveling through space. Thus, the cosmic microwave background acts like a lightemitting curtain, beyond which we cannot see. It is the oldest, largest, and furthest observable structure known to science.

4. The fourth line of evidence concerns the creation of the universe’s first chemical elements. The earliest kinds of ordinary matter, formed during the first thousandth of a second of the big bang, were protons, neutrons, and electrons. Conditions during just the next 3 minutes were right for protons and neutrons to “fuse” together into more complex nuclei. After these 3 minutes, the universe was too cool and too dilute for protons and neutrons to continue fusing. Well-developed and highly reliable nuclear physics calculations predict that at the end of these 3 minutes, 75% of the original protons still remained, while 25% of the original protons had fused with neutrons to form four other types of nuclei, labeled 21H, 32He, 42He, and 73Li . The remaining single protons are ordinary hydrogen nuclei, labeled 11H. The universe was then made of two different types of hydrogen, two types of helium, and one type of lithium, in the proportions stated in Table 1. Astronomers have made measurements of the light or “spectra” from the oldest stars, stars that presumably formed from the original material created in the big bang and that have changed little since that time. These measurements show relative amounts of the five isotopes that are in excellent agreement with the theoretically predicted amounts of Table 1. This detailed quantitative agreement between observations and the big bang theory’s predictions for five different nuclear types is strong evidence for the theory. The prediction and confirmation of 21H is especially compelling, because nuclear physics predicts that there is essentially no process anywhere in the universe, other than the big bang, that could have made this material. The big bang is as real as the “snow,” or interference, that you can see on an old analog (pre-digital) TV screen when the power is on with no station tuned in. Cosmic microwave background radiation causes some of this interference. The echo of the big bang is all around you!

Table 1 Predicted nuclear composition of universe at about 3 minutes after start of big bang. Current observations of the oldest material in the universe agree well with these predictions. Nuclear type

Relative concentration by mass

1 1H 2 1H 3 2He 4 2He 7 3Li

75% 5–10 parts in 100,000 2–5 parts in 100,000 25% 2–5 parts in 10 billion

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Einstein’s Universe and the New Cosmology In the patterns of the subtle temperature differences in the cosmic microwave background in different directions we are learning to read the Genesis story of the expanding universe. The resulting origin story will be the first ever based on scientific evidence and created by a collaboration of people from different religions and races all around the world, all of whose contributions are subjected to the same standards of verifiability. Nancy Ellen Abrams, Lawyer and Writer, and Joel Primack, Astrophysical Theorist, Writing in the Journal Science

Figure 14

The two-dimensional surface of an expanding balloon is a twodimensional representation of the expansion of the three-dimensional universe. As space, represented by the balloon’s surface, expands, the galaxies, represented by flat raisins glued to the balloon, move farther apart. Although this twodimensional analogy has the universe expanding into the empty space outside the balloon, there is no space outside the real threedimensional universe.

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The universe is still made mostly of hydrogen and helium, although heavier elements created since the big bang now contribute a small percentage. Nearly all the hydrogen and helium can be traced back to the big bang. Although our bodies contain no helium, the hydrogen forged 14 billion years ago in the big bang is one of the most prevalent elements in your body and in all living organisms. CONCEPT CHECK 3 The gold nuclei in the universe were (a) all created in the big bang; (b) all created sometime after the big bang; (c) created partly during the big bang, and partly after.

3 THE POSSIBLE GEOMETRIES OF THE UNIVERSE The expansion of the universe may be the most important fact ever discovered about our origins. The key to understanding it is to not take the term “big bang” too seriously. It was not like the explosion of a bomb that happened in time and space. Rather, the big bang created time and space. Time and space are part of the universe, not the other way around. As the universe expands, it makes its own time and space. The universe is expanding, but it is not expanding into anything because there is no space “outside” of the universe. So space started small and has been geting bigger ever since. One of the predictions of general relativity is that three-dimensional space can’t remain static but must always either expand or contract. It’s remarkable that even space itself must continually change. Everything, it seems, is active and changing: The stars are born and die, life on Earth evolves, you and I are born and will die, and even space itself must always expand or contract. Direct evidence for the expansion of the universe comes from astronomical observations of other galaxies outside our Milky Way galaxy. Distant galaxies are moving away from us, and the more distant galaxies move away faster. But the galaxies are not just moving away from our particular galaxy; they are all moving away from one another. Regardless of which galaxy you live in, you will observe the other galaxies moving away from you. It’s like a loaf of raisin bread expanding as it bakes: If you were standing on any one of the raisins observing the other raisins, you would observe all of them moving away from you, and more distant raisins would move away from you faster. To visualize the expansion of our entire curved three-dimensional universe, we must imagine a two-dimensional analogy as in Section 1. Imagine that the surface of a partially inflated balloon is a two-dimensional universe, similar to the ant and globe analogy of Section 1. To represent the galaxies, imagine two-dimensional (flat) raisins glued to the surface. Remember that, in this two-dimensional analogy, you must imagine that the inside and outside of the balloon don’t exist; only the twodimensional surface of the balloon is supposed to exist. Now imagine that the balloon inflates (Figure 14), representing the expansion of the universe. Note that, as the balloon expands, the distance between all the raisins increases. No matter which raisin you are standing on, all the other raisins move away from you. No raisin is at the center of this balloon universe, in fact the surface of the balloon has no center. In agreement with the philosophy of the Copernican revolution, this universe is, on average, the same all over. Note that the galaxies are at rest relative to the balloon’s surface. It’s not really the raisins that are moving; instead, the space between the raisins is expanding. In

Einstein’s Universe and the New Cosmology

the real three-dimensional universe, gravity holds each galaxy (also each star and each planet) together in a relatively fixed size and shape, while the space between the galaxies expands. Note also that no galaxy is at the edge of the balloon universe, because the balloon universe has no edge. And neither does the real three-dimensional universe. According to general relativity, the possible shapes or geometries for the largescale structure of the three-dimensional universe fall into three categories. Figure 15 shows the two-dimensional analogs of these three-dimensional geometries. A closed universe bends back on itself to form a sphere. If you lived in a closed universe, you could detect this from the fact that straight (that is, straightest) lines that start out parallel eventually meet (Figure 9), and the angles of a triangle add up to more than the normal 180°, as you can see from Figure 15. Although a closed universe has only a finite extent, the other two geometries have infinite total extents and so only a portion of these surfaces can be shown in the figure. A flat universe has no overall large-scale curvature (in all three geometries there will be smaller-scale warps caused by stars, black holes, galaxies, and other objects) and has the normal Euclidean geometry with which you are familiar—parallel lines remain parallel, and the angles of a triangle add up to 180°. An open universe is analogous to a saddle-shaped surface; in such a universe, straight lines that start out parallel eventually diverge from each other, and the angles of a triangle add up to less than 180°. Regardless of which of the three geometries our actual universe might have, if you follow a straight (straightest) path, you will never come to an edge or to the center of the universe. In the open and flat universes, this is because the universe is infinite in extent. In the closed universe, it is because straight (straightest) lines simply curve back to where they started. In such a universe, if you head in an absolutely straight line for many billions of light-years, you will reach your starting point. CONCEPT CHECK 4 The universe is expanding. Is everything in the universe expanding? (a) Yes. (b) No, the distances between the galaxies are not expanding. (c) No, the Milky Way galaxy is not expanding. (d) No, our solar system is not expanding. (e) No, Earth is not expanding.

4 THE SHAPE OF THE UNIVERSE The revolution in cosmology has been driven by a revolution in observational techniques. Until 1992, observations having cosmological significance were few and far between and highly imprecise. Cosmology was of necessity highly theoretical and conjectural. The age of precision cosmology began in 1992 with the first observation of the details of the cosmic microwave background, similar to the far more detailed Figure 13. Figure 13 is a “microwave photograph,” similar to an infrared photograph, showing the temperature variations in the background radiation emitted by the big bang. The light regions are slightly warmer than the dark regions. This radiation was emitted just 400,000 years after the big bang. Before that time, the temperature of the universe was so high that protons and electrons moved too rapidly to stick together to form hydrogen atoms. The resulting mix of electrically charged protons and electrons immediately absorbed any radiation present. At 400,000 years, the universe had cooled enough for electrons to combine with protons to form neutral hydrogen atoms, and radiation propagated through space for the first time. The microwaves

(a) Closed geometry

(b) Flat geometry

(c) Open geometry

Figure 15

Two-dimensional analogs of the possible large-scale geometries of the three-dimensional universe, as predicted by general relativity. A closed universe bends back on itself to form a three-dimensional spherical space; in such a universe, the angles of a triangle add up to more than the normal 180° and the total volume is finite. A flat universe has no overall largescale curvature; it has the normal Euclidean geometry where the angles of a triangle add up to 180°, and an infinite total volume. An open universe is analogous to a saddle-shaped surface; in such a universe, the angles of a triangle add up to less than 180° and the total volume is infinite.

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that made Figure 13 traveled through nearly empty space for 14 billion years before entering the microwave detectors that created this map. You are looking at the image of the earliest light in the universe, a 14-billion-year-old “fossil.” From this map, showing details of the waves of matter and energy (similar to sound waves in air) that sloshed around in the early universe, scientists conclude that the large-scale geometry of the universe is flat rather than closed or open (Figure 15). Here’s how we know.

If you’re religious, it’s like looking at God. George Smoot, Leader of the Team That Announced in 1992 the Discovery of the Ripples in the Cosmic Microwave Background

How do we know the shape of the universe? With their knowledge of the physical nature of the hot, dense, and electrically charged early universe, scientists can predict the maximum distance that wavelike disturbances in this material could travel during the 400,000 years between the big bang and the release of the light seen in this map. Astronomers can also directly observe this distance in the cosmic microwave background, based on the average size of the observed hot or cool regions seen in the map (Figure 16). However, such observations are distorted by the geometry of the space through which the microwave radiation travels on its long journey to Earth, and this distortion enabled scientists to determine that geometry. As shown in Figure 16, a typical wavelike disturbance, as observed today from Earth, should make an angle of about 1° if the universe is flat, while a closed universe would warp the radiation into an angle larger than 1° and an open universe would warp it into an angle smaller than 1°. The observed angle was about 1°—fairly conclusive evidence that the overall geometry of the universe is flat or at least very close to it.

CONCEPT CHECK 5 Since there is evidence that the universe is flat, does this mean that there is no such thing as curved or warped space? Defend your answer. (a) Yes. (b) No. Figure 16

A single typical disturbance—a region of hotter or cooler temperature—in the nearly uniform early universe

A typical wavelike disturbance in the material of the early universe, as observed today from Earth, should make an angle of about 1° if the universe is flat. A closed universe would warp the radiation into an angle larger than 1°, while an open universe would warp it into an angle smaller than 1°. The observed angle was about 1°— strong evidence that the overall geometry is flat.

Hot, dense, charged material prior to 400,000 years after the big bang

Distance that any single disturbance could spread during 400,000 years following the big bang

14 billion light-years

Angle of less than 1 (open geometry)

Angle of 1 (flat geometry)

Angle of more than 1 (closed geometry) Observer

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11.5 DARK MATTER We’re accustomed to thinking that the universe is made mostly of the visibly luminous (shining) stars and a few nonluminous objects such as planets. But we’ve learned during the past 20 years that the universe is made of many more kinds of things than this. First, an enormous amount of the hydrogen and helium created in the big bang has neither gathered into stars nor collected within the visible galaxies but instead lies in the vast regions between the galaxies where it is invisible and nearly undetectable. Astronomers first detected it by observing how the light traveling to Earth from distant objects is partly absorbed as it passes through intergalactic space. The mass of this invisible intergalactic gas is now known to be 10 times larger than the mass of all the stars, planets, and luminous gas in the universe! Stars, planets, and intergalactic gas are made of atoms of ordinary matter, just like your chair. But there are other kinds of matter, matter that is not formed into atoms. One example is the neutrino. There are a vast number of individual neutrinos flying through the universe with a total mass estimated at one-quarter of the total mass of all the stars. Another kind of nonatomic matter is the black hole. Judging from what’s known about the massive black holes at the centers of galaxies, these are estimated to contribute a total mass about one-tenth as large as the mass of all the stars. That’s pretty fantastic, in my opinion. But there’s more. During the past few decades, scientists have learned that there is another kind of matter, matter not made of protons, neutrons, electrons, neutrinos, or any of the other particles currently known. Nobody knows what it’s made of, although there are several hypotheses. It doesn’t interact with electromagnetic radiation, so it can’t be detected by emitted light (like stars) or reflected light (like planets) or absorbed light (like intergalactic gas), and nobody has yet detected it in the laboratory. But we know it’s there because of its gravitational effects on the stars in galaxies, and we know there’s a lot of it. The total mass of this so-called dark matter is 60 times larger than the mass of all the stars!

How do we know that dark matter exists? Several independent methods of observation show that most galaxies, including our own, are made mostly of dark matter. One method is based on the fact that galaxies are rotating structures, with stars and gas orbiting the center. Like planets orbiting the sun, the stars and gas are held into their roughly circular orbits by the gravitational pull of the massive center of the galaxy. When astronomers observe stars and gas clouds orbiting the centers of their galaxies, their speeds turn out to be so high that the galaxies would fly apart unless held together by the gravitational pull of many times more matter than we actually see. So galaxies must contain invisible matter. But how can astronomers measure orbital speeds around distant galaxies where it’s difficult to pick out individual stars let alone measure their speeds? Looking at galaxies that could be seen “edge on” (Figure 17) from Earth, Vera Rubin (Figure 18) compared the light coming from points on one side of the galaxy’s bright center with the light coming from points on the other side. Since the galaxy is rotating, the stars on one side were moving toward Rubin’s telescope, and the stars on the other side were moving away. The frequency of the light coming from the stars moving toward the telescope was higher than the frequency of the light moving away, for the same reason a police siren shifts to a higher pitch as the police car approaches you and then to a lower pitch as it recedes from you while you listen from the sidewalk. From the difference between the two frequencies, Rubin was able to calculate the speeds of the stars.

California Inst. of Technology/ Palomar/Hale Observatory Figure 17

A galaxy, full of stars and gas and dust, viewed “edge on.”

John Irwin collection/AIP/Photo Researchers, Inc. Figure 18

Vera Rubin. She made pioneering discoveries that contributed to understanding the existence and amount of dark matter by observing frequency shifts of stars in galaxies.

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Einstein’s Universe and the New Cosmology In a second method of observation, light reaching Earth from distant galaxies is warped as it passes through the gravitational fields of galaxies that lie in the path of the light. By analyzing this bending, called “gravitational lensing” (Figure 19), astronomers can deduce that the intervening galaxies contain far more matter than can be seen. By the time you read these words, dark matter might be discovered in the laboratory. The biggest particle accelerator in history is coming online in 2009 or 2010 at the European Organization for Nuclear Research, or “CERN,” near Geneva, and physicists believe that it will be able to spot the predicted candidates for dark matter, if they exist.

W. Couch/R. Ellis/NASA Headquarters Figure 19

Warped light. To make this photograph, the Hubble Space Telescope peered straight through the center of a distant cluster of galaxies. The rounded objects in the photo are galaxies in this cluster. The stretched-looking objects are other galaxies lying at great distances behind the “foreground” cluster of galaxies. The light from these more distant galaxies is gravitationally warped as it passes through the foreground cluster. The warped light in this photograph comes from galaxies lying many billions of light-years away; some of this light originated when the universe was barely a quarter of its present age! A photograph such as this is direct visual evidence for the general theory of relativity.

From such observations, we know that our galaxy, and most other galaxies, is immersed in a giant spherical cloud of dark matter whose diameter is many times the diameter of the visible galaxy (Figure 20). What, then, is this dark matter? No known form of matter can account for it. Scientists expect that entirely new forms of matter will be discovered, and there have been several theoretical suggestions about what form it might take. It must interact only weakly with ordinary matter, or we would have discovered it by now. Whatever it is, it’s all around us: There are probably billions of dark matter particles passing through your body every second, but leaving no effect on your body. Dark matter has inspired many searches among cosmic rays (particles from space) and in high-energy physics experiments. A parallel situation existed during 1914 to 1955 when theory suggested that an unobserved particle was created during beta-decay, but no such particle could be detected until 1955, when physicists discovered the neutrino. The laboratory discovery of dark matter would be momentous.

Luminous matter Dark matter

Figure 20

Dark matter forms a giant invisible spherical “cloud” around each visible galaxy. The small object at the center of the figure is a spiral galaxy, like our Milky Way galaxy, seen edge-on (compare Figure 17).

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Einstein’s Universe and the New Cosmology

CONCEPT CHECK 6 What’s so unusual about dark matter? (a) It exerts no gravitational force. (b) Its gravitational force pushes (or repels) instead of pulling. (c) It is made of material that has never been observed in our laboratories. (d) It moves faster than lightspeed. (e) It does not interact with electromagnetic radiation.

6 THE ACCELERATING UNIVERSE AND DARK ENERGY Will the universe expand forever, or will it eventually collapse back inward on itself? This is similar to asking what happens to an object that is thrown upward from Earth’s surface. If you throw a ball upward, it slows as it comes to a momentary stop at its maximum height and then immediately accelerates downward to the ground. But if NASA “throws” a space vehicle upward faster than 11 km/s (25,000 mph), it slows as it rises but instead of returning to the ground it keeps rising and escapes from Earth. Like the rising ball and the space vehicle, it stands to reason that the universe’s expansion should be slowing down. Just as the ball and the space vehicle are slowed by the backward pull of Earth’s gravity, the universe’s expansion should be slowed by the inward gravitational pull of all the matter in the universe. It’s important to quantitatively measure this deacceleration of the universe, because a sufficiently large deacceleration would imply that the universe is like the upward-thrown ball in that it will eventually stop expanding and then immediately begin collapsing on itself in an ultimate “big crunch.” On the other hand, if the deacceleration is sufficiently small, then the universe is like the upward-thrown space vehicle and will continue expanding forever. But it’s difficult enough to measure the expansion rate of the universe, let alone the rate at which that expansion rate is slowing down, so for many decades cosmologists didn’t know whether the universe would eventually collapse or would continue expanding forever. During the 1990s, cosmologists managed to measure that deacceleration. The result, in 1998, was a shocker: The universe’s expansion isn’t slowing at all. It’s speeding up.

Not only are we not at the center of the universe, we aren’t even made of the same stuff the universe is. Joel Primack, Astrophysical Theorist, University of California

How do we know the universe is accelerating? First, let’s see how scientists measure the speeds at which the galaxies are moving apart. Light waves stretch as they travel through the universe, because of the stretching of space during the time of travel. Thus, light from distant galaxies arrives at Earth with a longer wavelength than it had when it left its home galaxy; it is shifted toward the long wavelength or red end of the electromagnetic spectrum. This redshift of the light from distant galaxies, first discovered during the 1920s, was the earliest evidence of the big bang and the expansion of the universe. Scientists can measure the amount by which a galaxy’s light is redshifted and from this deduce the galaxy’s speed. But in order to use redshifts to confirm that the universe is expanding, one needs to know that it actually is the more distant galaxies that are redshifted the most. Such distances are not easy to determine. You can’t just stretch a tape measure out to a distant galaxy! Nevertheless, astronomers have for many years had methods for determining such distances and have amply confirmed that more distant galaxies are more redshifted in just the way expected in an expanding universe. Recently, astronomers developed an especially powerful method of determining such distances, along with speeds. Large modern telescopes can detect a particular type of supernova explosion (an explosion of a star) in far-distant galaxies. These “Type 1a supernovas” are bright enough to be seen even at distances greater than halfway across the observable universe. Also, it’s known that all Type 1a supernovas are nearly identical, and all shine with the same brightness during their roughly one-month period of maximum

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Einstein’s Universe and the New Cosmology intensity following the explosion. Since they all have the same actual brightness, more distant ones always appear dimmer from Earth, and from their observed brightness one can deduce how far away they must be. Thus, Type 1a supernovas are our most accurate markers for determining expansion speeds and distances across most of the universe. They’re sufficiently accurate to determine not only the speeds but also the rate of change of the speeds—the accelerations—of distant parts of the universe. In 1998, these observations revealed that the expansion of the universe is actually speeding up.

This was not expected. If you threw a silver dollar up into the air and, instead of slowing and coming back down, it sped up until it rose out of sight, you’d say that’s a pretty mysterious way to lose a dollar. You’d probably want to know what pushed it into outer space. In the same way, the gravitational pull of all the matter in the universe should slow the universe’s expansion. But it’s speeding up. What’s pushing on it? Recall that the receding galaxies are not really moving at all, but are simply remaining roughly at rest in space while space itself expands, like the raisins in our expanding balloon analogy in Section 3. Since accelerations are caused by forces, the accelerating expansion means that something is pushing outward on the fabric of space. What can it be? It’s certainly not matter of either the ordinary or the dark type, because the force of gravity from both ordinary and dark matter can only pull, not push. Scientists believe that all of space, including even “empty” space or vacuum, must contain some new form of nonmaterial energy that pushes outward. It’s called dark energy. This astonishing new concept burst upon the physics community in 1998 with the discovery of the acceleration of the universe. Nobody knows what dark energy is, although some theories relate it to the energy of the field that “inflated” the universe during the early moments of the big bang (next section). Dark energy is more mysterious than dark matter: We have evidence that it’s there, but little idea what it is. Dark energy must influence the shape of the universe, because Einstein says that all forms of energy have mass and because mass affects the curvature of space. It happens that it’s possible to infer the amount of dark energy present in the universe from the details of the cosmic microwave background. When the mass of this dark energy is added to the masses of the luminous matter, nonluminous ordinary matter, and dark matter in the universe, the total comes out to be precisely the amount needed to flatten the overall geometry of the universe! Thus, the flatness of the universe, dark matter, the acceleration of the universe, and dark energy all fit together in a beautifully consistent but totally unexpected picture of the universe. All of this provides a new answer to the ancient question “What is the universe made of?” Observations of the cosmic microwave background, and of the universe’s acceleration show that it’s made mostly of dark energy! The other ingredient is matter, the great bulk of which is dark matter. In more detail, the universe is 73% dark energy, 23% dark matter, nearly 4% nonluminous “ordinary” matter (including intergalactic gas, neutrinos, and black holes), and only 0.4% (less than half a percent) ordinary visible matter (Figure 21). The universe is stranger than you or I could have imagined: 96% of it is made of completely unknown matter and energy, most of the remaining 4% is invisible, and only a fraction of 1% is normal visible matter. The universe we can see is only a tiny fraction of all that is! To return to the question that began this section: If the universe continues its accerating expansion, it will not only expand forever but will expand faster and faster forever. But this assumes that the universe does keep accelerating, and, given the surprises of the past few years, few cosmologists would bet much on any particular long-term scenario.

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Einstein’s Universe and the New Cosmology Figure 21

Dark energy (identity unknown): 73%

Dark matter (identity unknown): 23%

Luminous matter: stars and luminous gas 0.4%

What is the universe made of? These numbers show that, whatever may be the nature of the unknown dark energy and dark matter, the universe is not made primarily of the same stuff that we are made of!

Other nonluminous components: intergalactic gas 3.6% neutrinos 0.1% Supermassive BHs 0.04%

CONCEPT CHECK 7 Type 1a supernova explosions make excellent markers for measuring the universe’s acceleration because (a) they all emit about the same amount of light; (b) they are all the same distance away from us; (c) they can be seen from immense distances; (d) they are all moving away from us at the same speed. CONCEPT CHECK 8 Dark energy (a) is made of some unknown form of matter; (b) has mass; (c) is made of invisible electromagnetic radiation; (d) pushes on space.

7 COSMIC INFLATION AND A BRIEF HISTORY OF THE UNIVERSE Alan Guth (Figure 22) bicycled hurriedly to the Stanford Linear Accelerator Laboratory (SLAC) to start work on the morning of December 7, 1979, breaking his personal speed record with a time of 9 minutes and 32 seconds. Working late the previous night, he had begun to understand a new and extraordinary cosmological phenomenon, and he was anxious to get back to thinking about it. He checked his calculations from the night before and found them exactly on target. Several weeks later the young physicist apprehensively presented his new idea to a packed audience at SLAC. The response was overwhelmingly favorable, exceeding Guth’s wildest expectations. The hypothesis of cosmic inflation was born. Guth had combined ideas from general relativity, quantum physics, and highenergy physics to explain how the matter and energy in the universe could have been created from nearly nothing by a high-energy submicroscopic event occurring in nearly empty space. Guth’s hypothesis does not explain how the universe actually got started, but it does explain how, starting from a tiny fragment of spacetime containing a minuscule amount of matter and energy, the universe expanded enormously while filling with matter and energy. Briefly, Guth’s hypothesis says that the universe started out unimaginably small, far smaller than a proton, and immediately expanded. Very early, at a trillionth of a trillionth of a trillionth (!) of a second after the beginning, the universe experienced an even more rapid period of expansion at speeds greatly exceeding lightspeed, during which it stretched by a factor of 1025 during only 10–35 seconds. To see how mind-boggling this is, try writing out these two numbers. This breakneck expansion is called, appropriately, “inflation” (recall Figure 14).

Donna Coveney/Massachusetts Institute of Technology News Office Figure 22

In 1979, high-energy particle physicist Alan Guth made a monumental cosmological discovery: the idea of cosmic inflation. The hypothesis has significant experimental support in satellite observations of the cosmic background radiation and in other cosmological observations. Cosmic inflation is the best scientific explanation to date of the details of the first few moments of time.

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Einstein’s Universe and the New Cosmology Nowhere is the inherent unity of science better illustrated than in the interplay between cosmology, the study of the largest things in the universe, and particle physics, the study of the smallest things. Rocky Kolb, Physicist at Fermilab

Where the telescope ends, the microscope begins. Which of the two has the grander view? Victor Hugo

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This sounds pretty bizarre, but it’s been receiving some observational confirmation lately. One response to this hypothesis is “How can the universe expand at faster than lightspeed, since special relativity predicts that nothing goes faster than light?” We’ve already dealt with this in Section 3: Special relativity predicts that no object can move through space faster than light. But general relativity tells us that the expansion of the universe is an expansion of the fabric of space itself, and there is no speed limit on this. The expansion carries galaxies and other objects along with it while those objects remain at rest relative to the space around them. According to Guth’s hypothesis, our universe started out so small that quantum effects such as the uncertainty principle dominated. One implication of this principle is that in every region of space, the energy in the region fluctuates randomly (or unpredictably) up and down around its average value, a little like the surface of a small portion of a lake fluctuates up and down due to wind rippling its surface. Even in supposedly “empty” space, such energy fluctuations are still required by the uncertainty principle. At extreme submicroscopic sizes, it’s thought that space and time do not exist as we know them but are instead broken up or “quantized” into tiny separate fragments having durations of about 10–49 s (that’s short!) and diameters of about 10–35 m (that’s small!). According to the inflation hypothesis, an unusually large energy fluctuation occurred in just such a fragment. This fluctuation had an energy of only some 109 joules, about the energy of one automobile tank of gasoline. According to E = mc2, the mass of this much energy is 0.01 milligrams—about as massive as a grain of dust. This doesn’t sound like enough energy to start a universe, but amazing things can happen when it’s all crammed into such a tiny region. One of those amazing things was that so much energy in such a small region created an enormous temperature of some 1032 degrees (try writing it out). Our universe immediately began expanding simply because it was so hot (this is also the reason ordinary explosions expand), and the expansion cooled it from its initial 1032 degrees down to around 1028 degrees. A major theme of modern physics, already encountered in our discussion of gravitational and electromagnetic fields, is that the universe is made of just a few kinds of fields that extend throughout all space and time. The cosmic inflation hypothesis is based on a new type of field, not yet observed in nature, called the inflation field. When our then-tiny universe had expanded and cooled to 1028 degrees, the inflation field developed something called a “false vacuum” that amounts to a gravitational force that strongly repels instead of attracting like the gravity that we know. This repulsive force sent the universe into a brief period of rapidly accelerating expansion or “inflation” up to speeds far faster than lightspeed. The expansion was actually “exponential”—that is, it had a fixed doubling time. Exponential growth can be surprising. Although this inflationary period began at 10–36 s into the big bang and lasted only until 10–34 s into the big bang, the universe’s size doubled nearly 100 times, resulting in a universe that was about 1025 (10 trillion trillion) times larger than it was before inflation. Even after inflation our universe was only a millimeter across but nevertheless the expansion was enormous. Think of a balloon being filled by a fire hose. Physicists believe that there are just four types of fundamental force fields: the gravitational field, electromagnetic field, “weak force” field, and “strong force” field. The last two are apparent only at the level of the atomic nucleus, in connection with nuclear forces. But in the fires of the early universe, the four fundamental forces were all “melted together” and indistinguishable. There was only one force, not four.

Einstein’s Universe and the New Cosmology

Physicists say that the four forces had the same “symmetries” and so did not exist individually. As the universe cooled, the gravitational force suddenly “froze out” of the unified force; it lost the symmetry that had unified it with the other forces and took on its own distinctive gravitational properties. This “symmetry breaking” is analogous to the loss of symmetry when water freezes: All directions are equivalent inside water, but ice crystals line up in specific directions—a loss of symmetry. As the universe continued cooling, the strong force froze out and formed its own unique patterns such as the quark-gluon plasma simulated in Figure 23. Finally, the weak force and the electromagnetic force froze out also, leaving us with the four forces that have their four distinct sets of properties that we observe today. But where did all the mass and energy in the universe come from, if energy is conserved and if everything developed from an energy fluctuation having the mass of a dust grain? Here’s where: The gravitational energy of any isolated lump of matter such as a star, that is held together only by gravity, is negative (less than zero), because work must be done on (rather than can be gotten from) the star in order to pull it apart into separated pieces. In the same way, the gravitational energy of the entire universe, due to the attraction between all its parts, is enormously negative. Inflation didn’t alter the universe’s net energy, but instead created negative energy (gravitational) and positive energy (kinetic, radiant, and the energy needed to create matter) in equal amounts. It’s like a man who spends a lot of money by going into debt; he spends like a millionaire, but his net financial balance remains zero. Thus the universe’s net energy remains very close to zero, balanced between an enormous negative gravitational energy and a slightly more enormous (by one gasoline tank) positive energy. The positive energy of matter and motion that you see today was scavenged in the early universe from gravity. As Alan Guth puts it, cosmic inflation is “the ultimate free lunch”: That gasoline tank’s worth of energy was the seed for everything. It’s a powerful story of how things came to be. Figure 24 shows some of the details of the time sequence. The time line is plotted in powers of 10, rather than simply in seconds, because a lot happens fast in the early universe due to the high energies involved!

It is said that there’s no such thing as a free lunch. But the universe is the ultimate free lunch. Alan Guth, Originator of the “Inflation” Idea That Explains How the Big Bang Could Have Created Our Universe out of a Vacuum

Figure 23

Simulated “snapshot” of two lead nuclei colliding at very high energy. The simulation portrays the nuclei just 6 * 10 - 24 seconds after impact, showing protons and neutrons in white. The smaller particles portrayed in darker hues are “quarks,” the particles of which protons and neutrons are made. This is a simulation of a real experiment that reproduced the theoretically predicted “quark plasma” that is believed to have existed at 10 microseconds after the big bang.

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Einstein’s Universe and the New Cosmology time in seconds 10⫺42

10⫺40

R ⫽ 10⫺35 m This is the shortest time, and the smallest distance, that can exist. All forces unified, dominated by quantum gravity. An energy fluctuation of 109 J having a mass of 0.01 milligrams occurs.

10⫺22

10⫺38

10⫺36

Gravity “freezes out” as a separate force.

10⫺20

10⫺18

T ⫽ 1028 K R ⫽ 10⫺28 m The universe begins “inflating” at speeds far larger than c, doubling 100 times and becoming 1025 times larger. 10⫺16

10⫺34

10⫺32

10⫺30

10⫺28

10⫺26

10⫺24

10⫺8

10⫺6

10⫺4

T ⫽ 1026 K Strong force freezes out. R ⫽ 1 mm Inflation ends, universe resumes its “normal” expansion at speed c.

10⫺14

10⫺12

10⫺10

T ⫽ 1015 K R ⫽ 3 centimeters Weak forces freezes out from EM force, so that four separate forces are now apparent. EM force dominates universe. 10⫺2

100

102 T ⫽ 109 K R ⫽ 10 million km Universe is cool enough for protons and neutrons to form into nuclei.

104

106

T ⫽ 108 K R ⫽ 40 million km Nuclei in Table 1 formed during the past few minutes. Universe is now too cool and diffuse for further nuclei to form.

108

1010

1012

30,000 years Gravity amplifies the lumpiness that was formed by quantum fluctuations prior to the inflationary expansion.

1014

400,000 years T ⫽ 3600 K. First atoms. Cosmic background radiation released: first light in universe (Fig. 13).

T ⫽ 1012 K R ⫽ 3 km It is now cool enough for quarks to form into protons and neutrons as shown in Figure 23.

1016

1018

30 A few 9 14 million hundred billion billion years million years years First years Sun, T ⫽ 3 K. stars. First Earth, You galaxies. planets are formed. born.

Figure 24

A really brief history of the universe. All numbers are only approximate, and the first millionth of a second is hypothetical (not yet checked directly by observation)! Temperatures are in degrees above absolute zero, or Kelvins, abbreviated K. The radius of the observable universe is abbreviated as R. For all times after the end of inflation, the universe is 1025 (10 trillion trillion) times larger than the observable universe, because the universe expanded far faster than lightspeed during inflation so that nearly all of it is so far away that light cannot reach here from there during the entire history of the universe.

How do we know that cosmic inflation occurred? Cosmic inflation has already passed several observational tests. First, it provides a convincing explanation of the origin of the large-scale gathering or “clumping” of stars into galaxies, of galaxies into clusters of galaxies, and even of clusters into superclusters, seen in today’s universe. It’s not hard to understand how any initial lumpiness would be amplified by gravitational forces into today’s quite “lumpy” universe of stars and galaxies—just as gravity can create stars out of diffuse clouds of gas and dust. But prior to the inflationary hypothesis, the big bang model offered no clue as to what created the initial lumpiness. Cosmic inflation’s answer is that quantum uncertainties during the big bang caused microscopic lumps that were then stretched by the expansion of the universe. Without inflation, the amount of stretching would be far too small for quantum fluctuations to explain the vast lumps (clusters of

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Einstein’s Universe and the New Cosmology galaxies, etc.) seen today. Inflation resolves this problem: Inflationary expansion stretches the initial quantum lumps enormously, and gravity works on these stretched lumps to produce precisely the clumping observed today. Second, Guth’s hypothesis predicts and explains the observed flatness of our universe. The reason is simple: Inflationary expansion plus additional "normal" expansion since that time stretched the universe so hugely that any overall curvature is now stretched flat, the way that the surface of an expanding balloon gets flatter and flatter as perceived by an ant on the balloon’s surface. It’s surprising that our universe should be flat, because a flat universe represents a delicate balance right at the borderline between the finite closed geometry and the infinite open geometry of Figure 15. Without inflation, there is no convincing explanation for why the universe should be so delicately poised. Guth predicted a flat universe more than a decade before the first observation, in 1992, of the patterns in the cosmic microwave background suggested that the universe really is flat. In 2001, more accurate observations of these patterns provided further confirmation of Guth’s prediction.

It appears that, without initial energy fluctuations and inflation, our universe could not have developed the patterns seen today in the layout of the galaxies. The great clusters of galaxies stretching across the universe still retain the microscopic pattern of those initial quantum fluctuations occurring in an unimaginably tiny lump of energy that started all of this. It all sounds too amazing to be true, but the truly amazing thing is that it’s been checked in some detail by specific observations. Just as ice crystals freeze along a direction that is previously undetermined or random, so cosmic inflation predicts that the specific “direction” in which the inflation field “froze” during the big bang was also random. But when I speak of “different directions” of inflation-field freezing, I really mean different properties of the various fundamental forces as they froze out of the preexisting symmetric unified force. In this process, basic properties of our universe such as the masses and charges of the fundamental particles might have been determined randomly. It’s even possible that ours is just one of many universes created in similar processes, each born in a new toss of the quantum dice and each characterized by different physical properties. According to the inflationary view, it’s possible that in our universe the numbers turned out to have just those values that allowed intelligent animals to evolve. In any other universe, in which these numbers were very different, life and intelligence might have been physically impossible. Our own existence might turn out to be the best explanation we have for these numbers having the values that they do have. This idea, that our universe must be organized in the way that it is because any other organization would not allow intelligent beings to be here to ask the question in the first place, is called the anthropic principle. And this outrageous but plausible connection between the big bang and our lives on Earth is a good place to end our excursion into cosmology. CONCEPT CHECK 9 Can anything go faster than light? (a) Yes, space can expand at faster than lightspeed. (b) Yes, certain subatomic particles can move through space at faster than lightspeed. (c) No, special relativity forbids it. (d) No, general relativity forbids it.

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© Sidney Harris, used with permission.

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Einstein’s Universe and the New Cosmology Problem Set Answers to Concept Checks and odd-numbered Conceptual Exercises and Problems can be found at the end of this section.

Review Questions EINSTEIN’S GRAVITY: THE GENERAL THEORY OF RELATIVITY 1. List two experiments you could do in a spaceship accelerating at 1g through outer space that might make you think you are at rest on Earth. 2. According to the equivalence principle, to what is acceleration equivalent? 3. In your own words, state the equivalence principle. 4. Give one piece of evidence showing that gravity bends light. 5. As observed in an accelerating reference frame, does a light beam bend? What does this tell us about the effect of gravity on light beams? 6. According to Newton, gravity is a force exerted by material objects on other material objects. What is gravity according to Einstein?

THE BIG BANG 7. About how old is the universe? 8. Describe two different pieces of evidence supporting the big bang. 9. Of what element is the universe mostly made? 10. Following up on the preceding question, what is the second most prevalent element in the universe? 11. Name two elements that were not made in the big bang. Name two that were.

THE GEOMETRY OF THE UNIVERSE 12. Give an example of a flat two-dimensional space, a curved two-dimensional space, and a two-dimensional space of finite extent. 13. How might we tell from inside our actual three-dimensional space whether space is curved? 14. List the three possible large-scale geometries of the uni