College Physics A Strategic Approach (Volume 1) [2 ed.] 0321611144, 9780321611147

Table of contents :
Front Cover
Brief Contents
Title Page
Copyright Page
About the Authors
Preface to the Instructor
Preface to the Student
Real-World Applications
CONTENTS
VOLUME 1 (ch.1-16)
Part I Force and Motion
Overview: Why Things Change
Chapter 1: Representing Motion
1.1 Motion: A First Look
1.2 Position and Time: Putting Numbers on Nature
1.3 Velocity
1.4 A Sense of Scale: Significant Figures, Scientific Notation and Units
1.5 Vectors and Motion: A First Look
1.6 Where Do We Go From Here?
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 2: Motion in One Dimension
2.1 Describing Motion
2.2 Uniform Motion
2.3 Instantaneous Velocity
2.4 Acceleration
2.5 Motion with Constant Acceleration
2.6 Solving One-Dimensional Motion Problems
2.7 Free Fall
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 3: Vectors and Motion in Two Dimensions
3.1 Using Vectors
3.2 Using Vectors on Motion Diagrams
3.3 Coordinate Systems and Vector Components
3.4 Motion on a Ramp
3.5 Relative Motion
3.6 Motion in Two Dimensions: Projectile Motion
3.7 Projectile Motion: Solving Problems
3.8 Motion in Two Dimensions: Circular Motion
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 4: Forces and Newton's Laws of Motion
4.1 What Causes Motion?
4.2 Force
4.3 A Short Catalog of Forces
4.4 Identifying Forces
4.5 What Do Forces Do?
4.6 Newton's Second Law
4.7 Free-Body Diagrams
4.8 Newton's Third Law
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 5: Applying Newton's Laws
5.1 Equilibrium
5.2 Dynamics and Newton's Second Law
5.3 Mass and Weight
5.4 Normal Forces
5.5 Friction
5.6 Drag
5.7 Interacting Objects
5.8 Ropes and Pulleys
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 6: Circular Motion, Orbits, and Gravity
6.1 Uniform Circular Motion
6.2 Speed, Velocity. and Acceleration in Uniform Circular Motion
6.3 Dynamics of Uniform Circular Motion
6.4 Apparent Forces in Circular Motion
6.5 Circular Orbits and Weightlessness
6.6 Newton's Law of Gravity
6.7 Gravity and Orbits
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 7: Rotational Motion
7.1 The Rotation of a Rigid Body
7.2 Torque
7.3 Gravitational Torque and the Cenrer of Gravity
7.4 Rotational Dynamics and Moment of Inertia
7,5 Using Newton's Second Law for Rotation
7.6 Rolling Motion
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 8: Equilibrium and Elasticity
8.1 Torque and Static Equilibrium
8.2 Stability and Balance
8.3 Springs and Hooke's Law
8.4 Stretching and Compressing Materials
SUMMARY
QUESTIONS AND PROBLEMS
SUMMARY Force and Motion
ONE STEP BEYOND Dark Matter and the Structure of the Universe
PROBLEMS
Part II Conservation Laws
Overview: Why Some Things Stay the Same
Chapter 9: Momentum
9.1 Impulse
9.2 Momentum and the Impulse-Momentum Theorem
9.3 Solving Impulse and Momentum Problems
9.4 Conservation of Momentum
9.5 Inelastic Collisions
9.6 Momentum and Collisions in Two Dimensions
9.7 Angular Momentum
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 10: Energy and Work
10.1 The Basic Energy Model
10.2 Work
10.3 Kinetic Energy
10.4 Potential Energy
10,5 Thermal Energy
10.6 Using the Law of Conservation of Energy
10.7 Energy in Collisions
10.8 Power
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 11: Using Energy
11 .1 Transforming Energy
11.2 Energy in the Body: Energy Inputs
11.3 Energy in the Body: Energy Outputs
11 .4 Thermal Energy and Temperature
11.5 Heat and the First Law of Thermodynamics
11.6 Heat Engines
11.7 Heat Pumps
11.8 Entropy and the Second Law of Thermodynamics
11.9 Systems, Energy, and Entropy
SUMMARY
QUESTIONS AND PROBLEMS
SUMMARY Conservation Laws
ONE STEP BEYOND Order Out of Chaos
PROBLEMS
Part III Properties of Matter
Overview: Beyond the Particle Model
Chapter 12: Thermal Properties of Matter
12.1 The Atomic Model of Matter
12.2 The Atomic Model of an Ideal Gas
12.3 Ideal-Gas Processes
12.4 Thermal Expansion
12.5 Specific Heat and Heat of Transformation
12.6 Calorimetry
12.7 Thermal Properties of Gases
12.8 Heat Transfer
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 13: Fluids
13.1 Fluids and Density
13.2 Pressure
13.3 Measuring and Using Pressure
13.4 Buoyancy
13.5 Fluids in Motion
13.6 Fluid Dynamics
13.7 Viscosity and Poiseuille's Equation
SUMMARY
QUESTIONS AND PROBLEMS
SUMMARY Properties of Matter
ONE STEP BEYOND Size and Life
PROBLEMS
Part IV Oscillations and Waves
Overview: Motion That Repeats Again and Again
Chapter 14: Oscillations
14.1 Equilibrium and Oscillation
14.2 Linear Restoring Forces and Simple Harmonic Motion
14.3 Describing Simple Harmonic Motion
14.4 Energy in Simple Harmonic Motion
14.5 Pendulum Motion
14.6 Damped Oscillations
14.7 Driven Oscillations and Resonance
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 15: Traveling Waves and Sound
15.1 The Wave Model
15.2 Traveling Waves
15.3 Graphical and Mathematical Descriptions of Waves
15.4 Sound and Light Waves
15.5 Energy and Intensity
15.6 Loudness of Sound
15.7 The Doppler Effect and Shock Waves
SUMMARY
QUESTIONS AND PROBLEMS
Chapter 16: Superposition and Standing Waves
16.1 The Principle of Superposition
16.2 Standing Waves
16.3 Standing Waves on a String
16.4 Standing Sound Waves
16.5 Speech and Hearing
16.6 The Interference of Waves from Two Sources
16.7 Beats
SUMMARY
QUESTIONS AND PROBLEMS
SUMMARY Oscillations and Waves
ONE STEP BEYOND Waves in the Earth and the Ocean
PROBLEMS
Appendices
A: Mathematics Review
B: Periodic Table of Elements
C: ActivPhysics OnLine Activities
D: Atomic and Nuclear Data
Answers to Odd-Numbered Problems
Credits
INDEX
Inside Back Cover (Quick Reference Data)

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Brief Contents Force and Motion CHAPTER 1

Representing Motion

2

CHAPTER

2

Motion in One Dimension

CHAPTER

3

Vectors and Motion in Two Dimensions

30

CHAPTER

4

Forces and Newton's Laws of Motion

CHAPTER

5

Applying Newton's Laws

CHAPTER

6

Circular Motion, Orbits, and Gravity

CHAPTER

7

Rotational Motion

CHAPTER

8 Equilibrium and Elasticity

67

102

131 166

200 232

It4;i'" Conservation Laws CHAPTER CHAPTER CHAPTER

9

Momentum

260

10 Energy and Wotk 289 11 Using Energy 322

Properties of Matter CHAPTER

12 Thermal Properties of Matter

CHAPTER

13 Fluids

362

405

1!Xi;i"'. Oscillations and Waves CHAPTER CHAPTER CHAPTER

14 Oscillations 444 15 Traveling Waves and Sound 477 16 Superposition and Standing Waves

507

1!Xi;i.. Optics CHAPTER

17

CHAPTER

18 Ray Optics 574 19 Optical Instruments

CHAPTER

Wave Optics

544

609

Electricity and Magnetism CHAPTER

20 Electric Fields and Forces

CHAPTER

21

CHAPTER

22 Current and Resistance

Electric Potential

642

675 7 12

23 Circuits 739 CHAPTER 24 Magnetic Fields and Forces CHAPTER

CHAPTER

25

CHAPTER

26 AC Electricity

776

Electromagnetic Induction and Electromagnetic Waves

816

852

'itd;i"... ' Modern Physics CHAPTER CHAPTER CHAPTER CHAPTER

27 28 29 30

Relativity

886

Quantum Physics

922

Atoms and Molecules Nuclear Physics

991

954

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c COLLEGE

...... ., . RANDALL D . KNIGHT California Polytechnic State Uni ve rsity. San Luis Obispo

'I)

BRIAN JONES Colorado State Un i vers ity

STUART FIELD Col o rado State University

Addison·Wesley Boston Colum bu s Indianapo lis New Yo rk San Francisco Upper Sad dle Rive r Amste rda m Cape Town Dubai London Mad rid Mi lan Munich Paris Montreal Toronto

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Real-World Applications Applications of biological or medical interest are marked BID in the li st below. MeAT-style Passage Problems are marked BID below. Other end-of-chapter problems of biological or medical interest are marked BlO in the chapter. "Try It Yourself' experiments are marked TIY .

Chapter 1 Deprh gauges 6 BID Scales of nerve cells vs galaxies 11

Accuracy oflongjumps 12 Tty How raU are you reaUy! 12 Mars Climate Orbiter: unit error

15 Navigating geese 23

Chapter 2 BID Tree rings 34

Rocker propulsion 122- 123 A mountain railway 123

Chapter 5 TIY Physics scudents can't

jump 141 Weighrless astronauts 142 Anri -lock brakes 146 Skydiver rerminal speed 150 BID Tracrion 154-155 Sropping disrances 157

Crash cushions 42

Solar sails 45 BID Swan's rakeoff 47 BID Chameleon rongues 48 Runway design 51 Braking distance 52 TIY A reaccion rime challenge 53

BID A springbok's pronk 55 BID Cheerah vs. gazelle 57

Chapter 3 BID Fish shape for lungingvs. veering 72 Designing speed-ski slopes 80 Oprimizingjavelin throws 82 TIY A game of catch in a moving

vehicle 86 Hollywood srunrs 86-87 Physics of fielding 89 Designing roller coasters 92 BID Record-breaking frogjumps 93

Chapter 4 Voyager and Newton's first

law 103 Tty Gerring rhe kerch up our 104 Searbelrs and Newton's first

law 104 Racing bike drag 111 Tty Feel rhe difference (ine"ia) 115 Race~car driver mass 116

Chapter 6 Clockwise clocks 170 Rotation of a compact disc 172 Scottish heavy hammer throw 173 Car cornering speed 176-177 Wings on Indy racers 177 Banked racetrack turns 177 BKl Maximum walking speed 178 BID How you sense "up" 180 Fast-spinning plancts 180 BID Cenrrifuges 181 TIV Human centrifuge 182 Rotating space stations 184 Variable graviry 187 Walking on rhe moon 188 Huming wirh a sling 191-192

Chapter 7 Srarring a bike 206 Designing wheelchair hand-rims

206 Turning a capstan 208 Camera stabilizers 211 TIV Hammering home inertia 214 Golf putter moment of inertia 217

Rolling vs. sliding: ancient movers

Chapter 8 BID Muscle forces 233 BID Finding rhe body's cenrer of graviry 236 Rollover safery for cars 237 Tty Balancing soda can 238 BID Human srabiliry 239 Tty Impossible balance 239 Elasriciry of a golf ball 240 BID Spider silk 244 BID Bone strength 244-245

Chapter 9 BID Oprimizing frogjumps 263 Tty Warer balloon carch 265

BID Ram skull adaprarions 265 BID Hedgehog spines 265 BID Squid propulsion 273 Ice-skating spins 278 Hurricanes 279 Aerial 6refighring 280

Chapter 10 Flywheel energy storage on the ISS 300 Why racing bike wheels are lighr 301 BID Energy srorage in rhe Achilles rendon 304 TIV Agitating Jroms 305

BID Jumping locusrs 307 Crash helmers 312 Runaway-truck ramps 314

Chapter 11 BID Energy in rhe body: inpurs 326 BID Calorie conrenr of foods 327 BID Energy in the body: omputs 327

BID Daily energy use for mammals

221 and rep riles 329 Bullers and Newron'srhird law 122 Spinning a gyroscope 222 BID Energy and locomorion 331 Get complete eBook Order by email at [email protected]

xvi

Get complete eBook Order by email at [email protected] Real-World Applications

Optical molasses 333 Temperature in space 334 TI Y Energetic cooking 335 Refrigeracors 341 Revecsible heat pumps 342 TIY Typing Shakespeare 345 BlO En[[opy in biological syscems 347 Efficiency of an automobile 348

Chapter 12 ~O Infrared images 362,393,394

Frost on Mars 366 ~O Swim-bladder damage co caugh~ fish 368 ~O Diffusion in the lungs 370 Chinook winds 377 Thermal expansion joints 379 TIY Thermal expansion to the rescue 380

BID Survival of aquatic life in winter 381 Hurricane season 382 Tin pest 383

BID Frogs that survive freezing 383 ~O Keeping cool 384 Gasoline engines 390 Carpet vs. tile for comfort 392 ~O

Penguin feathers 393 Heat transfer on earth 394 ~O Breathing in cold air 395 Ocean temperature 403

Chapter 13 Submarine windows 409 Pressure zones on weacher maps

411 Tire gauges 411 Barometers 412 BIO Measuring blood pressure 414 BIO Blood pressure in giraffes 414 BIO Body-fat measurements 416 Floating icebergs and boats 417 Hot-ait balloons 419 TIY Pressure forces 424

BIO BID BID BID BID BID

Airplane lift 424 Prairie dog burrows 424 Measuring arterial pressure 425 Cardiovascular disease 429 Intravenous transfusions 430

Chapter 14 BID Heart rhythms 445

BID TIY

BID

BIO TIY

BID

BID BID

Metronomes 451 Bitd wing speed 452 SHM in your microwave 454 Swaying buildings 455 Measuring mass in space 457 Weighing DNA 458 Car collision rimes 460 Pendulum prospecting 460 Animallocomotion 462 How do you hold your arms! 462 Gibbon brachiation 462 Shock absorbers 464 Tidal resonance 465 Musical glasses 466 Hearing (resonance) 467 Springboard diving 467 Spider-web oscillations 476

Chapter 15 BID Echolocation 477,488, 506 BID Ftog wave-sensors 480 BIO Spider vibration sense 481 TIY Distance co a lightning strike 483

Controlling exhaust noise 526

BKl The bat detector 528 Dogs' growls 529 Harmonics and harmony 537 Tsunamis 539

Chapter 17 TIY Observing interference 551

CD colors 556

BID Iridescent feathers 558 Antireflection coatings 558

Colors of soap bubbles and oil slicks 560 TIY Observing diffraction 565 Laser range finding 566 BID The Blue Morpho 573

Chapter 18 Shadow in a solar eclipse 578 Anti~gravity

mirrors 580

Optical image stabilization 583 Binoculars 584

Snell's window 585 Optical fibers 585 BKl Arthroscopic surgery 585 BIO Mirrored eyes (gigantocyptis) 594

BKl Range of hearing 488

Supermarket mirrors 596

Sonar imaging 488 BKl Ultrasound imaging 488 BID Owl ears 490 BID Blue whale vocalization 492

Optical fiber imaging 601 Mirages 608

Hearing in mice 493

BKl Hearing (cochlea hairs) 494 Solar surface waves 495 Red shifts in astronomy 497

BID Wildlife tracking with weathet radar 497 BID Doppler ultrasound imaging 498 Earthquake waves 499

Chapter 16 BID Shock wave lithotripsy 509 T I Y Through the glass darkly 512

The Tacoma bridge standing wave

515 Suing musical instruments 515 Microwave cold spots 516 BID Resonances of the ear canal 519 Wind musical instrumencs 520

BID Speech and hearing 521 Synthesizers 521 BID Vowels and formants 522

xvii

Chapter 19 BKl The Anableps "fout-eyed" fish 609 BKl A Naurilus eye 610 Cameras 610

BID The human eye 613 TlY Inverted vision 613

BID Seeing underwater 614 BKl Neat- and fatsightedness 615 Forced perspective in movies 6 17

BKl Microscopes 618-620,627 Telescopes above the atmosphere

621 Rainbows 623 BKl Absorption of chlorophyll 624 Fixing the HST 625

BID Optical and eieerron micrographs 627

BKl Visual acuity for a kestrel 629 BIO The blind spot 631 BID Surgical vision correction 635 BID Scanning confocal microscopy 637

Chapter 20

BID Saying "ah" 522 Blood pressure and flow 437 BKl Gel electrophoresis 642,664 Get complete by email at [email protected] Aerive noise reduerion 524 Scales of living creatutes 439 eBook Order TIY Charges on tape 645

xviii

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Real~World Applications

TlY Pulling water 648 ~O Bees picking up pollen 648 ~O Hydrogen bonds in DNA 651 ~O Separating sperm cells 652

Electrostatic precipitators 660 ~

Electric field of the heart 661 Static protection 663

Lightning rods 663 ~O Electrolocation 663 Cathode-ray tubes 665 ~O Flow cytometry 674

Chapter 21 BID Electropotencials around the

~O ~O ~O ~O

brain 675 Cause oflightning 679,711 Membrane potential 680 Medical linear accelerators 683 Shark elecrroreceptors 692 The electrocardiogram 694 Random-access memory 697

Camera flashes 700 00 Defibrillators 700 Fusion in the sun 702

Chapter 22 BID Percentage body-fat measurement

712,730-731 Monito ring corrosion in power

lines 714 TIY Listen to your potencial 718

The electtic torpedo tay 719 Fuel cells 719 Lightbulb filaments 722 Testing drinking water 723 00 Impedance tomography 723 Photoresistor night lights 725 Cooking hot dogs with electricity

728 Lightbulb failure 738

Chapter 23 00 Electric fish 739,775 Christmas- tree lights 745

H eadlight wiring 747 Transducers in measuring devices

749 Flashing bike light 755 Inrermittenr windshield wipers 757 BID Electricity in the nervous

system 757-764 BID Electrical nature of nerve and

muscle cells 758

Get complete BID Interpreting brain clecrrical potentials 762

Soil moisture measuremenr 765

The ground fault interrupter 871

BKl Cardiac defibrillators 774

Halogen bulbs 879 T he greenhouse effect 881

Chapter 24 BIO Magnetic resonance

imaging 776,789 TIY Buzzing magnets 780 Hard disk data storage 781 BKl Magnetocardiograms 787 The au rora 793

BID Mass spectrometers 794 BID Electromagnetic flowm eters 795 TlY Magnets and TV screens 795 Electric motors 802 BK) Magnetotactic bacteria 804 Loudspeaker cone function 805

The velocity selector 814 Ocean potentials 815

Chapter 25 BlO Color vision in animals 816,840 BID External pacemaker programming

817 BKl Shatk navigation 819 Generators 821 TlY Dynamo flashlights 821 Credit card readers 826

Magnetic braking 828 BID Transcranial magnetic stimulation

829 Radio transmission 830

The solar furnace 832 Polarizers 833 Polarization analysis 834

00 Honeybee navigation 834 TlY Unwanced tran smissions 837

Colors of glowing objects 838 BIO Infrared sensors in snakes 839 Astronomical images 841 Tethered satellite circuits 841

Metal detectors 851

Chapter 26 Charging electric tOothbrushes 856 Transformets 855-857 Power transmission 858

Household wiring 859 BlO Electrical safety 861 The lightning crouch 862 TIV TestingGFI circuits 863 Laptop trackpads 864

Chapter 27 Global Positioning Systems 886, 902, 913-914 The Stanford Linear Accelerator

905 Hyperspace in movies 910 Nuclear fission 913

00 Pion therapy 921

Chapter 28 00 Electron microscopy 922,936 00 X-ray imaging 923 BlO X-ray diffraction of DNA 925 BlO Biological effects of UV 928 00 Frequencies for photosymhesis 929 BIO Waves, photons and vision 931 TIV

High-energy moonlight 932 Photographing photOns 933 Scanning tunneling microscopy

943 BID Magnetic resonance imaging 945

Chapter 29 00 Spectroscopy 955 Colors of nebulae 956 Sodium filters for telesco pes 977 00 Fluorescence 979 BlO LASIK surgery 982 Compact fluorescent lighting 982

Light-emitting diodes 990

Chapter 30 00 Bone scans 991, 1010 Measuring past earth temperature

993 Nuclear fusion in the sun 996

Nuclear power 997 Plutonium "batteries" 1006

00 Radiocarbon dating 1007 BIO Radioactive isotopes for medicine

1002 BID Gamma-ray medical sterilization

1008 BIO Radiation dose from environmenral, medical sources 1009

00 Nuclea r medicine 1009 00 Nuclear imaging, PET scans 1010-1012

Under-pavement car detectors 866 Cerenkov radiation 1015 Cleaning upby computer 867 eBook Order email power at [email protected] Nuclear fission 1022 BIO Nuclear magnetic resonance 870

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Detailed Contents

c Preface to the Instructor

IV

Studying fot and Taking the MCAT Exam

x

P reface to the Student

XIV

Real-World Applications

XVI

'4#·'40'"

3.1 Using Veccots 3.2 Using Veccots on Motion Diagrams 3.3 Coordinate Systems and Vector

II14;i.1 Force and Motion OVERVIEW

Components

3.4 Motion on a Ramp 3.5 Relative Motion 3.6 Motion in Two Dimensions: Projectile

1

W hy Things Change

Vectors and Motion in Two Dimensions

Motion

ft

0

3.7 Projectile Motion: Solving Problems 3.8 Motion in Two Dimensions: Circular Motion SUMMARY

. ~(

QUESTIONS AND PROBLEMS

·y:t·,;;:Ut' 14U·!;llill Representing Motion 1.1 Motion: A First Look 1.2 Position and Time: Putting Numbers on Nature

1.3 Velocity 1.4 A Sense of Scale: Significant Figures,

2 3 6 9

Scientific Notation, and Units

11

1.5 Vecmrs and Morion: A First Look 1.6 Where Do We Go From Here?

17 22 24 25

SUMMARY QUESTIONS AND PROBLEMS

·4n·';·4;iJ 2.1 2.2 2.3 2.4 2.5 2.6

Motion in One Dimension Describing Motion Uniform Motion Instantaneous Velocity Acceleration Motion with Constanc Acceleration Solving One-Dimensional Motion

Problems 2.7 Free Fall SUMMARY QUESTIONS AND PROBLEMS

30 31 36 39 42 44 48 52 58 59

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Forces and Newton's laws of Motion What Causes Motion? Force

A Short Catalog of Forces Identifying Forces What Do Fo rces Do? Newmn's Second Law Free-Body Diagrams Newton's Third Law SUMMARY QUESTIONS AND PROBLEMS

·4n·';·441 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Applying Newton's laws Equilibrium Dynamics and Newton's Second Law Mass and Weight Normal Forces Friction

Drag Interacting Objects

Ropes and Pulleys SUMMARY QUESTIONS AND PROBLEMS

67 68 71

74 79 82 84 86 89 94 95

102 103 104 107 III 113 115 118 120 124 125

131 132 135 138 142 143 148 150 153 158 159

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xx

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Conservation Laws OVERVIEW

.41t,'1·4;11

Circular Motion, Orbits, and Gravity

6.1 Uniform Circular Mo tion 6.2 Speed, Velocity. and Acceleration 6.3 6.4 6.5 6.6 6.7

in U niform Circular Motion Dynamics of Unifo rm Circular Motion Apparenr Forces in Circular Motion Circular Orbits and Weighrlessness Newton's Law of Gravity Gravity and Orbits SUMMARY QUESTIONS AND PROBLEMS

19U"114;D

Rotational Motion 7.1 The Rotation of a Rigid Body 7.2 Torque 7.3 Gravitational Torque and the Cenrer of Gravity 7.4 Rotational Dynamics and Moment of Inertia 7.5 Using Newton's Second Law for Rotation 7.6 Rolling Motion SUMMARV QUESTIONS AND PROBLEMS

19161114;·:' 8.1 8.2 8.3 8.4

Equilibrium and Elasticity Torque and Static Equilibrium

Stability and Balance Springs and Hooke's Law Stretching and Compressing Materials SUMMARY QUESTIONS AND PROBLEMS

PART I SUMMARY ONE STEP BEYOND

PART I PROBLEMS

Force and Motion Dark Matter and the Structure of the Universe

Why Some Things Stay the Same

259

166 167 171 173 179 182 185 189 193 194

200 201 204 209 213 217 220 224 225

232 233 237 239 242 247 248 254 255 256

·91t·'1·400'

Momentum 9.1 Impulse 9.2 Momenmm and the Impulse. Momentum Theorem

260 26 1 262

9.3 Solving Impulse and Momentum Problems

9.4 Conservation of Momenrum 9.5 Inelastic Collisions 9.6 Momentum and Collisions in Two Dimensions

9.7 Angular Momentum SUMMARY QUESTIONS AND PROBLEMS

191!"1141111Energy and Work 10.1 10.2 10.3 10.4 10.5 10.6

The Basic Energy Model Work Kinetic Energy Potential Energy

Thermal Enetgy

266 268 274 275 276 28 1 282

289 290 294 298 301 304

Using the Law of Conservation

of Energy 10.7 Energy in Collisions 10.8 Power SUMMARY QUESTIONS AND PROBLEMS

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306 309 312 315 316

Get complete eBook Order by email at [email protected] Deta il ed Con te nts

'A#,'I·§;II'

Co

11 .1 11.2 11 .3 11.4 11.5 11.6 11.7 11 .8 11.9

Using Energy Transforming Energy Energy in ,he Body: Energy Inpurs Energy in ,he Body: Energy Ourpurs Thermal Energy and Temperature Heat and the First Law of Thermodynamics Heat Engines Heat Pumps Enrropy and ,he Second Law of Thermodynamics Sysrems, Energy, and Enrropy SUMMARY QUE STIONS AND PROBLEMS

PART II SUMMARY ONE STEP BEYOND

Conservation Laws Order Out of C haos

PART II PROBLEMS

322 323 326 327 331

334 338 341

'AiOI·§;I@1 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Fluids

405

Fluids and Densi,y Pressure Measuring and Using Pressure Buoyancy Fluids in Motion Fluid Dynamics Viscosity and Poiseuille's Equation

406 407 411 415 419 422 427 431 432

SUMMA RY QUESTIONS AND PROBLEMS

343 346 349 350 356 357 358

xxi

PART HI SUMMARY ONE STEP BEYOND

Properties of Matter Size and Life

PART III PROBLEMS

438 439 440

Oscillations and Waves OVERVIEW

MOtion That Repeats Again and Again

443

Properties of Matter OVERVIEW

Beyond , he Particle Model

36]

'AiOI.§;1 tI

Oscillations

14.1 Equilibrium and Oscillation 14.2 Linear Restoring Forces 14.3 14.4 14.5 14.6 14.7

and Simple Harmonic Motion D escribing Simple Harmonic Motion Energy in Simple Harmonic Motion Pendulum Motion Damped Oscillations Driven Oscillations and Resonance SUMMARY QUESTIONS AND PROBLEMS

'A#·'I·§jlfA 12.1 12.2 12.3 12.4 12.5

Thermal Properties of Matter

T he Aromic Model of Marrer The Aromic Model of an Ideal Gas Ideal~ Gases Processes Thermal Expansion Specific Heat and Heat ofTransformJtion 12.6 Calorimetry 12.7 Thermal Properties of Gases 12.8 Heat Transfer

362 363 365 371 378

'A#·'I'§jl,., Traveling Waves and Sound 15.1 The Wave Model 15.2 Traveling Waves 15.3 Graphical and Mathematical D escrip tions of Waves

15.4 Sound and Ligh, Waves 381 15.5 Energy and Inrensiry 385 15.6 Loudness of Sound 387 390 15.7 The Doppler Effecr and Shock Waves SUMMARY 396 SUMMARY QUESTIONS AND PROBLEMS QUESTIONS AND PROBLEMS 397 Get complete eBook Order by email at [email protected]

444 445 447 449 455 460 463 465 469 470 477 478 479

483 487 490 492 495 500 501

xxii

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18.6 Image Formation with Spherical

Iln'!il;ll~ Superposition and Standing 507 508 509 511 516 520

Waves 16.1 The Principle of Superposition 16.2 Standing Waves 16.3 Standing Waves on a String 16.4 Standing Sound Waves 16.5 Speech and Hearing 16.6 The Interference of Waves from

SUMMARY QUESTI ONS AND PR O BLEM S

PART IV SUMMARY

Oscillations and Waves Waves in rhe Earth and the Ocean

PART IV PROBLEMS

Light is a Wave

i4#·!i·§jiQI

Optic al Instruments 19.1 The Camera 19.2 The Human Eye 19.3 The Magnifier 19.4 The Microscope 19.5 The Telescope 19.6 Color and Dispersion 19.7 Resolution of Optical Instruments

7

538

SUMMARY QUESTIONS AND PROBLEMS

539 540 PART

11,*1•• Optics OVERVIEW

SUMMARY QUESTIONS AND PROBLEMS

523 527 530 531

Two Sources

16.7 Beats

ONE STEP BEYOND

Mirro rs

18.7 The Thin-Lens Equation

v SUMMARY

ONE STEP BEVOND

Optics Scanning Confocal Microscopy

PART V PROBLEMS

593 597 602 603

609 610 613 616 618 620 622 624 630 631 636 637 638

543

lectricity and Magnetism OVERVIEW

Charges, Currents. and Fields

Electric Fields and Forces 20.1 20.2 20.3 20.4 20.5 20.6 20.7

Charges and Forces Charges, Atoms, and Molecules

Coulomb's Law The Concept of the Electric Field Applications of the Electric Field Conduccors and Electric Fields

Wave Optics 17.1 What Is Light! 17.2 The Interference of Light 17.3 The Diffraction Grating 17.4 Thin-Film Interference 17.5 Single-Slit Difftaction 17.6 Circular-Aperm re Diffraction SUMMARY QUESTIONS AND PROBLEMS

&#·'4·5;11:1 Ray Optics 18.1 18.2 18.3 18.4 18.5

544 545 548 553 556 560 564 567 568

642 643 649 651 655 658 662

Forces and Torques in Eleccric

Fields

·;U4414;IU

641

SUMMARY QUESTIONS AND PROBLEMS

Electric Potential

663 667 668

675

21 .1 Electric Potemial Energy and Electric Potemial

21 .2 Sou rces of Eleccric Potemial 21 .3 Electric Potemial and Conservation of Energy 21 .4 Calculating The Electric Pocemial 21 .5 Connecting Potemial and Field 21 .6 The Electrocardiogram 21 .7 Capacitance and Capacicors 21 .8 Dielectrics and Capacitors 21.9 Energy and Capacitors

574 The Ray Model of Light 575 Reflection 578 Refraction 581 Image Formation by Refraction 586 SUMMARY Thin Lenses: Ray Tracing 587 QUESTION S AND PROBLEMS Get complete eBook Order by email at [email protected]

676 678 680 684 691 694 695 698 699 703 704

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24.3 Electric Currents Also Create Magnecic Fields 24.4 Calculating the Magnetic Field Due to a Current

xxiii

782 785

24.5 Magnetic Fields Exert Forces on Moving Charges

789

24.6 Magnetic Fields Exert Forces on Currents

795

24.7 Magnetic Fields Exert Torques on Dipoles

799

24.8 Magnets and Magnetic Materials SUMMARY QUESTIONS AND PROBLEMS

i4:h';:'if f1 i4:£·'1·8#jA

Current and Resistance 22.1 A Model of Current 22.2 Defining and Describing Current

22.3 Batteries and emf 22.4 Connecting Potential and Current 22 .5 Ohm's Law and Resistor Circuits 22.6 Energy and Power SUMMARY QUESTIONS ANO PROBLEMS

712

713 715 717 720 724 727 732 733

Electromagnetic Ind u ction and Electromagnetic Waves

25.1 25.2 25.3 25.4 25.5

Induced Currents Motional emf Magnetic Flux Faraday's Law Induced Fields and Electromagnetic Waves 25.6 Properties of Electromagnetic Waves

23.1 23.2 23 .3 23 .4 23 .5 23.6

Circuits

739

Circuit Elements and Diagrams

740 741 743 748 750

Kitchhoffs Laws

Series and Parallel Circuits Measuring Voltage and Current More Complex Circuits CapacitOrs in Parallel and Series 23 .7 Circuits 23 .8 Electricity in the Nervous System SUMMARY QUESTIONS AND PROBLEMS

752 755 757 766 767

Electromagnetic Waves

25.8 The Electromagnetic Specrrum SUMMARY

i4#·"·8if .lld

AC Electricity

817 818 821 825 829 831 835 836 843 844 852

26.1 Alternating Current 26.2 AC Electricity and

853

Transformers 26.3 Household Electricity

855 859

26.4 Biological Effects and Electrical Safety 26.5 Capacitor Circuits 26.6 Inductors and Inductor

861 863

Circuits

26.7 Oscillation Circuits SUMMARY QUESTIONS AND PROBLEMS

i4:£·'1·8if 4 Magnetic Fields and ' Forces

816

25.7 The Photon Model of

QUESTIONS AND PROBLEMS

i4:hl·8#"

803 806 807

PART VI SUMMARY

Electricity and Magnetism

776 ONE STEP BEYOND The Greenhouse Effect and Global Warming 24.1 Magnetism 777 VI PROBLEMS 24.2 The Magnetic FieldeBook Order by email 778 at PART Get complete [email protected]

865 867 873 874 880 881 882

xxiv

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'40,'4'4i*;:1 Atoms and Molecules

Modern Physics OVERVIEW

New Ways of Looking ar rhe World

29.1 Spectroscopy 29 .2 A toms 29.3 Bohr's Model of Aromic

885

Quantization

29.4 The Bohr Hydrogen Arom 29.5 The Quantum -Mechanical Hydrogen Arom 29.6 Multielectron Atoms 29.7 Excited States and Spectra 29.8 Molecu les 29.9 Stimulated Emission and Lasers SUMM ARY QUESTIONS AND PROBLEMS

14#·"·ii;'14 27 .1 27 .2 27 .3 27 .4 27 .5 27 .6 27 .7 27.8

Relativity

886

Relariviry: Wh ats Ir All About!

887 887 891 894 897 899 904

Galilean Relativity Eins tein's Principle of Relariviry Events and Measurements

The Relariviry of Simulraneity Time Dilation Length Contraction

Velocities of Objects in Special Relarivity

27 .9 Relativistic Momentum 27 .10 Relativistic Energy SUMMARY QUESTIONS AND PROBLE MS

906 908 910 915 916

1·'t.!414;iUI 30.1 30.2 30.3 30.4 30.5 30.6

28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8

X Rays and X-Ray Diffracrion The Phococleccric Effecc Phocons Maner Waves

Energy Is Quantized Energy Levels and Quantum Jumps The Uncertainty Principle Applicarions and Implicacions

of Quanrum Theory SUMMARY QUESTIONS AND PROBLEMS

922 923 925 931 933 936 939 940 943 946 947

955 957 960 963 969 971 974 978 980 984 985

Nuclear Physics

991

Nuclear Strucrure

992 994 997 999 1003

Nuclear Srabiliry Forces and Energy in the N ucleus Radiati on and Radioac tivity

Nuclear Decay and H alf-Lives Medical Applicarions of Nuclear Physics 30.7 T he Ulrimare Building Blocks of Mactcr SUMMARY

QUESTIONS AND PROB LEM S

PART VII SUMMARY ONE STEP BEYOND

'AiBA,Sifi :1 Quantum Physics

954

Modern PhysiCS The Physics of Very Cold Aro ms

PART VII PROBLEMS

1007 1011 1016 1017 1023 1024 1025

Appendix A Machemacics Rev iew

A-1

Appendix B Periodi c Table of che Elements Appendix C AccivPhysics OnLine Accivicies

A-3

Appendix D Acomic and Nuclear D aca

A-5

Answers ro Odd-Numbered Problems Credits

A-9

Index

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A-4

C-1

I-I

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Force and Motion

The cheetah is the fastest land animal, able to run at speeds exceeding 60 miles per hour. Nonetheless, the rabbit has an advantage in this chase. It can change its motion more quickly and will likely escape. How can you tell, by looking at the picture, that the cheetah is changing its motion?

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Why Things Change Each of the seven parts of this book opens with an overvi ew that gives you a look

ahead, a glimpse of where your journey wilJ take you in the nex t few chapters. It's easy to lose sight of the big picture while you ' re busy negotiating the terrain of each chapter. In Part T, the big picture is, in a word , change. Simple observations of the world around YOLI show that most thin gs change. Some changes. slich as aging. are biological. Others, such as sugar dissolving in yo ur coffee, are chemical. We will look at changes that involve mOlion of o ne form or another- running and jumping, tllrowin g ball s, lifting weights. There are two big questions we must tackle to study how thin gs change by moving:

• How do we describe motion? How should we measure Of characterize the motion if we want to analyze it mathematically? • How do we explain motion? Why do objects have the particular motion they do? Wh y, when you toss a ball upward, does it go up and th en come back down rather than keep going up ? What are th e "laws of nature" that allow us to predi ct an object's motion ? Two key co ncepts that will help answer these questions are force (the "cause") and acceleration (the "effect"). Our basic tools will be three laws of motion elucidated by Isaac Newton. Newton's laws relate force to acceleration, and we will use them to ex plain and explore a wide range of problems. As we learn to solve problems dealin g with motion , we will learn basic tec hniques thal we can apply in all the pans of this book.

Simplifying Models Reality is extremely complicated. We would never be able to develop a science if we had to keep track of every detail of every situation. Suppose we analyze the tossing of a ball. 1s it necessary to analyze the way the atoms in the ball are connected? Do we need to analyze what you ate for breakfast and the biochemi stry of how that was translated into muscle power? These are interesting questions, of course. But if our task is to understand the motion of the balJ , we need to simplify! We can do a perfectly tine analysis if we treat the ball as a round solid and yo ur hand as another solid that exerts a force on the ball. This is a model of the situation. A model is a simplifi ed desc ription of reality- much as a mode l airplane is a simplifi ed version of a real airplane-that is used to red uce the complexity of a problem to the point where it can be analyzed and understood. Model building is a major part of the strategy that we will develop for solving problems in all parts of th e book. We will introduce different models in different part s. We will pay c lose attention to where simplifying assumptions are be ing made, and why. Learnin g how to simplify a situation is the esse nce of sllccessful modeling-and successful problem solving.

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LOOKING AHEAD .. The goa ls of Chapter 1 are to introduce the fundamental concepts of motion and to review the related basic mathematical principles.

The Chapter Preview

Describing Motion

Each c hapter will start wi th an overview of the material to comc. You shou ld read these chapter previews carefull y to gel a se nse of the conten t and structu re of the chapter.

Pictures like the one above or the o ne at right give us val uable clues about motio n.

This picture shows successive images of a frog jumping. Th e images of t he f rog are getting farther apart, so t he frog mu st be speedi ng up.

A rrows wi ll show t he connections and flow between different topics in the preview.

... -----Looo_ ...[ .... •

- ~ - , - ~ -----.-

...

~­

==~,- -

..... ~ ,-_.,--__ _•

=-...:.:t=---... .-..... :"-:,..::::''',:::..''!:..,

--...-_----.--.-. -----

You will learn to make much simpler pictures to describe the key features of moti on.

f

= 0s is

2s

3s

4s

5 s 6 s 7 58s

• • • • / • • ••• Car starts brak ing here

This diagram tel ls us everythin g we need to know about the mot ion of a ca r.

Numbers and Units

Vectors

For a full desc ription of motion, we need to ass ign num bers to physical quantities suc h as speed .

Num bers alone aren ' t enough, somet imes the directio n is important too. We ' ll use vectors to represe nt suc h quantiti es.

A chapter preview is a visua l presentat ion t hat outlines t he big ideas and t he organization of

the chapte r to come .

The chapter previews not only let you know what is comin g, but also help you make con nections with materia l you have already seen.

Looking Back ..

Get prev complete We'll tell you what material from ious chapters is especially important to review to best understand the new materia l.

This speedometer gives speed in both m iles per hour and kilometers per h our. You w ill learn how t o use and convert unit s and how to describe large and sm all numbers.

When you push a swing . the direction of the f orce makes a difference.

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You wi ll see how to do simpl e math wi th vectors.

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Motion: A First Look

1.1 Motion: A First Look The concept of motion is a th eme that will appear in one form or another throughout thi s entire boo k. You have a well-developed intuition about motion , based on your experiences, but you ' ll see that some of the most important aspects of motion can be rath er subtle. We need to develop so me tools to help us ex plain and understand motion , so rath er than jumping immediately inro a lot of mathematics and calcula-

tions, this first chapter focuses on visualizing motion and becoming farruliar with the concepts needed to desc ribe a mo ving object. One key diffe rence between phys ics and other scie nc es is how we set up and solve problems. We' ll often use a two-s tep process to solve motion proble ms. The first step is to develop a simplified represelllalioH of the motion so that key ele ments sta nd out. For exampl e, the photo of the ski er at the start of the chapter allows us to observe his position at many successive times. It is precisely by considering thi s so rt of picture of motion that we will begin our study of thi s topic. The seco nd ste p is to analyze the moti on with the language of mathe matics. The process of putting numbers on nature is often the most challenging aspect of the problems you will solve. In thi s chapter, we will ex plore the ste ps in thi s process as we introduce the basic co ncepts of motion.

Types of Motion As a startin g point , let's defin e motion as the c hange of an object's position or orientation with time . Examples of motion are easy to li st. Bicycles, baseball s, cars, airplanes, and rockets are all objects that move. The path along whi ch an object moves, which might be a straight line o r might be curved, is called the object's trajectory. FIGURE 1.1 shows four basic types of motion that we will study in thi s book. In thi s chapter, we will start with the first type of motion in the fi gure, motion along a strai g ht line. In later c hapters, we will learn about circ ular motion, whi c h is the Illation of an object alon g a c ircular path ; proj ec til e motion , th e Illation of an objec t through th e air; and rotational motion, the spinnin g of an object about an aX1S.

FIGURE 1.1 Four basic types of motion.

Projectile motionGet

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3

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CHAPTER

1

Representing Motion

1.2 Several frames from th e video of a ca r.

FIGURE

Making a Motion Diagram An easy way to study motion is to record a video of a moving object with a stati onary camera. A video camera takes images at a fi xed rate, typ ically 30 images every second. Each separate image is caJl ed aframe. As an example, FIGURE 1.2 shows severaJ fram es from a video of a car going past. Not surprising ly, the car is in a differe nt position in each frame.

III!JD .. It's important to keep the camera in ajixed position as the object moves by. Don't "pan" it to track the movin g object. ... Suppose we now edit th e video by layering the frames on top of each other and then look at th e final result. We end up with the picture in FIGURE 1.3 . This composite image, showin g an object's positions at several equally spaced installts of time, is calJed a motion diagram. As simple as motion diagrams seem, they will turn out to be powerful tools for analyzing motion. Now let's take our camera out into the world and make so me moti on diagrams. The followin g table illustrates how a motion diagram shows important features of differe nt kinds of motion. A motion diagram of th e ca r shows all the frames si multaneously.

FIGURE 1.3

The ba ll is in Ihe same posi lion in ~ ~ all four frames. ~

. ~.

~...

Examples of motion diagrams

•

/t

T ht.:: samc amount of time elapscs bclwcen each image and the nex !.

An object that occupi es only a single po:,·ition in a motion diagram is af rest .

A stationary ball on the ground. Images that are equally spaced indi cate an object movin g with cOllstant speed.

A skateboarder roUing down the sidewalk. An increasing distance between the im ages shows th at the object is speeding III'.

A sprinter starting the lOO-meter dash . A decreasing distance between the images shows lhat the object is slowing do wl!.

A car stopping for a red light. A more co mplex moti on diagram shows changes in speed and direclion.

A basketball free throw.

We have defined several concepts (at rest, co nstant speed, speedin g up, and slowing down) in terms of how the mo ving object appears in a motion di agram. These are called operational definitions. meaning that the concepts are defi ned in terms of a parti cular procedure or operation perform ed by the in ves ti gator. For exa mple, we could answer the question ]s the airp lane speeding up? by checki ng whether or not the images in the plane's motion diagram are getting farther apru1. Man y of the concepts Get completeineBook email at [email protected] ph ys ics Order will be by introduced as operational defi nitions. This reminds us that phys ics is an experimental science.

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Motion: A First Look

5

STOP TO THINK 1 1 Which car is going faster, A or B? Assume there are equal intervals of time between the frames of both videos.

Car A

Car B

ImJD .. Each chapter in this textbook has several Slap to Think questions. These question s are designed to see if you've understood the basic ideas that have just been presented. The answe rs are given at the end of the chapter, but you should make a serious effort to think about these questions before turning to the answers. If you answer correctly and are sure of your answer rather than just guessing, you can proceed to the nex t section with confidence. But if you answer incorrectly, it would be wise to reread the preceding sections carefully before proceeding onward . ...

The Particle Model For many objects, the motion of the object as [f whole is not infiuenced by the details of the object's size and shape. To describe the object's motion , all we really need to keep track of is the motion of a si ngl e point: You could imagine looking at the motion of a dot painted on the side of the object. In fact, for the purposes of analyzing the motion, we can often consider the object liS flit were just a si ngle point, without size or shape. We can also treat the object as ifaU of its mass were concentrated into thi s single point. An object that can be represented as a mass at a single point in space is called a particle. If we treat an object as a particle, we can represent the object in each frame of a motion diagram as a simple dot. FIGURE 1.4 shows how much simpler motion diagrams appear when the object is represe nted as a particle. Note that the dots have been numbered 0, I, 2,. . to tell the seque nce in which the frames were ex posed. These diagrams still convey a complete understanding of the object's motion. Treatin g an object as a particle is, of course, a simplification of reality. Such a simpLification is called a model. Models allow us to focus on the important aspects of a phenomenon by excluding those aspects that play only a minor role. The particle model of motion is a simplification in which we treat a moving object as if all of its mass were concentrated at a single point. Using the particle model may allow us to see connections that are very important but that are obscured or lost by examining aU the parts of an extended, real object. Consider the motion of the two objects shown in FIGURE 1.5. These two very different objects have exactly the same motion diagram. As we will see, all objects falling under the influence of gravity move in exactly the same manner if no other forces act. The simplification of the particle model has revealed somethjng about the physics that underljes both of these situations. Not all motions can be reduced to the motion of a single point, as we'll see. But for now, the particle mode l will be a useful tool in unde rstanding motion.

Three motion diagrams are shown. Which is a du st particle settling to the floor at constant speed, whi c h is a ball dropped from th e roof of a building, and which is a descending rocket slowing to make a soft landing on Mars?

A.

o.

,.

B.

o.

2.

,.

3.

2.

4.

C.

FIGURE 1.4 Simplifying a motion diagram

using the particle model. (a) Molion d iagram of a car stoppi ng

(b) Same motion diagram usi ng the particle model The same :HTlOun l o f time elapses between each frame and the nex t.

•

\ \.

\

Numbers show (he order i n w hich the frames were lal\cll.

4.

3. 4.

•2 •3

A single d Ol is used 10

represent the object.

falling objects.

.0

i

""""' ""~,,,,-,G~ we sec lhal a f'liling baseba ll and a diver have exactly the same Illotion diagram.

,. 2.

\ \.

FIGURE 1.5 The particle model for two

o.

3.

.'\

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CHAPTER 1

Representing Motion

1.2 Position and Time: Putting Numbers on Nature To develop our understanding of motion further, we need to be able to make quantitative measurements: We need to use numbers. As we analyze a motion diagram, it is useful to know where the object is (its positioll) and when the object was at that position (the tim e) . We' ll start by considering the motion of an object that can mo ve only along a straight line. Examples of this one-dimensional or " 1-0" motion are a bicyclist moving along the road, a train moving on a long straight track, and an elevator moving up and down a shaft.

Position and Coordinate Systems Suppose you are driving along a long, straight country road, as in FIGURE 1.6, and your friend calls and asks where you are. You might reply that you are 4 miles Origin (post office) east of the post office, and your friend would then know just where yo u were. Direction Your position Your location at a particular in stant in time (when your friend phoned) is called W-E your position. Notice that to know your position along the road, your friend needed three pieces of information. First, you had to give her a reference point 4 miles (the post office) from which all di stances are to be measured. We call this fixed reference point the origin. Second, she needed to know how far you were from that reference point or origjn~in thi s case, 4 miles, Finally, she needed to know which side of the origin you were on: You could be 4 miles to the west of it or 4 miles to the east. We will need these same three pieces of information in order to specify any object's position along a line. We first choose our origin, from which we measure the position of the object. The position of the origin is arbitrary, and we are free to place it where we li.ke. Usually, however, there are certain points (such as the well-known post office) that are more convenient choices than others. In order to specify how far our object is from the origin, we lay down an imaginary axi s along the line of the object's motion . Like a ruler, thi s axis is marked off in equal ly spaced divi sions of distance, perhaps in inches, meters, or miles, dependjng on the problem at hand. We place the zero mark of th.i s ruJer at the origin, aLlowing us to locate the position of our object by reading the ruler mark where the object is. Finally, we need to be ab le to specify which side of the origin our object is on. To do this, we imagine the axis extending from one side of the origin with increasing Sometimes measurements have a positive markings; on the other side, the axi s is marked with increasing I/.egative natural origin. This snow depth gauge numbers. By reporting the position as either a positive or a negative number, we has its origin set at road level. know on what side of the origin the object is. These elements~all origin and an axis marked in both the positive and negative di.rection s~ca n be used to unambiguously locate the position of an object. We call this a coordinate system. We will use coordinate systems throughout this book, and we will soon develop coordinate systems that can be lIsed to describe th e posiFIGURE 1.7 The coordinate system used tions of objects moving in more complex ways than just along a line. FIGURE 1.7 to describe objects along a country road. shows a coordinate system that can be used to locate various objects along the country road discussed earlier. The post offi ce defi nes the zero, or origi n. of the Although our coordinate system works well for desc ribing the position s of coordinutc !.ystcm. ) objects located along the axis, our notation is somewhat cumbersome. We need to kee p say ing things like "the car is at pos ition +4 miles." A better notation, and , ,'!?', , one that will become particularly important when we study motion in two djmen,~'" ,~, - 61 5- 4 - 3- 2- 1 0 2 3 /' 4 Smiles sio ns, is to use a sy mbol such as x or y to represent th e position along th e axis. Th is cow is at Your car is at Then we can say "the cow is at x = -5 miles." The sy mbol that represe nts a posiposi ti on - S miles. position + 4 mi les. tion along an axis is called a coordinate. The introduction of symbo ls to represen t position s (and, later, velocities and accelerations) also allows us to work with Get completethese eBook Ordermathematically. by email at [email protected] quantities FIGURE 1.6 Describing your position.

nnb

""

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Position and Time: Putting Numbers on Nature

FIGURE 1 .8 below shows how we would set up a coordinate system for a sprinter mnning a 50-meter race (we use the standard symbol " m" for meters). For hori zontal motion like this we usuaJJy use the coordinate x to represent the position. FIGURE 1.8 A coordinate system for a 50-meter race.

I' 0

10

f , 20

\. The start of the race is a natura l choice for the origin .

30

~1ilISh~f 40

FIGURE 1.9 Examples of one-dimens ional motion. For ven ica! motion. we' ]] use the coord inate y . .' .;..

Positi ve values of y ex tend y (feel) upward .

Th is is the symbol. or coordinate . used to represent pos it ions along the axis "\

,/ Stan,

7

\

50

~ .../ (In)

6

i

5 4 3

The un its in which x .J is measured go here.

Motion along a straight line need not be horizontal. As shown in FIGURE 1.9, a rock falling vertically downward and a skier skiing dow n a straight slope are al so examples of straight-Line or one-dimensional motion.

For 111otion along a strai ght slope, we'll usc the coord inate x .

o

10

30

I

/'

o

\

20

2 c

.t (111)

40

Positi ve values of x extend to the ri ght.

Time The pictures in Figure 1.9 show the position of an object at just one instant of time . But a full motion diagram represe nts how an object moves as time progresses. So far, we have labeled the dots in a motion diagram by the numbers 0, 1,2, ... to indicate the order in which the frames were ex posed. But to fully describe the motion , we need to indicate the rime, as read off a clock or a stopwatch, at which each frame of a video was made. Thi s is important, as we can see from the motion diagram of a stoppin g car in FIGURE 1.10 . If th e fram es were taken 1 second apart, thi s motion diagram shows a leisurely stop; if 1/ 10 of a second apart, it represe nts a screechin g halt. For a complete motion diagram , we thus need to label each frame with its corresponding time (sy mbol f) as read off a clock . BUI when should we start th e clock? Which frame should be labe led I = 07 This choi ce is much like choosing the origin x = 0 of a coordinate system: You can pick an y arbitrary point in the motion and label it "{ = 0 seconds." This is simply th e instant you dec ide to start your clock or stopwatch, so it is the origin of your time coordinate. A video frame labeled " I = 4 seconds" means it was taken 4 seconds after you started your clock . We typically choose I = 0 10 represent th e "beginning" of a problem, but the object may have been movin g before then. To illustrate, FIGURE 1 . 11 shows the motion diagram for a car movin g at a co nstant speed and then braking to a hall. Two possible choices for th e fram e labeled 1= 0 seconds are shown; our choice depends on what part of the motion we're interested in. Each successive position of the car is then labeled with the clock reading in seconds (abbrev ialed by the symbol "s").

FIGURE 1 .10 Is this a leisurely stop or a screech in g halt?

•o

•

•

•3 •• 45

2

I

FIGURE 1 .11 Th e motion diagram ofa car that travels at constant speed and then brakes to a halt. Ifwc' re interested in the entire ..... motion of the car. we ass ign ."" this point the time 1 = 0 s.

I ="'OS I s

2s

3s

4s

5s 6 s7s8s

• • • • / • • ••• Car starts braking here

\

I = OS I s 2s3s4s

• • • • 1• • ••• Ifwc're interested in only the braking pan of the motion . we assign I = 0 s here.

Changes in Position and Displacement Now that we've seen how to measure position and time, let's return to the problem of motion. To desc ribe motion we' Ll need to measure the changes in position that occur with time. Consider the followin g: Sam is standing 50 feet (ft) east of the corner of 12th Street and Vine. He then walks to a second point 1.50 ft east of Vine. What is Sam's change of position ?

FIGURE 1 .12 Sam undergoes a displacement Ax from position position X f .

to

J~::~:~I====;5~~0~~~1~6~0~~·~I~:~O· Thi s is Sam 's di splacement d x.

.~

>

Stan x .

shows Sam's motion on a map. We've placed a coordinate system on the map, using the coordinate x. We are free to place the origin of our coordinate system whereve r we wish, so we have placed it at the intersection. Sam's initial position is complete eBook Order by Sam email at [email protected] then at Xi = 50 ft. Get The positive value for Xi tell s us that is east of the origin.

FIGURE 1.12

Xi

I'

1

I

x

i'

End

x cft)

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CHAPTER

1

Represe nting Motion

III.!ID .... We will labe l special va lues of x or y with subscript s. The value at the start of a proble m is usuall y labeled with a subscript Hi," for initial, and the value at the end is labeled with a sub script "f, " for final. For cases hav ing several special values, we wiJlusuall y use subscripts" I," "2," and so on. .... Sam's final position is Xf = 150 ft, indi cating that he is 150 ft eas t of th e ori gin . You can see th at Sam has changed positi on, a nd a challge of positio n is called a displacement. His di spl aceme nt is the di stance labe led d x in Figure 1.12. The G reek letter delta (d ) is used in math and scie nce to indicate the change in a quan tity. Thus d x indicates a chan ge in the position x.

ImlD .... .6. x is a sillgle sy mbol. You cann ot cancel out or re move the .6. in algebraic operations . .... Th e size and the direction of th e displacement both matter. Roy Riegels (pu rsued above by team mate Benn y Lam) found thi s out in dramatic fashion in the 1928 Rose Bowl when he recove red a fumble and ran 69 yardstowa rd his own tea m 's end zo ne. An impress ive distance, but in the w ro ng di rectio n! FIGURE 1.13 A displacement is a signed

quantity. Here ll X is a negative number. A fi nal position to the left of the initial position gives a negative di splaccme nt.

lJtl

1

6

To get from the 50 ft mark to the J 50 ft mark, Sam clea rl y had to walk JOO ft, so the change in hi s position- hi s di splace ment- is 100 ft. We ca n think abo ut di splaceme nt in a more general way, however. Displacement is the difference between a fin a] position Xf and an initial position Xi' Thus we can write ~ x = x r - X; =

Displacement is a siglled quaHliry; that is, it can be either positi ve or negative. If, as shown in FIGURE 1.13 , Sam's final position Xf had been at the ori gi n instead of the 150 ft ma rk , his di splacement would have been ~x = xl' - X ; =

50 ft = - 50 ft

Change in Time

12th Street

,

,

, x (fl)

100

50

150

bicycle mov ing to th e right at a co nstant spee d.

Os

I ,

2,

3s

4s

5s

A displac eme nt is a chan ge in positio n. In order to quantify motio n, we' ll need to also co nsider changes in time, which we call time intervals. We've see n how we can label eac h frame of a motion diagram with a specific time, as determined by our stopwatch. FIGURE 1 .14 shows th e moti on diagram of a bicycle moving at a constant speed, with the times of the measured poi nts indicated. The di splacement between the initial position Xi and the final position Xf is ~x

I

I

0

20

40

60

80

100

120

= Xr - x ; = 120 ft -

~I

x (ft )

/

Final pos ition x f

a ft =

J 20 ft

Simjlarl y, we defi ne the tim e interval between th ese two points to be

6s

•, •, •, •, •, • •

EXAMPLE 1 1

a ft -

The negative sign tells us that he moved to the left along the x-ax is, or 50 ft lVesl.

1

In itial posi tion Xi

lOa ft

ImID .... A ge neral principle, used throughout thi s book, is that the chan ge in any quantity is the fjnal value of the quantity minus its injtial vaJue . ....

FIGURE 1. 14 Th e motion diagram of a

\

J50 ft - 50 ft =

as = 6 s

6s-

= Ir - I; =

t.t measures tbe elapsed time as an object moves from an initial position X i at time ti to a final position X f at time t f o No te that, unlike .6.x, .6.1 is always positi ve because If is always greater than Ii' A time interval

How long a ride?

Carol is enjoy in g a bicycle ride on a country road that run s eas(-

west past a water tower. Define a coord inate system so that inc reasing x means mov in g east. At noon, Caro l is 3 mile s (m i) east of the waler tower. A half-hour later, she is 2 mi west of the water tower. What is her displacement during that half-hour? PREPARE Allhough it may seem like overki ll for s lic h a s imple problem, you should start by making a drawing, like the one in FIGURE 1.15, with the x-axi s along the road. Di stances are measured with respect to the water Get tower,complete so it is a natural origin for the eBook Order by

FIGURE 1.15 A drawing of Carol's motion.

End

x

,

•• , -2

, - I

!

Stan X,

dx

,

,

•,

0

2

3

x(mi)

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coordinate system. Once the coordinate system is established , we can show Carol's initial and final positions and her displacement between the two. SOLVE We've spec ified values for Carol's initiaJ and final positions in our drawing. We can thus compute her displacement: 6.x

=Xr-Xj

= (-2 mi) -

1.3 Velocity

9

ASSESS Once we've co mplete d the so lution to the probl e m, we need to go back to see if it makes sense. Carol is movin g to the west, so we ex pect he r displacement to be negative-and it is. We can see from Ollr drawing in Figure 1.15 that she has moved 5 miles from her starting position, so our answer seems reasonable.

(3 mi ) = - 5 mi

ImD] .... All of the numerical examples in the book are worked out with the same three-ste p process: Prepare, Solve, Assess. It's tempting to cut corners, especiaLly for the simple problems in these early chapters, but you should take the time to do all of these ste ps now, to practice your problem-solving tec hnique. We' ll have more to say about our general problem-solving strategy in Chapter 2 . .... STOP TO THINK 1 3 Sarah starts at a positive position along the x-ax is. She then undergoes a negative di splacement. Her final position

A. Is positive.

B. Is negative.

C. Could be either positive or negative.

1.3 Velocity We aU have an intuitive sense of whether something is moving very fast or just cruising slowly along. To make this intuitive idea more precise, let's start by examining the motion diagrams of so me objects moving along a st raight line at a constant speed, objects that are neither speeding up nor slowing down. This motion at a co nstant speed is called uniform motion. As we saw for the skateboarder in Section 1.1 , for an object in uniform motion, successive frames of the motion diagram are equally spaced. We know now that this means that the object's displacement d x is the same between successive frames. To see how an object's displacement between successive frames is related to its speed, consider the motion diagrams of a bicycle and a car, traveling along the same street as shown in FIGURE 1.16. Clearly the car is moving faster than the bicycle: In any I-seco nd time inte rval , the car undergoes a displacement dx = 40 ft, while the bicycle's displacement is ollly 20 ft. The di sta nces trave led in ] second by the bicycle and the car are a measure of their speeds. The greater the distance traveled by an object in a given time interval , the greater its speed. Thi s idea leads us to define the speed of an object as distance trave led in a given time interval speed = - - - - - - - - - - " - - - - - time interval

~. FIGURE 1.16 Motion diagrams for a car

and a bicycle. During each second, (he car moves tw ice as far as the bicycle. Hence (he

0,

\\

I , .

\

Os

l s\

2s

3s

•

.....

•

(1.1)

For the bicycle, this equation gives

20 ft ft Iss

= - - = 20-

while for the car we have 40 ft ft 1s s The speed of the car is twice that of the bicycle, which seems reasonable. speed

h

•

Bicycle

3, •

Car

.;.,---;,;.--.:---;-,---;,---"CC{-,=,,-~,- x (ft) o 20 40 60 80 100 f20

Speed of a mov ing obj ect

speed

..

car is moving at a greater speed.

= - - = 40-

ImD] .... The division gives units that are a fraction: ft/s. This is read as "feet per more familiar "miles per hour." .... second," just Like Getthecomplete eBook Order by email at [email protected]

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10

CHAPTER 1

Representing Motion

FIGURE 1.17 Two bi cyc le s travel ing at th e sa me speed, but w ith d iffe re nt ve lociti es.

Bike 1 is moving to the righl.

--

Bi ke 2 is moving the left .

to

l 1 2•, • • - 16, 5 , 2, .i-

Os

3,

I ,

4,

4,

3,

•, •, •, •, •, 0

20

40

60

80

5s

6s

Is

0s

,

,

100

120

--

Bike I Bike 2 x ( fI)

To full y charac terize the moti on of an object, it is importa nt to specify not o n.l y the object's speed but also the directioll in which it is moving. For exampl e, FIGURE 1.17 shows the moti o n di agrams of two bicycles trave li ng at th e sam e speed of 20 ft/s. The two bicycles have the same speed, but somethin g about their motion is differentthe direction of their motio n. The problem is th at the " di stance trave led" in Eq uati on L I does n' t capture a ny informatio n about the di rec ti on of trave l. But we've seen that the disp/acemellf of an obj ect does co nta in this inform ati on. We can the n introduce a new qu a ntity, the velocity, as

.6. x .6.1

displacement velocity = - ' - - - - - time interval

@

( 1.2)

Ve loc ity o r a mov ing obj ec t

The velocity of bicycle I in Figure 1.17, computed usi.ng the I seco nd tim e interva l between the 1 = 2 sand t = 3 s positions, is

.6. x X3 - X 2 = M 3s-25

v= -

60 ft - 40 ft I

5

ft + 20s

whil e the ve locity fo r bicycle 2, durin g th e same tim e inte rval, is Ll x

v= -

Ll t

=

X, -x,

3s- 2s

60 ft - 80 ft is

ft - 20s

lmID ... We have used X 2 fo r the position at time t = 2 seco nds and X3 fo r the positio n at time t = 3 seco nd s. The subscripts serve the same ro le as beforeidenti fy ing parti cul ar positions-but in thi s case the positions are iden ti fied by the time at which each position is reached . ..... The two ve loci ti es have opposite signs because the bicycles are trave li ng in opposite directi ons. Speed measures only how fast a n object moves, but velocity tells us both an object's speed and its direction. A positi ve ve locity indicates moti on to th e right or, for vertical motion, up ward . Similarl y, an obj ect movi ng to the left , or dow n, has a negati ve velocity.

lmID ... Learn.ing to d istin gui sh between speed, whi ch is always a positi ve number, and velocity, which can be e ither positive or negati ve, is one of the most importan t tasks in the anal ys is of moti on. ..... The ve locity as defined by Equati on 1.2 is actu all y what is called the average ve locity. On average, over each I. s interval bicycle I. moves 20 ft, but we don' t kn ow if it was moving at exactl y the sa me speed at every mome nt during this time interval. In Chapter 2, we' ll develop th e idea of instantalleolls ve locity, the velocity of a n obj ect at a parti cul ar instan t in time. S ince our goal in thi s chapter is to visualize motion with moti on diag rams, we ' U somew hat blur the di stincti on between average and instantan eous quantities, refining these defi niti ons in C hapter 2, whe re our goal will be to deve lop th e mathemati cs of motjon .

EXAMPLE 1 2

Finding the speed of a seabird

Albatrosses are seab irds that spend most of their li ves fl ying over the ocean looking for food. Wi th a stiff tail wi nd . an albatross can fly at hi g h speeds. Satell ite data o n o ne pani cula rl y speedy a lbatross showed it 60 mil es eas t of its roost at 3:00 PM a nd th en, at 3:20 PM, 86 m iles east of its roost. What was its veloc ity? PREPARE The state men t or th e proble m prov ides us with a nat ural coord in ate system: We can measure di stan ces with respect to th e roost, with d istances to the east as

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A Sense of Scale: Significant Figures, Scientific Notation, and Units

11

positive. With this coordinate system, the motion of the albatross appears as in FIGURE 1.18. The motion takes place between 3:00 and 3:20, a time interva l of 20 minutes, or 0.33 hour. FIGURE 1.18 The motion of an albatross at sea.

.

Stan

Roost

xj

...............

......... ax

End ~

-' f

-"--T'--T'- - r ,--,,--x (mi)

o

25

50

75

100

know the initial an d final positio ns, and we know the time interval, so we can calculate the velocity:

SOLVE W e

6.x

Xf -

Xi

26 mi

v = t;; = 0.33 h = 0.33 h = 79 mph ASSESS The velocity is positive, which makes sense becau se Figure 1.1 8 shows that th e motion is to the right. A speed of 79 mph is certainly fast, but the problem said it was a " particul arly speedy" albatross, so our answer seems reasonable. (Indeed, albatrosses have been observed to fly al such speeds in the very fast winds of the Southern Ocean. Thi s problem is based on real observations, as will be our general practice in this book.)

The "Per" in Meters Per Second The units for speed and velocity are a un.it of di stance (fee t, meters, miles) divided by a unit of tim e (seconds, hours). Thus we could measure velocity in units of m/s or mph , pronounced " meters per second" and " mjles per hour." The word " per" wiiJ often arise in physics when we consider the ratio of two quantities . What do we mean, exactly, by "per"? If a car moves with a speed of 23 mIs , we mean that it travels 23 meters for each I seco nd of e lapsed time. The word " per" thu s associates the number of units in th e num erator (23 m) with one unit of th e de nominator (I s) . We' ll see man y other examp les of thi s idea as the book progresses. You may already know a bit about dellsity; you can look up the density of gold and you'll find that it is 19.3 glcm' ("g rams per cub ic cent ime ter"). Thi s mean s that the re are 19. 3 grams of gold/or each 1 cubic centimeter of the metal. Thinking about the word "per" in thi s way will help you better understand physical quantities whose units are the ratio of two other units.

1.4 A Sense of Scale: Significant Figures, Scientific Notation, and Units Physics attempts to explain the natural world, from the very small to the exceedingly large. And in order to understand our world, we need to be able to measure quantities both minuscule and enormous. A properly reported measurement has three elements. First, we can measure our quantity with only a cerlain precision. To make this preci120 um = 1.2 x 10- 4 III sion clear, we need to make sure that we report our measureme nt with th e correct From galaxies to cells ... BID In science, number of signijicallt figures. we need to express numbers both very large Second, writing down the really big and small numbers that often come up in and very small . The top image is a computer physics can be awkward. To avoid writing all those zeros, scie ntists use scieHt~fic simulation of the structure of the universe. Bright areas represent regions of clustered Hotation to express numbers both big and small. Finally, we need to choose an agreed-upon set of ttl/irs for the quantity. For speed, galax ies. The boltom image is cortical nerve cells. Nerve cells re lay signals to each other common units include meters per second and miles per hour. For mass, the kilogram through a complex web of dendrites. These is the most commonly used unit. Every physical quantity that we can measure has an images, though si milar in appearance. differ Get complete eBook Order by email at [email protected] associated set of units. in scale by a factor of about 2 X I 028 !

12

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Representing Motion

FIGURE 1.19 The precis ion of a meas urem ent depends on th e in strum en t used to make it. T hese cal ipers have a precision o r O.OI mill .

Walter Davis's best long jump on this day was reported as 8.24 m . Thi s imp lies that the actual length of the jump was betwee n 8.235 m and 8.245 m , a spread of only 0.01 m, w hi ch is 1 c m. Does thi s clai m ed accuracy seem rea so nable?

TRY IT YOURSELF

Measurements and Significant Figures Whe n we measure any quantit y, such as th e le ngth of a bo ne o r th e weight of a spec ime n, we can do so with onl y a certain precision. The di gital calipe rs in FIGURE 1.19 can make a meas ure ment to within ± 0.01 mm, so they have a precisio n of 0.0 1 111111. If you made the meas ure me nt with a rul er, you probabl y co uldn ' t do bette r than about ± I mm, so th e prec ision of the rul e r is about I mm . The precision of a meas ureme nt can also be affected by the skill or judgmen t of the perso n perfo rmin g the meas ure me nt. A stopwatch mi ght ha ve a precision of 0.001 s, but, due to yo ur reaction tim e, your measure me nt o f th e tim e of a sprinte r would be muc h less precise. It is important that yo ur measure me nt be reported in a way that reflec ts its actual precisio n. Suppose yo u use a rul er to meas ure the le ngth of a parti cular spec ime n of a newl y di scove red spec ies of frog. You judge that you can mak e thi s measure ment with a precisio n of about I mm, or 0.1 c m. In thi s case, th e frog's length should be reported as , say, 6.2 cm. We inte rpret this to mean that the actual value fall s betwee n 6.15 e m a nd 6.25 e m and thus rounds to 6.2 em. Reportin g the frog's le ngth as simpl y 6 c m is saying less than you know ; you are withholding information. On th e other hand , to re po rt th e numbe r as 6.213 c m is wrong. Any person revie win g your work wou ld interpret th e number 6.2 13 cm as meaning that the actual le ngth faUs between 6.2125 em and 6.2135 em, thus rounding to 6.21 3 em. In thi s case, yo u are claiming to have knowledge and information that you do not reall y possess . The way to state your knowledge precisely is through the prope r use of significant figures . You can think of a significant figure as a di g it that is re liabl y known. A measure ment such as 6.2 cm has fwo signjfi ca nt fi gures, the 6 and the 2. The nex t decimal place-th e hundredths-is not re liabl y known and is thu s not a significant figure. Similarly, a time measure ment of 34.62 s has four significant fi gure s, impl ying that the 2 in the hundredth s place is reliabl y known . When we pe rform a calculation sLlch as adding or muJtipl y ing two or more meas ured numbe rs, we can' t c laim more accuracy for th e res ult than was prese nt in the in.itial measurem en ts. N ine out of ten numbers used in a calcul ation might be kno wn with a precision of 0.0 I %, but if the tenth number is poorly known , with a prec ision of on ly 10%, the n the result of the calculation ca nn ot possibl y be more precise than 10%. Determining the proper number o f significant fi gures is strai ghtforward, but th ere are a few definite rul es to follow. We will often spe ll out suc h tec hni cal details in what we call a "Tacti cs Box." A Tacti cs Box is designed to teac h you particu lar skills and technjqu es . Each Tactic s Box will use th e il icon to designate exercises in the Student Workbook that you can use to practice these skill s.

TACTICS BOX 1. 1

How tall are you really? U you measure your height in the moming,just after you wake up, and then in the evening, after a full day of activi ty, you' ll find that your even ing height is shorfer by as much as 3/4 inch. Your height decreases over the course of the day as gravity compresses and reshapes your spine. lf you give your he ight as 66 3/ 16 in, you are claiming more sig nificant fi gures than are truly warranted; the 3/ 16 in isn' t really reliably known because your height can vary by more than this. Expressi ng your he ight to the nearest inch is plenty!

o

Using significant figures

When you multiply or divide several numbers, or when you take roots, the number of significant figures in the answer should match the number of significant figures of th e least precisely known numb er used in th e calculation: Three signifi cant fi gures

"-

3.73 X 5.7 = 21 /

Two significan t figures

1'".

A n ~wc r should

.......... the

(W O,

have Ihe lower of or two significalH fi gure~ .

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Contilllled

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f}

A Sen se of Scale: Significant Figure s, Sci entific Notation , and Units

When you add or subtract severaJ numbers, the number of decimal places in the answer should match the smallest number of decimal places of any number used in the calculation: 18.54 -Two decimal places

+ I06.6-0ne decimal place 125.1 Co"

€)

.......... Answer should have the (ower of thc two. or one deci ma l place.

Exact numbers have no unce rtainty and, when used in calculations, do not change the number of significant figures of meas ured numbers. Examples of exact numbers are 7T and the number 2 in the relation d = 2r between a circle's diameter and radiu s.

There is one notable exception to these rules: • It is acceptable to keep one or two ex tra digit s during illfermediate steps of a calculation. The goal here is to minimize round-off errors in the calculation. But the/inal answer must be reported with the proper number of signifi cant figures. Exercise 15

EXAMPLE

Measuring the velocity of a car

13

To measure the ve locity of a car, cloc ks A and B are set up at two po ints along the road, as show n in FIGURE 1.20 . Clock A is precise to 0.0 1 s, whil e B is precise to on ly 0. 1 s. The di stan ce between these two clocks is carefu lly measured to be .1 24.5 m. The two clocks are automaticall y started when the car passes a trigger in the road; eac h clock SLOpS automaticall y when the car passes that cl oc k. Aft er the car has passed bot h cloc ks, clock A is fo und to read fA = 1.22 s, and clock B to read Is = 4 .5 s. The time fro m the lessprec ise c loc k B is correctly reported wiLh fewer sign ifican t fi gures th an that from A. What is the ve locity of the car, and how should it be reported with the cOlTect number of signifi cant fi gures? FIGURE 1 .20

II

A

~

B

Q

't5

'\

Both clocks Sian whcn the

'fh il> number ha!> one decim:.ll pl ace. tl l ~ IB - IA ~

T hi s num ber has two decima l places.

\

/

(4.5 s) - ( 1.22 s)

~

3.3 s

;' By rule 2 of Tactics Box 1.1 . the result should have one deci mal place.

We can now calculate the veloc ity with the di splacement an d the time in terva l: The displaceme nt has fo ur significant fig ures .~

Meas uri ng th e ve locity of a ca r.

(l1

SOLVE The time interval is:

v

~

-tl.x

~

tl l

The time interval has / two significant fig ures.

124.5 Tn 3.3 s

~

38 Tn/s

;'

By rule I of Tactics Box 1 1. the result should have fl\ '() significa nt fi gures .

6.x - 124 .5 m

car cro~ses thi s trigger. PREPARE To calculate the veloc ity, we need the d isplace me nt 6. x and the time in te rval 6. 1 as the car moves between the two cloc ks. The di splacement is given as 6.x = 124.5 m; we can calculate the time in terval as the di ffe rence between the two m e a ~ sured times.

ASSESS O ur fi nal value has two signifi cant fi gures. S uppose you had bee n hired to measure the speed of a car thi s way, and you reported 37.72 m/s. It would be reasonable for someo ne lookin g at your result to assume that the meas urements you used to arri ve at thi s value were correct to four sign ificant fi gures and th us that you had measured time to the nearest 0.001 second. Our calTect result of 38 tnls has all of the accuracy that you can claim, but no more!

Scientific Notation It's easy to write down measure ment s o f ordin ary-sized objects: Your height mi ght be 1.72 meters, the we ight of an apple 0 .34 pound. But the radiu s of a hydroge n atom is 0.000000000053 m, and the di stan ce to the moon is 384000000 m. Keeping track of all those zeros is quite cumbersome.

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13

14

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1

Representing Motion

Beyond requiring yo u to deal with all th e zeros, writing quantities thi s way makes it un c lear how many significant figures are involved. In the di sta nce to the moon g ive n above, how man y of those digits are sign ifi ca nt ? Three? Four? All nine? Writing numbers using scientific notatio n avoids both these probl e ms. A value in scie ntifi c notation is a number with one di gi t to the left of the decimal point and zero or mo re to th e right of it, multiplied by a power of te n. This solves the proble m of all the zeros and makes the number of significant figures immed iately apparent. In scientific notation , writing the distance to the sun as 1.50 X 1011 m implies that three digits are signifi ca nt ; writing it as 1.5 X 1011 m implies Ihat only two digits are. Even for smaller va lues, sc ientific notation can clarify the number of significant figures. Suppose a di stanc e is re ported as 1200 m. Ho w many s ignifica nt figures does thi s meas urement have? It 's ambiguous , but us ing scie ntific notation can remo ve any ambiguity. If this di stan ce is known to within I m, we can write it as 1.200 X 10 3 m, showing that all four digits are s ignificant ; if it is accurate to only 100 m or so, we can report it as 1.2 X 103 m, indicating two s ignifi cant fi gu res. Tactics Box 1.2 shows how to co nvert a number to scie ntific notation, and how to co rrectly indicate the number of significant figures.

TACTICS BOX 1.2

Using scientific n otati on

To convert a number into scie ntific notation:

o

For a number greater than 10, move the decimal point to the left until only one digit remain s to the left of the decimal point. The remaining number is the n multiplied by 10 to a power; thi s power is given by the numbe r of spaces the decimal point was moved. Here we convert the diameter of the earth to scientific notation: We move the decimal poi nt unti l there is only one digit to its lef~ , counti ng the number of steps .

Since we moved the decimal point 6 steps, the power of ten is 6.

\·. ~ ilZe8ee In = ~ X 10";;;,-TIle number of d igits here equals the number of significant figu res.

6 For a number less than I, move the decimal point to the right until it passes the first di git that isn't a zero. The remainin g number is the n multip'-ied by IO to a negative power; the power is given by the number of spaces the decimal point was moved. For the diameter of a red blood cell we have: We move the decimal poi nt until it passes the first digi t lhat i!> not a zero. counting the number of steps.

'\.. ~ gg~g~~ 5

In

=

7J

Since we moved the deci nKII poi nt 6 steps. the power of ten is -6.

X

w, l;(

"- - The nl1mber of d igits here equal s the number of significant figures. Exercise 16

II

Prope r use of significant figures is part o f the "culture" of science . We will frequently emphasize these "cultural iss ues" because you must learn to speak the same language as th e natives if you wish to communicate effecti ve ly ! Most students know the rules of significant figures, ha ving learned them in hi gh school, but many fail to

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15

A Sense of Scale: Significant Figures, Scientific Notation, and Units

apply them. It is important that you understand the reaso ns for significant figures and that you get in the habit of using them properly.

Units As we have see n, in orderto measure a quantity we need to give it a numerical value. But a measure ment is more than just a number-it requires a lIlIil to be given. You canlt go to the deli and ask for " three quarters of cheese." You need to use a unithere, one of weight, such as pounds-in addition to the number. In your daily life, you probably use the English system of units, in which di stances are measured in inches, feet, and miles. These units are we ll adapted for daily life. but they are rarely used in scientific work. Given that science is an international discipLine, it is also important to have a sys tem of units that is recognized around the wor ld . For these reasons, scientists use a system of units called Ie Systeme Illternarial/ale d'Ullites. commonly referred to as 51 units. S1 units were originally developed by the French in the late 1700s as a way of standardizing and regularizing numbers for commerce and science. We often refer to these as metric ullits because the meter is the basic standard of length. The three basic SI quantities, shown in Table 1.1, are time, length (or distance), and mass. Other quantities needed to understand motion can be expressed as combinations of these basic units. For example, speed and velocity are expressed in meters per seco nd or m/s. This combination is a ratio of the le ngth unit (the meter) to the time unit (the second).

The importance of units In 1999, (he $ 125 million Mars Climate Orbiter burned up in

rhe Martian atmosphere instead of entering a safe orbit from which it could perfo rm observations. The problem was faulty units ! An engineering tcam had provided critical

data on spacec raft performance in English units, but the navigation learn assumed these data were in metric units. As a consequence, the navig'lti on team had the spacecrafl fly too close to the planet. and it burned up in the

atmosphere.

Using Prefixes We will have many occasions to use lengths, times, and masses that are either much less or much greater than the standards of I meter, I seco nd, and I kilogram. We will do so by using prefixes to denote various powers of ten. For instance, the prefix "kilo" (abbreviation k) denotes /03, or a factor of 1000. Thus I km equals 1000 OJ , I MW equals 10 6 wallS, and I 11- V equals 10-6 V. Table 1.2 lists the common prefixes that wilJ be used frequently throughout this book. A more extensive li st of prefixes is shown inside the cover of the book. Allhough prefixes make it easie r to talk about quantities, the proper SI units are meters, seco nds, and kilograms. Quantities given with prefixed units must be converted to base SI units before any calculations are done. Thus 23.0 em must be converted to 0.230 m before starting calcu lations. The exception is lhe kilogram, which is already the base SJ unit.

Unit Conversions Although SI units are our standard, we cannot entirely forget that th e United Slates still uses Engl ish units. Even after repeated exposure to metric units in classes, most of us "think" in English units. Thus it remains important to be able to convert back and forth betwee n SI units and English units. Table 1. 3 shows some frequently used conversions that will come in handy. One effective method of performing un.it conversions begins by noticin g that since, for example, I mi = 1.609 km , the ratio of th ese two di stances-illcludillg their units-is equal to I , so that I mi

1.609 km

1.609 km I mi

~~"' =

1

A ratio of values equal to I is called a conversion factor. The following Tactics Box shows how to make a unit conversion .

TABLE 1.1 Common $1 units

Quantity

Unit

Abbreviation

lime

second

s

length

meter

m

mass

ki logram

kg

TABLE 1.2 Common

prefixes

Prefix

Abbreviation

mega-

M

kilo-

k

centi-

c

10- 2

milli -

m

mi cro-

Iin

10- 3 10- 6 10-'

nano-

Power of 10

106 10'

TABLE 1.3 Useful unit conversions

I inch (in)

= 2.54 e m

I foot (ft) = 0.305

In

= 1.609 kill I mile per hour (mph) = 0.447 m/s

I mile (mi)

I In

=

39.37 in

I km = 0.62 1 mi

I Ill/s = 2.24 mph

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16

Get complete eBook Order by email at [email protected] CHAPTER 1

TACTICS BOX 1 . 3

Representing Motion

Making a unit conversion

o Stan wi th the

6

quantifY you wish (0 convert. \

"-....

9

0

Mult iply by the appropriate conversion fac(Or. Because Ihis conversion factor is equal fO I, mult ip lying by it does not change the val ue of the quantity-only iL