Mechanics: educational manual

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G. Sh. Yar-Mukhamedova K. Sh. Shunkeyev N. N. Zhanturina

MECHANICS Educational manual

Almaty «Qazaq University» 2018

UDC 531 (075) LBC 22.2 я 73 Y 26 Recommended for publication by the decision of the Academic Council of the Faculty of Physics and Technology, Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol №2 dated 03.11.2017) Reviewers: Doctor of Physical and Mathematical Sciences, professor T.A. Kozhamkulov Doctor of Technical Sciences, professor S.M. Kozhakhmetov Doctor of Physical and Mathematical Sciences, professor B.N. Mukashev

Y 26

Yar-Mukhamedova G.Sh. Mechanics: educational manual / G.Sh.Yar-Mukhamedova, K.Sh. Shunkeyev, N.N. Zhanturina. – Almaty: Qazaq University, 2018. – 132 p. ISBN 978-601-04-3049-5 The content of the workbook corresponds to the curriculum (syllabus) and the teaching and methodological complex of the discipline «Mechanics», which is part of the compulsory component of the preparation of bachelors of the Physics and Technics Faculty. There are main laws of mechanics in tutorial. It consist 7 parts, at the end of them are control questions, problems. The parts are kinematics, Dynamics of material point and translational motion of body, work and energy, Mechanics of solid state, Gravity. Elements of the field theory, Elements of Fluid Mechanics, elements of special relativity theory. The tutorial can be used by students of physical specialties. The workbook is intended for students by specialties 5В060400 «Physics», 5В060500 «Nuclear Physics», 5B071000 «Material Science and Technology of New Materials», etc. Published in authorial release.

UDC 531 (075) LBC 22.2 я 73 ISBN 978-601-04-3049-5

 Yar-Mukhamedova G.Sh., Shunkeyev K.Sh., Zhanturina N.N., 2018 © Al-Farabi KazNU, 2018



INTRODUCTION ................................................................................... 5 Part 1. KINEMATICS ............................................................................. 7 1. Models in mechanics. Reference frames. Trajectory, the path length of the displacement vector .............................. 7 2. Velocity ................................................................................................. 10 3. Acceleration and its components ........................................................... 13 4. Angular velocity and angular acceleration ............................................. 16 Control questions by the theme and problems ........................................... 19 Part 2. DYNAMICS OF MATERIAL POINT AND TRANSLATIONAL MOTION OF BODY .................................. 21 5. First newtons law. Mass. Forse……………………………................... 21 6. Second newtons law………………………………………………........ 22 7. Third newtons law…………………………………………….. ............ 25 8. Friction…………………………………………………………. ........... 25 9. Conservation of momentum. Center of mass……………….. ............... 28 10. Equation of motion of variable mass body…………………….. ......... 31 Control questions by the theme and problems ........................................... 32 Part 3. WORK AND ENERGY…………………………………….. ..... 34 11. Energy, work, power………………………………………................. 34 12. Kinetic and potential energy…………………………… ..................... 36 13. The energy conversation law………………………………… ............ 40 14. Graphical representation of the energy………………………. ............ 43 15. The collision of absolutely elastic and nonelastic bodies… ................. 47 Control questions by the theme and problems………… ........................... 52 Part 4. MECHANICS OF SOLID STATE……………………… ......... 54 16. Moment of inertia…………………………………… ......................... 54 17. Kinetic energy of rotation……………………………………....... ...... 56 18. Moment of force. The equation of dynamics of rotational motion of solid body……………………………………….. ..................... 58 19. Momentum and the law of its conservation……………….................. 61 20. Free axys. Gyroscope………………………………………… ............ 64 21. Deformations of solid body……………………………… .................. 68 Control questions by the theme and problems……….……… ................... 72


Part 5. GRAVITY. ELEMENTS OF THE FIELD THEORY ............. 74 22. Kepler's laws. Law of gravity………………………………….. ......... 74 23. The force of gravity and weight. Weightlessness……………........ ..... 76 24. Field of gravity and its strength…………………………………… .... 78 25. Work in the field of gravity. Potential of gravity field ......................... 79 26. Escape velocities……………………………………………….. ......... 82 27. Non-inertial frame of reference. Forces of inertia…………… ............ 83 Control questions by the theme and problems………………. ................... 90 Part 6. ELEMENTS OF FLUID MECHANICS ................................... 92 28. Pressure in gas and liquids…………………………………… ............ 92 29. The continuity equation……………………………… ........................ 94 30. Bernoulli equation and its consequences………………………. ......... 95 31. Viscosity (internal friction). Laminar and turbulent flow of liquids .................................................................................................... 101 32. Methods of viscosity determining…………………………… ............ 104 33. Motion in liquids and gases………………………… .......................... 105 Control questions by the theme and problems…… ................................... 108 Part 7. ELEMENTS OF SPECIAL RELATIVITY THEORY ............ 110 34. Galileo transformations. Mechanical principle of relativity…....... ...... 110 35. Postulates of special relativity theory………………………… ........... 112 36. Lorentz transformations………………………………………… ........ 114 37. Consequences from the lorentz transformations………………… ....... 117 38. Interval between events………………………………………...... ...... 122 39. Main law of relativistic dynamics of a material point…… .................. 124 40. The law of relationship of the mass and energy……………… ........... 126 Control questions by the theme and problems problems… ........................ 129 REFERENCES ........................................................................................ 131



The world around you, all that exists around us and finds us through the senses is a matter. An essential property of matter and form of its existence is the movement. Stir in the broadest sense of the word – it is all possible changes of matter – from the simple to the most complex moving processes of thinking. Various forms of motion of matter being studied by various sciences, including physics. The subject of physics, for that matter, and any science, can be opened only when its detailed presentation. Post a strict definition of the object of physics is quite difficult because the boundaries between physics and a number of related disciplines conditional. At this stage of development can not be saved definition of physics just as the science of nature. Akademik Ioffe (1880-1960, Russian physicist) defined physics as a science that studies the general properties of matter and the laws of motion and the field. Currently, it is generally accepted that all interaction is done through the fields, such as gravitational, electromagnetic, nuclear force fields. Golf, along with a substance is a form of matter existence. The inseparable connection between the field and the substance, as well as differences in their properties will be considered as the study of the course. Physics – the science of the simplest and yet the most common forms of movement of matter and their mutual transformations. Studied physics forms of motion (mechanical, thermal, and others.) Are present in all higher and more complex forms of motion of matter (chemical, biological, etc.). Therefore, they being the most simple, are at the same time the most common forms of motion. The higher and more complex forms of motion of matter – the subject matter of other sciences (chemistry, biology, etc.). Physics is closely related to the natural sciences. This close connection of physics with other branches of natural science, as noted by Academician Vavilov (1891-1955, Russian physicist and public fig5

ure), led to the fact that the physics of the deepest roots grown into astronomy, geology, chemistry, biology and other natural sciences. As a result a number of new related disciplines such as astrophysics, biophysics and others. Physics is closely connected with the technique, and this relationship is two-way. Physics grew out of technology needs (development of the mechanics of the ancient Greeks, for instance, was caused by the demands of construction and military equipment of the time), and Technology, in turn, determines the direction of physical research (for example, at the time the task of creating the most efficient heat engines caused a rapid thermodynamics development). On the other hand, the development of physics depends on the technical level of production. Physics – the basis for the creation of new branches of technology (electronic engineering, nuclear engineering, etc.). The rapid pace of development of physics, its growing ties with the technique point to an important role of physics course in technical colleges: a fundamental basis for the theoretical training of the engineer, without which it is impossible successful activity




1. Models in mechanics. Reference frames. Trajectory, the path length of the displacement vector Mechanics – the part of physics that studies the laws of mechanical motion and causes or alter the movement. Mechanical motion – a change of the position of a body with time in space. Development of mechanics as a science begins with the III. BC. e., when an ancient Greek scientist Archimede (287-212 BC. e.) formulated the law of equilibrium of the lever and the laws of equilibrium of floating bodies. The basic laws of mechanics were installed by the Italian physicist and astronomer Galileo G. (1564-1642) and were finalized by the English scientist Isaac Newton (1643-1727). Mechanics of Galileo-Newton is called classical mechanics. It examines the laws of motion of macroscopic bodies which velocities are small in comparison with the speed of light in vacuum. The laws of motion of macroscopic bodies at velocities comparable with the velocity of light are studied in relativistic mechanics, based on the special theory of relativity formulated by Albert Einstein (18791955). To describe the motion of microscopic bodies (individual atoms and elementary particles), the laws of classical mechanics do not apply – they are replaced by the laws of quantum mechanics. In the first part of our course, we will study the mechanics of Galileo-Newton, consider the motion of macroscopic bodies at velosities much lower than the light speed. In classical mechanics generally accepted concept of time and space, developed by Newton and dominated by the natural sciences during the XVII-XIX centuries. Mechanics of Galileo-Newton considers space and time as objective forms of existence of matter, but in isolation from each other and from the movement of material bodies, which is corresponded to the level of knowledge at the time. 7

Mechanics is divided into three sections: 1) kinematics; 2) dynamics; 3) Static. Kinematics studies the movement of bodies without considering the reasons that cause this movement. Dynamics studies the laws of motion of bodies and the reasons that cause a movement or its change. Static studies the laws of equilibrium of the system of bodies. If you know the laws of motion of bodies, some of them can be set the laws of equilibrium. Therefore, static laws apart from the laws of dynamics physics does not consider. Mechanics describes the motion of bodies, depending on the conditions of specific tasks using various physical models. The simplest model is a material point – a body having mass, the size of which can be neglected in comparison with the distance on which it is considered. The concept of the material point – the abstract, but it facilitates the introduction of the solution of practical problems. For example, by studying the motion of planets in their orbits around the sun, you can take them as material points. Arbitrary macroscopic body or system of bodies can be mentally divided into small interacting parts, each of which is considered as a material point. Then study the motion of an arbitrary system of bodies reduced to the study of a system. Under the influence of bodies on each other's the body may be deformed, i.e. it can be changed its shape and size. Therefore, in the mechanics is introduced another model – absolutely solid body. Absolutely solid body is called the body, which under no circumstances can not be deformed and under all conditions, the distance between two points (or more precisely between two particles) of the body remains constant. Any movement of a rigid body can be represented as a combination of translational and rotational movements. Translational motion – a movement in which every line is rigidly associated with a moving body, it remains parallel to its original position. The rotational movement – a movement in which all points of the body are moving in circles, the centers of which lie on the same straight line, called the axis of rotation. The movement of bodies takes place in space and time. Therefore, it is necessary to know in order to describe the motion of 8

a material point, the places where space, this point was, and at what times it was held a certain position. Mass point position is determined with respect to any other arbitrarily selected body, called the reference body. Since it is bound reference system – a set of coordinate system, clock and the reference body. In the general case the motion is determined by scalar equations

x  x(t ) , y  y(t ) , z  z (t ) or r  r (t ) .


Equations (1.1) are called the kinematic equations of motion of a material point. The number of independent coordinates, completely defining the position of a point in space, are called the number of freedom degrees. If material point moves freely in space, then, as has been said, it has three freedom degrees, if it moves along a line, then it has one freedom degree.

Figure 1. Freedom degrees of a material point

Eliminating the equations (1.1), we obtain the equation of the trajectory of motion of a material point. The trajectory of a material point is the line described this point in space. Depending on the shape of the motion the path may be straight or curved. Let’s consider the motion of a point along an arbitrary trajectory (Fig. 2). The clock will start from the time when the point is in position A. The length of the portion of the trajectory AB is called the length of the path: s = s(t). Vector r = r – r0, drawn from the initial position of the moving point in its position at a given time (the 9

increment of the radius vector of a point in the reporting period of time), is called the displacement.

Figure 2. The motion of a point along an arbitrary trajectory In linear motion the displacement vector coincides with the corresponding portion of the path and the module |r| is equal to the traveled distance s

2. Velocity Velocity is a vector quantity that refers to "the rate at which an object changes its position." Imagine a person moving rapidly – one step forward and one step back – always returning to the original starting position. While this might result in a frenzy of activity, it would result in a zero velocity. Because the person always returns to the original position, the motion would never result in a change in position. Since velocity is defined as the rate at which the position changes, this motion results in zero velocity. If a person in motion wishes to maximize their velocity, then that person must make every effort to maximize the amount that they are displaced from their original position. Every step must go into moving that person further from where he or she started. For certain, the person should never change directions and begin to return to the starting position. Velocity is a vector quantity. As such, velocity is direction aware. When evaluating the velocity of an object, one must keep 10

track of direction. It would not be enough to say that an object has a velocity of 55 m/h. One must include direction information in order to fully describe the velocity of the object. For instance, you must describe an object's velocity as being 55 m/h, east. This is one of the essential differences between speed and velocity. Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity and is direction aware. Let the mass point moves along a curved path so that at time t the corresponding position vector r0 (Fig. 3). For the small amount of time t point will pass a way s and receive elementary (infinitesimal) displacement r.

Figure 3. Displacement vector of the point

The vector average velocity is the ratio of the increment r radius vector of the point to the time interval t:

v 

r . t


Average velocity vector coincides with the direction г. Average velocity decreases and indefinitely tends to a limiting value, which is called the instantaneous velocity v:

r dr .  r 0 t dt

v  lim



Instantaneous velocity v, therefore, is a vector quantity that is equal to the first derivative of the radius vector of a moving point in time. Since the secant coincides with the tangent, the velocity vector v is tangential to the path in the direction of motion (Fig. 3). With decreasing t path s more will approach |г|, so the instantaneous velocity module r r s ds v  v  lim  lim  lim  t 0 t t 0 t t 0 t dt Thus, the instantaneous rate is the first derivative module paths over

v  ds dt


When nonuniform motion module instantaneous velocity varies with time. In this case, use a scalar – an average rate of nonuniform motion: Δss v  .


From Fig. 3 > |||, since s >|г|, and only in the case of rectilinear motion

s  r If the expression ds=vdt (. See (2.2)) integrated over time in the range from t до t+t, we find the length of the path traveled by the point in time t: t t+Δt t

 vdt .




In the case of uniform motion in a numeric value the instantaneous velocity is constant; then the expression (2.3) takes the form t t+Δt t


 dt  vt . t


The length of the path traveled by a point in the time interval from t \ to fa, is given by the integral t2

s   v  t  dt . t1

3. Acceleration and its components In the case of non-uniform motion is important to know how does the velocity change rapidly with time. Physical quantity that characterizes the rate of velocity change in magnitude and direction is called an acceleration. Let’s consider the plane motion. i.e. the movement in which all sections of the trajectory points lie in the same plane. During the time t moving point has moved to the position B and the acquired velocity v is different from both the unit and the direction and equal v1 = v + v. We transfer vector v1 to the point А and find v (Fig. 4).

Figure 4. Plane motion

An average acceleration of the uniform motion in the interval from t to t+t is the vector quantity equal: 13

a 

v . t

The instantaneous acceleration of the material point at time t is the limit of the average acceleration: v dv  . dt t 0 t

a  lim a  lim t 0

Thus, the acceleration is a vector equal to the first time derivative of velocity. Let’s expand the the vector v into two components. To do this, from the point A (Fig. 4) in the direction of the velocity v postpone the vector AD with module equal v1. Obviously, the vector equal

CD , equal v determines the rate of velocity change over time t

by module: v = v1 – v. The second component vn of the vector v represents the change in velocity during t by direction. The tangential component of the acceleration

v v dv  lim  dt t 0 t t 0 t

a  lim

i.e. is the first time derivative of the velocity, thereby determining the speed of the rate of velocity change by module. Let’s find a second component of acceleration. Assume that the point B is close enough to the point A, so we can assume s the arc of a circle of radius r, which is little different from the chord AB. Then, from the similarity of triangles AOB and EAD should be vn AB  v1 r , but as AB=vt, the

At limit t  0 we obtain v1  v. 14

Because the v1 = v, an angle EAD tends to zero, due to the triangle EAD is isoscele, the ADE angle between v and vn tends to direct angle. Therefore, at t  0 the vectors vn and v are mutually perpendicular. The second component of the acceleration equal to

is called the normal component of acceleration and is perpendicular by the normal to trajectory to the center of curvature (it is also called the centripetal acceleration). The full acceleration of the body is the geometric sum of the tangential and normal components (Figure 5.):

Figure 5. Components of acceleration

Thus, the tangential component of the acceleration characterizes the rate of velocity change by modulus (tangential to the path), and the normal component of acceleration – the rate of velocity change by direction (toward the center of curvature of the path). Depending on the tangential and normal components of the acceleration the motion can be classified as follows: 1) а = 0, an = 0 – uniform motion; 2) a = a = const, an = 0 – nonuniform motion. 15

At such motion a  a 

v v 2  v1  . t t 2  t1

If at initial moment of the time t1 = 0 and initial velosity v1=v0, then, take t2=t and v2 = v, we obtasin a=(v—v0)/t, then

v  v0  at . Integrating this equation in the range from zero to an arbitrary time t, we find that the length of the path traveled by a point, in the case of nonuniform motion t




s   vdt   v0  at dt v0 t 

at 2 . 2

3) a = f(t), an = 0 – linear motion with variable acceleration; 4) a = 0, an = const. At a = 0 the velocity by module does not change, but changes in direction. From formula an =v2/r. It follows that the radius of curvature should be a constant. Consequently, the rotational motion is uniform; 5) a = 0, an  0 – uniform curvilinear motion; 6) a = const, an  0 – curved uniformly accelerated motion; 7) a = f(t), an  0 – curvilinear motion with a variable acceleration.

4. Angular velocity and angular acceleration Consider a rigid body that rotates around a fixed axis. Then the individual points of this body will describe a circles of different radii centers of which lie on the axis of rotation. Suppose a point moves along a circle of radius R (Fig. 6). Its position after a time t moves angle . Elementary (infinitesimal) rotations can be regarded as vectors (or they are designated as  are d ). The module of vec16

tor  is equal to the angle of rotation and its direction coincides with the direction of forward motion of the tip of the screw, which head rotates in the direction of the point on the circle, i.e. obeys the right-hand screw (see Fig. 6). Vectors, directions of which are associated with the direction of rotation are called pseudovectors or axial vectors. These vectors have no certain points of application: they can be deposited from any point of the axis of rotation. Angular velocity is a vector quantity equal to the first derivative of the angle of rotation of the body by time:

   d .   lim  dt t 0 t

The vector  is directed along the rotation axis by right-handed  screw rule, i.e. as vector d (Fig. 7). The dimension of the angular velocity

dim   T 1 and its unit – radians per second (rad/s).

Figure 6

Figure 7

The linear velocity of a point (Fig. 6)

s R   lim  R lim  R t 0 t t 0 t t 0 t

v  lim

v  R . The formula for the linear velocity in vector form can be written as a vector product: 17

 v  R . If  = const, then the rotation is uniform and can be characterized by a period of revolution T – the time at which the point makes one full rotation, i.e. rotates to an angle 2. Since the time interval t = Т corresponds  = 2, then  = 2/Т, where T

2 .

The number of complete rotations made by the body when it moves uniformly by circle per unit of time is called the frequency:

from this . The angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect by time: . The tangential component of the acceleration

. Normal acceleration .


When the body rotates about a fixed axis angular acceleration vector is directed along the rotation axis at the direction of the elementary increment of angular velocity. At accelerated motion the   vector  has the same direction as vector  (Figure 8), at decelerated motion – is opposite to it (Figure 9.)

Figure 8

Figure 9

Thus, the connection between the linear (the path length s, the linear velocity v, tangential acceleration а, normal acceleration аn ) and the angular values (rotation angle , the angular velocity , angular acceleration ) is expressed as follow formulas:

In the case of uniformly accelerated motion of point by circle (-const.)

where 0 – initial angular velocity. Control questions by the theme 1. What is the material point? 2. What is the difference between the displacement and path? 3. Define the concepts of «mechanical motion», «body of reference» and «material point». 4. The difference between the concepts «trajectory», «path» and «displacement». 5. What is the forward motion of the body? 6. What is the uniform rectilinear motion of a body (particle)?


7. 8. 9. 10.

What does the speed of uniform motion show? What movement is called equally accelerated? What is acceleration and what does it show? The unit of acceleration. Draw graphs showing the dependence of velocity on time with uniform motion, uniformly accelerated motion with initial velocity and without it (there should be three graphs in total). 11. What is the free fall of bodies? What is the acceleration of gravity g near the Earth's surface. 12. The law of addition of velocities. An example of the law. 13. The motion of a material point along a circle. Definitions of the period of revolution, rotation speed. Formulas for centripetal acceleration and velocity for a uniform motion of a point along a circle. Problems 1.1. The dependence of the traversed path of the body from time is given by the equation s=A+Bt+Ct2+Dt 3(С=0,1 м/с2, D=0,03 m/s3). Define: 1) time after the beginning of the motion, through which the acceleration of a body is equal 2 m/s 2; 2) average acceleration of body through time. [1) 10 s; 2) 1,1 m/s2] 1.2. Neglecting air resistance, determine the angle at which the body is thrown to the horizon, where the maximum lifting height of the body is equal to 1/4 of the range of its flight. [45°] 1.3. Wheel of radius R = 0,1m rotated so that the angular velocity dependence on time is given by the equation  = 2At + 5Bt4 (А = 2 rad/s2 и В = 1 rad/s5). Determine the total acceleration of the wheel rim points after t=1 s after the start of rotation and the number of revolutions made by the wheel during this time. [а=8,5 m/s2; N = 0,48] 1.4. Normal acceleration of the point moving by a circle of radius r = 4 m, is given by equation an=A+Bt+Ct2 (A = 1 m/s2, B = 6 m/s2, С = 3 m/s2). Define: 1) tangential acceleration of a point; 2) the path traversed by the point after time t1=5 s after the start of motion; 3) full acceleration for time t2=1 s. [1) 6 m/s2; 2) 85 m; 3) 6,32 m/s2] 1.5. Wheel rotation frequency speed at uniformly decelerated motion in t = 1 min decreased from 300 to 180 minutes -1. Define: 1) angular acceleration of the wheel; 2) The number of complete rotations made by the wheel during this time. [1) 0,21 rad/s2; 2) 240] 1.6. Disc of radius R = 10 cm rotates around a fixed axis so that dependence of the angle of rotation of the disk radius on time is given by the equation =A+3t+Ct2+Dt3 (B = 1 rad/s, С = 1 rad/s2, D = 1 rad/s3). Determine for the points on the rim of wheel at the end of 2nd second after the start of motion: 1) tangential acceleration %; 2) normal acceleration аn; 3) full acceleration а. [1) 1,4 m/s2; 2) 28,9 m/s2; 3) 28,9 m/s2]




5. First Newtons law. Mass. Forse Dynamics is the main part of mechanics, based on three Newton's Laws, formulated in 1687. Newton's Laws play a crucial role in the mechanics and are (like all the laws of physics) a generalization of the results of vast human experience. They are considered as a system of interrelated laws and subjected to experimental verification of every single law, and the whole system. Newton's first law: An object in motion with a constant velocity will continue in motion unless acted upon by some net external force. A body at rest is just a special case of Newton’s first law with zero velocity. Newton’s first law is often called the law of inertia and is also used to describe inertial reference frames. Mechanical motion is relative, and it depends on the nature of the reference system. Newton's first law does not apply in any frame of reference, and those systems with respect to which it is performed are called inertial frames. Inertial reference frame is a frame of reference with respect to which a material point, free from external influences, either at rest or at uniform motion by a straight line. Newton's first law asserts the existence of inertial reference systems. Laws of Nature are essentially mathematical postulates that permit us to understand natural phenomena and make predictions concerning the time evolution or static relations between the coordinates associated with objects in nature that are consistent mathematical theorems of the postulates. These predictions can then be compared to experimental observation and, if they are consistent (uniformly successful) we increase our degree of belief in them. If they are inconsistent with observations, we decrease our degree of belief in them, and seek alternative or modified postulates that work better. 21

The entire body of human scientific knowledge is the more or less successful outcome of this process. This body is not fixed – indeed it is constantly changing because it is an adaptive process, one that self-corrects whenever observation and prediction fail to correspond or when new evidence spurs insight (sometimes revolutionary insight!) Newton’s Laws built on top of the analytic geometry of Descartes (as the basis for at least the abstract spatial coordinates of things) are the dynamical principle that proved successful at predicting the outcome of many, many everyday experiences and experiments as well as cosmological observations in the late 1600’s and early 1700’s all the way up to the mid-19th century. When combined with associated empirical force laws they form the basis of the physics you will learn in this course.Body weight – a physical quantity, which is one of the main characteristics of matter that determines its inertia (inertial mass) and gravity (gravitational mass) properties. It can now be considered proven that the inertial and gravitational masses are equal to each other (with an accuracy of not less than 10 values). To describe the effects mentioned in the first Newton's law, was introduced the concept of force. Under the action of body forces, or change the speed of motion, i.e. body acquire acceleration (dynamic display of power), or deformed, i.e. body change their shape and size (static display of forces). At any time, the power is characterized by a numerical value, direction in space and point of application. Thus, the force is a vector quantity, which is a measure of the mechanical effects on the body from other bodies or fields, in which the body becomes at-acceleration or changes its shape and size we can take that the inertial and gravitational masses are equal to each other (with an accuracy not less than the value of 10). 6. Second Newtons law Newton's second law – the fundamental law of dynamics of translational motion answers the question of how does change the mechanical motion of a material point under the influence of mechanical forces. If we consider the effect of various forces on the same body, then the acceleration acquired by the body, is always directly proportional to the resultant of applied forces: 22



The action of the same force on the bodies with different masses their acceleration are different, namely


1 (F= const). m


Using the expression (2.1) and (2.2) and taking into account that the force and acceleration are vector quantities, we can write


F. m


Equation (2.3) expresses the second Newton's law: The net force applied to an object is directly proportional to its acceleration in an inertial reference frame. The constant of proportionality is called the mass of the object. In the system SI proportionality coefficient k = 1. Then


F m

F  ma  m

dv . dt


Given that the mass of the material point (the body) in classical mechanics is a constant in the expression (6.4) it can be made under the sign of the derivative: d (2.5) F  mv . dt Vector quantity P = mv (2.6) numerically equal to the product of the mass of a particle and its velocity and has direction of the velocity is called momentum (momentum) of the material point. Putting (2.6) into (2.5), we obtain 23


dp . dt


This expression – a more general formulation of Newton's second law: the rate of change of momentum of a material point is equal to the acting at it force. Expression (2.7) is called the equation of motion of a material point. The unit of force in SI – newton (N): 1 N – force acting to the mass of 1 kg according to the acceleration of 1 m/s2 in the direction of the force: 1 N= 1 kg*m/S2. Newton's second law is valid only in inertial reference systems. Newton's first law can be obtained from the second. Indeed, in the case of zero resultant force (in the absence of effects on the body from other bodies) the acceleration (consider (2.3)) is also zero. However, Newton's first law is regarded as a separate law (rather than as a consequence of the second law), since it asserts about the existence of inertial reference systems, which only satisfied the equation (2.7). In mechanics of great importance is the principle of superposition: if on a material point act multiple forces, then each of these forces gives to a material point the acceleration according to Newton's second law, as if was not other forces. According to this principle, force and acceleration can be decomposed into components, the use of which leads to a significant simplification of solving problems. For example, in Fig. 10 acting force F = ma broken down into two components: the tangential force F (tangential to the trajectory) and normal Fn (directed along the normal to the center of curvature).

F ma  m

Fn  ma  Using expressions a 

dv dt

mv 2  m 2 . R

2 dv , an  v R , and v = Rv, we can right: dt


F  ma  m

dv ; dt

Fn  man  mv 2 / R  m 2 R .

If the material point is acted by some forces, according to the principle of force independence in Newton’s second law we mean resultant force.

7. Third Newtons law The interaction between particles (bodies) is defined by Newton's third law: The force exerted by body 1 on body 2 is equal in magnitude and opposite in direction to the force that body 2 exerts on body 1. If the force on body 2 by body 1 is denoted by F21, then Newton’s third law is written as:

F12   F21 ,


where F12 – the force acting on the first material point from the second; F21 – the force acting on the second material point from the first. These forces are applied to different material points (bodies), always operate in pairs and are a forces of nature. Newton's third law allows the transition from the dynamics of a separate material points to the dynamics of system of material points. This follows from the fact that the interaction of a system is lead to the pair of interaction forces between particles.

8. Friction Discussing still forces, we were not interested in their origins. However, in mechanics we will consider the various forces: friction, elasticity, gravity. From the experience we know that any body moving on a horizontal surface of the other body, in the absence of other forces slows down with time and eventually stops. This can be explained by the 25

existence of the friction that prevents sliding bodies in contact with each other. The friction depends on the relative velocity of bodies. The friction forces can be of different nature, but as a result of their actions the mechanical energy is transformed into internal energy of bodies in contact. There are external (dry) and internal (liquid or viscous) friction. External friction is called the friction that occurs in the plane of contact of two bodies in contact at their relative motion. If the bodies are motionless, we can talk about the friction of rest, if there is a relative motion of the bodies, then depending on the nature of their relative motion we can talk about sliding friction, rolling or spinning. Internal friction is called the friction between the parts of the same body, such as between the various layers of liquid or gas, the speed of which changes from layer to layer. In contrast to the external friction there is no static friction. If the bodies slide relative to each other and separated by a layer of viscous liquid (lubricant), the friction occurs in the lubricant layer. In this case we speak about of a hydrodynamic friction (lubrication layer is thick enough) and boundary friction (lubricant layer thickness 0,1 microns or less). Let us discuss some of the laws of the internal friction. This friction is due to the roughness of contacting surfaces; in the case of very smooth surfaces the friction is due to the forces of intermolecular attraction. Let’s consider the body lying in the plane (Fig. 10), to which is attached a horizontal force F. The body will move only when the applied force F is greater than the friction f. French physicists G. Amontons (1663-1705) and S. Coulomb (1736-1806) empirically established the following law: the friction is proportional to the normal force with which one body acts on another:

F fr  fN , where f – sliding friction coefficient, which depends on the properties of the contacting surfaces. Let us find the value of the coefficient of friction. If the body is on an inclined plane with an inclination angle  (Fig. 12), it starts to move, when only the tangential component of gravity force is greater 26

than the friction. Therefore, in the limiting case (start sliding body) F = Ffr or P sin 0  f  fN  fP cos 0 , then

f  tg 0

Figure 10

Figure 11

For smooth surfaces play a role the intermolecular attraction. The law of friction for them F fr  f  N  Sp 0  ,

where p0 – the additional pressure caused by the forces of intermolecular attraction, which rapidly decrease with increasing the distance between the particles; S – the square of contact between the bodies; f – coefficients of friction. Friction plays an important role in nature and technology. Due to friction move transport, the nail kept driven into the wall and so on. In some cases, the friction forces have a deleterious effect and therefore they must be reduced. For this purpose, a lubricant is applied to the friction surfaces (friction force is reduced by about 10 times) to fill irregularities between the surfaces and is located as the thin layer so that the surface would no longer like to touch each other and slide against each other as separate liquid layers. Thus, the external friction of solids replaced with much lower internal friction of liquid. A radical way to reduce the friction force is the replacement of sliding friction by rolling friction (ball and roller bearings and so on.). The strength of the rolling friction is determined according to the law, established by Coulomb: 27

F fr  fN / r ,


where r – radius of the rolling body; f – coefficient of rolling friction with the dimension dim fk = L. From (8.1) it follows that the force of rolling friction is inversely proportional to the radius of the rolling body.

9. Conservation of momentum. Center of mass To derive the law of conservation of momentum, let’s consider some definitions. The collection of material points, considered as a single entity, is called the mechanical system. The forces of interaction between the material points of the mechanical system are called internal. The forces with which act on the material points of the system the external body are called external. The mechanical system of bodies on which no external force is called a closed (or isolated). If we have a mechanical system consisting of many bodies, according to Newton's third law, the forces acting between the two bodies are equal and are opposite directed, i.e. geometric sum of the internal forces is zero. Consider a mechanical system consisting of n bodies, mass and the velosity of which are m1, m2, ..., mn and v1, v2, ..., vn. Let F1, F2, ..., Fn – resultant of internal forces acting on each of the bodies, and F1, F2, ..., Fn – resultant of external forces. Let’s write Newton's second law for each of the N bodies (mechanical system):

d m1v1   F1'  F1 , dt d m2 v2   F2\  F2 , dt d mn vn   Fn\  Fn . dt


Adding these equations term by term, we obtain

d m1v1  m2 v2  m3v3  vn   F1\  F2\  F3\  ...  Fn\  F1  F2  ...Fn . dt But as the geometric sum of the internal forces of the mechanical system is equal zero by the third law of Newton,

d m1v1  m2 v2  m3v3  vn   F1  F2  ...Fn , dt or

dp  F1  F2  ...Fn , dt



where p   mi vi – system momentum. Thus, the time derivative of i 1

the momentum of the mechanical system is the geometric sum of the external forces acting on the system. In the absence of external forces (consider a closed system)

dp n d   mi vi   0, p  dt i 1 dt


m v

i i

 const.

i 1

The last expression is the law of conservation of momentum: impulse of an isolated system is conserved, i.e., does not change over time. The law of conservation of momentum is valid not only in classical physics, though he received as a consequence of Newton's laws. Experiments show that it holds for closed systems of microparticles (they obey the laws of quantum mechanics). This law is universal, i.e., the law of conservation of momentum – fundamental law of nature. The law of conservation of momentum is a consequence of a certain symmetry properties of space – its homogeneity. The homogeneity of space is that the parallel displacement in the space of an 29

isolated system of bodies as a whole, its physical properties and the laws of motion do not change, in other words, do not depend on the choice of the position of the origin of the inertial reference system. Note that, according to (9.1), the momentum is maintained and to an open system if the geometric sum of all the external forces is zero. In the mechanics of Galileo-Newton due to the independence of the masses on velocity the impulse of the system can be expressed through the speed of its center of mass. The center of mass (or center of mass) of a system of material points is called an imaginary point C, the position of which describes the distribution of the mass of the system. Its radius vector is

where mi and ri – mass and radius vector of i – material point; n – n

number of material points in the system; m   mi , – system’s mass. i 1

Velosity of mass center

Taking into account that pi  mi vi the ,


 mi , we can right i 1


i.e. the momentum of the system is the product of a system’s mass and velocity of its center of mass. Substituting (9.2) into (9.1), we obtain (9.3) so center of mass moves as the point at which the mass of whole system is concentrated and on which the force is equal to the geometric 30

sum of all external forces applied to the system. Expression (9.3) is a law of motion of the mass center. In accordance with (9.2) from the law of conservation of momentum follows that the center of mass of a closed system moves uniformly or remains stationary.

10. Equation of motion of variable mass body The motion of some bodies is accompanied by changes in their weight, for example, the mass of a rocket decreases as a result of the expiration of gases produced by the combustion of fuel, and so on. We derive the equation of motion of variable mass body on the example of the rocket. If at time t the rocket’s mass is m, its speed v, then after the time dt its mass will decrease to dm, and the speed becomes equal to v + dv. Change in the momentum of the system in time dt where u – velocity of expiration of gas relative to the missile. Then . If the system is subjected to external force, dp = Fdt, therefore



dvc dm .  F u dt dt


The second term on the right side of (10.1) is called the reaction force Fp. If it is the opposite of the direction, the rocket is accelerating, and if coincides with v, then rocket braked. Thus, we have obtained the equation of motion of variable mass body (10.2) which was first displayed by Meshcherski (1859-1935). 31

The idea of using the reactive force to create aircrafts was voiced in 1881 by N.I. Kibalchich (1854-1881). Konstantin Tsiolkovsky (1857-1935) in 1903 published an article in which he proposed the theory of the rocket and the basis of the liquid jet engine theory. Therefore, he is considered as the founder of Russian cosmonautics. Let’s apply the equation (10.1) to the motion of the rocket, which is not affected by any external forces. Putting F = 0, and assuming that the rate of emitted gases relatively constant (rocket moves in a straight line), we obtain


dv dm ,  u dt dt

The value of the integration constant C is determined from the initial conditions. If at the initial time the velocity of rocket is zero, and its launch weight is m0 then C = ulnm0. Consequently, (10.3) This ratio is called the Tsiolkovsky rocket equation. It shows that: 1) the larger the final mass of the rocket m, the greater should be the starting mass m0; 2) the greater the velocity of the gases and the greater may be the final mass for this launch weight of the rocket. Equations (10.2) and (10.3) are obtained for the non-relativistic motions, i.e. for the cases where the velocities v and u are small compared with the speed of light in a vacuum. Control questions by the theme 1. Newton's first law, a system for which he is just. principle of relativity of Galileo. 2. Newton's second law. 3. What is the density of matter. Unit of density. 4. The third law of Newton. 5. How many kinds of forces are distinguished in mechanics? 6. Hooke's Law.


7. The law of sliding friction. The frictional force of rest. 8. What is body weight and weightlessness? What is the first cosmic speed? 9. Formulate Newton's third law, explain it. What forces are called external, and which are internal? What is called the center of mass (the center of inertia) of the mechanical system? When considering the motion of a mechanical system it is sufficient to consider the motion of only its center of mass? 10. What forces act on the load suspended on the thread? What is called body weight? In what case is the weight of a body numerically equal to the force of gravity? 11. How many equations are needed to describe a mechanical system consisting of n bodies that are acted upon by both internal and external forces? How do these equations form? What is the sum of all internal forces acting on a system of bodies? 12. Write down the law of conservation of momentum for two interacting bodies. Problems 2.1. The body slides by an inclined plane with an inclination to the horizontal 30 °. Determine the velocity of the body at the end of the third second from the beginning of sliding, if the coefficient of friction is 0.15. [10.9 m / s] 2.2. The plane describes the loop of radius 80 m. What should be the lowest speed of the aircraft to the pilot's seat does not cut out at the top of the loop? [28 m / s] 2.3. The unit is mounted on top of the two inclined planes constituting with the the horizon angles 30 ° and 45 °. Weights of equal mass (m1 = m2 = 2 kg) are connected by thread, thrown through the block. Considering the thread and block weightless, taking friction coefficients f1 = f2 = f = 0.1 and neglecting the friction determine 1) the acceleration of weights, 2) tension of the thread. [1) 0.24 m/s 2; 2) 12 H] 2.4. On the railway platform is mounted recoilless gun from which the shot along the track at an angle 45 ° to the horizon. The weight of platform with gun 20 tonns, projectile mass m = 10 kg, the coefficient of friction between the wheels and the rails of the platform f = 0,002. Determine the velocity of the projectile, if after the shot the platform shifted to the distance s=3 m. [ v0  M 2 fgs m cos   970 m/s] 2.5. On the boat with mass m = 5 tonns is the water jet, eject 25 kg / s of water at a speed of u = 7 m/s relative to the boat backwards. Neglecting the resistanse to the boats movement, define: 1) the speed of the boat after 3 minutes after the beginning of the motion, 2) the maximum possible speed of the boat. [1) v  u 1  exp  t m  = 4,15 м/с; 2) 7 m/s]




11. Energy, Work, Power Energy – an universal measure of different forms of motion and interaction. With different forms of motion of matter is binded various forms of energy: mechanical, thermal, electromagnetic, nuclear and so on. In one phenomena the matter motion form does not change (for example, hot body heats cold), in others – it goes into an another form (eg, as a result of friction mechanical motion is converted into heat). However, it is essential that in all cases the energy given up (in one form or another) by one body to another body, equal to the energy received by the latter body. The change in mechanical motion of the body is caused by the forces acting on it by other bodies. To quantitatively characterize the process of energy exchange between the interacting bodies in mechanics introduced the concept of the work of force. When the body moves rectilinearly and has a constant force F, which is at an angle а with a moving direction, then the work of this force is equal to the product of the projection of force F and the direction of movement, multiplied by the displacement of the point of force application . (11.1) In general, the force can vary as by mod, and direction, so the formula (11.1) can not be used. If, however, consider the elementary displacement dr, then the force F can be assumed to be constant, and the movement of its point of application – straight. The elementary work of the force F on the displacement dr is called a scalar quantity dA = Fdr = Fcos ds = F2ds, 34

where  – an angle between F and dr; ds=|dr| – elementary path; Fs – projection of vector F on dr (Fig. 12).

Figure 12

The force work in the area of the trajectory from point 1 to point 2 is equal to the algebraic sum of the elementary works on selected infinitesimal sections of the path. This amount is reduced to the integral (11.2) To calculate this integral it is necessary to know the dependence of the force Fs from the path along the trajectory 1-2. Let this dependence is shown graphically (Fig. 13), then the required work A is defined on the graph by the square of the shaded figure. If, for example, the body moves in a straight line, the force F = const and a = const, we obtain

where s – distance traveled by body. From (11.1) it follows that if  < /2 the force work is positive, and in this case, the F component is in the the same direction as the vector of velocity v (see. Fig. 13). If  > /2, the force work is negative. If  = /2 (the force is perpendicular to the displacement of) the force work is zero. 35

The unit of work is – Joule (J): 1 J – the work done by the force on the path 1 m (1 J=1 N-m). Fs

Figure 13. Calculation of the work by integration

In order to characterize the rate of work doing, introduce the concept of power: . During the time dt the force F does the work Fdr, and power developed by this force at a given time


Fdr  Fv dt


that is equal to the scalar product of the force vector to the velocity vector, with which moves the point of force application; N – the scalar quantity. The unit of Power is watt (W): 1 W – the power, at which on time 1 second is done the work 1 J (1 W = 1 J/s).

12. Kinetic and potential energy The kinetic energy of the mechanical system – the energy of the mechanical motion of the system. The force F, acting on a body at rest, and causing its motion, does work, and the energy of a moving body is increased by the 36

amount of expended work. Thus, the work dA of the force F on the path that the body passed during the ascending of the velocity from 0 to v, is to increase the kinetic energy of the body i.e. . Using Newton's second law F = ma and multiplying by displacement dr, we get . The v  r , then dA = mvdv = mvdv = dT, and

dt 

. Thus, the body of mass m moving with velocity v, has a kinetic energy . (12.1) Formula (12.1) shows that the kinetic energy depends only on the mass and velocity of the body. The kinetic energy of the system is a function of its state of motion. From the derivation of (12.1) it was assumed that the motion is considered in an inertial frame of reference, since otherwise it would be impossible to use Newton's laws. In different inertial reference systems moving relative to each other, the body speed, and consequently, its kinetic energy will vary. Thus, the kinetic energy depends on the choice of the reference system. Potential energy – mechanical energy of a system of bodies, determined by their mutual disposition and the nature of the interaction forces between them. Let the interaction of bodies is carried out by force fields (e.g., the field of elastic forces, the field of gravitational forces), characterized by the fact that the work done by the acting forces at body mo37

tion from one position to another, does not depend on a trajectory of this motion and itdepends only on initial and final positions. Such fields are called potential, and the forces acting in them – conserved. If the work done by the force depends on the path of the body from one point to another, such force is called dissipative; the example of it is friction. The body in a potential force field has potential energy P. The work of conservative forces in elementary (infinitesimal) change of the configuration of the system is equal to the increment of potential energy, taken with a minus sign, as the work is done by the loss of potential energy: .


The work dA work is expressed as a scalar product of the force F on the displacement dr and the expression (12.2) can be written as .


Therefore, if the function P(r) is known, then from the formula (12.3) we can find the force F in magnitude and direction. Potential energy can be determined on the basis of (12.3) as the , where C -constant of integration, i.e., the potential energy is defined with accuracy to some arbitrary constant. This, however, does not affect on the laws of physics, as they contain or the difference between the potential energies in the two positions of the body, or a derivative of P by coordinates. Therefore, the potential energy of the body in any particular situation is considered to be zero (the zero reference level is selected), and the energy of the body in other states is determined relative to the zero level. For the conservative forces .


Or in vector form , where

gradП 


   i j k, x y z


(i, j, k – the unit vectors of the coordinate axes). The vector defined by the expression (12.5) is called the gradient of a scalar P. For it, along with the designation grad P is also applied .  («nabla") refers to the symbolic vector, called Hamilton operator or Nablaoperator.    (12.6)  i j  k. x y z The particular form of the function P depends on the nature of the force field. For example, the potential energy of a body with mass m, raised to a height h above the Earth's surface, is equal to P =mg,


where the height H is measured from the zero level for which P0=0. Expression (12.7) follows directly from the fact that the potential energy is equal to the work of gravity force when the body dropped from a height h on the Earth's surface. Since the reference point is chosen arbitrarily, then the potential energy can be negative (the kinetic energy is always positive!). If we take the potential energy of the body on the ground surface as zero, the potential energy of the body, located on the bottom of the shaft (depth h1), P= –mgh'. Let’s find the potential energy of elastically deformed body (spring). The strength of the elastic deformation is proportional to: , where Fх elast – the projection of the elastic force on x-axis; k – the coefficient of elasticity (a spring – stiffness) and the minus sign indicates that the force Fх elast is directed oppositely to deformation x. 39

According to Newton's third law, the deforming force is equal in modulus to elastic force and is oppositely directed to her, i.e. . The elementary work dA, done by the force Fx elast at it infinitely small deform ation dx is equal

and is for the increase in potential energy of the spring. Thus, the potential energy of elastically deformed body P=


The potential energy of the system is a function of the system state. It depends on the system configuration and its position in relation to external bodies. The full mechanical energy of the system – the energy of mechanical motion and interaction: E=T+P, і.e. is the sum of kinetic and potential energies.

13. The energy conversation law The law of energy conservation – the result of summarizing of many experimental data. The idea of this law belongs to Mikhail Lomonosov (1711-1765), set out the law of conservation of matter and motion, and quantitative formulation of the energy conservation law was given by the german doctor Yu Mayer (1814-1878) and the german naturalist G. Gelmgoltz (1821-1894). 40

Let’s consider the system of material points with masses m1, m2,…, mn, move with velosities v1, v2,…, vn. F1, F2, ..., Fn – the resultant internal conservative forces acting on each of these points, and F1, F2, ..., Fn – the resultant of the external forces which will also be considered as conservative. In addition, we assume that also non conservative external forces act on material points; resultants of these forces acting on each of the material points is denoted f1, f2,…, fn. At v m2. The first ball continues to move in the same direction as before the collision, but at a slower speed (v1 < v1). Second 49

ball’s velocity after impact is greater than the velocity of the first collision velocity (v'2 > v1) (Fig. 19);

Figure 19

c) m1 < m2. The direction of motion of the first ball at collision t varies – the ball bounces back. The second ball moves in the same direction, in which was moving first ball before collision, but at a slower velocity, i. е. v'2 < v1 (Fig. 20);

Figure 20

d) m2 >> m1 (for example, a collision with the wall of the ball). Equations (15.8) and (15.9) should be, that v1   v1; v2  2m1 v1 / m2  0. 2. At m1 = m2 expressions (15.6) and (15.7) take the form ,


i.e. the balls of equal weight «exchange» by velocity. Absolutely nonelastic collision – a collision of two bodies, in which the bodies are together, moving on as a whole. We can demonstrate an nonelastic collision with the use the balls of plasticine (clay), moving towards each other (Fig. 21). 50

Figure 21

If the masses of balls m1 and m2, their velocities before collision v1 and v2, then using the law of conservation of momentum, we can write where v – velocity of the balls after the impact. Then


m1v1  m2v2 . m1  m2


If the balls are moving towards each other, then they will work together to continue to move in the direction of the moving ball with a greater momentum. In a particular case, if the masses of the balls are equal (m1 = m2), then

Let us see how does change the kinetic energy of the balls at the central nonelastic collision. Since at the process of collision of balls between them there are forces, which depend on their velocities,not on deformations only, we are dealing with forces like the friction, so the law of conservation of mechanical energy should not be respected. As a result of the deformation it is a "loss" of the kinetic energy passed into heat and other forms of energy. This "loss" can be determined from the difference of kinetic energy of bodies before and after impact: .


Using (15.10), we are taking

If the body was initially motionless (v2=0)

v T 

m1v1 , m1  m2

m2 m1v12 . m1  m2 2

When m2 >> m1 (the mass of a body at rest is very large), then v > m2), then v  v1 and virtually all of the energy is spent on the largest possible displacement of the nail, not the permanent deformation of the wall. Inelastic collision – an example of how there is a «loss» of mechanical energy under the influence of dissipative forces. Control questions by the theme 1. What is it work? 2. Why bodies do need in energy? 3. What is the impulse of the body? Which system is called closed? 4. Law of conservation of momentum. 5. Power, units of its measurement. 6. The concept of mechanical energy. Kinetic and potential energy. 7. What is the potential energy of an elastically deformed body (spring)? 8. The law of conservation of total mechanical energy. 9. What is the moment of force relative to the axis of rotation lying inside the body of revolution? 10. What determines the sign of the moment of the force lying in the plane of rotation? 11. When the body having the axis of rotation will be in equilibrium?


Problems 3.1. Determine: 1) the work of load raise on an inclined plane; 2) average, and 3) the maximum powers of the lifting device if the load weight is 10 kg, length of inclined plane 2 m, the angle of inclination is 45 ° to the horizon, coefficient of friction 0.1 and a rise time 2 s. [1) 173 J; 2) 86 W; 3) 173 W] 3.2. From the tower with height of 35 m horizontally was thrown stone with weight 0.3 kg. Neglecting air resistance, determine: 1) the speed with which the stone is thrown if in 1 second after the beginning of the movement, its kinetic energy became 60 J; 2) the potential energy of the stone in 1 second after the beginning of the movement. [1) 17.4 m / s; 2) J 88.6] 3.3. Neglecting the friction, determine the lowest height at which the carriage should slide down the chute with a man turning into a loop radius of 10 m, so that it made a complete loop and does not fall out of the gutter. [25 m] 3.4. The bullet mass m = 10 g, flying horizontally with a velocity v = 500 m / s, hits a ballistic pendulum length l=1 and m = 5 kg and becomes stuck in. Determine the angle of deviation of the pendulum. [18 ° 30 /] 3.5. The dependence of the potential energy of a particle in a central force field on the distance r from the center of the field is given by П (r ) 

A r2

B , where А r

and В positive constants. Determine the value of r0, corresponding to the equilibrium position of the particle. Is this the position of stable equilibrium position? [г0 = 2А/B] 3.6. At the center is absolutely elastic collision the moving body mass m1 hits the resting body mass m2, whereby the first body velocity is reduced in speed n=1,5 times. Define: 1) the ratio m1/m2; 2) kinetic energy Т'2 of the second body if the initial kinetic energy of the first body T1=1000 J. [1) 5; 2) 555 J] 3.7. The body of weight m1=4 kg moves with velocity v1 = 3 m/s and hits the motionless body of the same mass. Considering collision central and inelastic, determine the amount of heat released during impact. [9 J]




16. Moment of inertia In the study of the rotation of solid bodies we use the concept of moment of inertia. The moment of inertia of the system (the body) with respect to this axis is called a physical quantity, which is equal to the sum of products of mass n materials points of a system on the squares of their distances from the axis under consideration:

In the case of a continuous distribution of mass, this amount is reduced to the integral

where the integration is over the volume of the body. The value of r in this case is a function of position of the point with coordinates x, y, z. As an example, let’s find the moment of inertia of a uniform solid cylinder with height h and a radius R by its geometric axis (Fig. 22). We split the cylinder into separate hollow concentric cylinders with the infinitely small thickness dr with the internal radius r and external r+dr. The moment of inertia of each of the hollow cylinder dJ = r2dm (dr > S2, then a member v21/2 can be neglected and v 22  2 g h1  h2   2 gh, v 2  2 gh.

This expression has been called Torricelli formula. 100

31. Viscosity (internal friction). Laminar and turbulent flow of liquids Viscosity (internal friction) – a property of real fluids to resist movement of one part of liquid relative to another.At displacement of one layers of real liquid relative to the other there the internal friction force directed tangentially to the layer surface occurs. The action of these forces is reflected in the fact that the layer, moving faster acts with the accelerating force on a layer, moving slowly, accelerating force acts. From the side of the slow layer on a faster layer the braking force acts. The force of internal friction F is greater, the larger the surface area of the layer under consideration S (Fig. 50), and depends on how quickly changes the fluid flow velocity in the transition from one layer to another. On figure there are two layers at a distance х from each other and moving with velocities v1 and v2. In this case v1 - v2 = v. The direction in which the distance between the layers is measured is perpendicular to the the flow velocity of the layers. The quantity, showing how the speed changes rapidly in the transition from one layer to another in the x direction, perpendicular to the direction of layers motion is called velocity gradient. Thus, the internal friction force module F 

v S. x


Where the proportionality coefficient , depending on the nature of the liquid, is called the dynamic viscosity (or viscosity).

Figure 50


The unit of viscosity – pascal-second (Pas): 1 Pas is the dynamic viscosity of the medium in which at laminar flow and the velocity gradient with the module of 1 m/s to 1 m, there is a force of internal friction 1 N per 1 m2on touch of layers surface (1 Pas=Ns / m2). The higher the viscosity, the more liquid is different from the ideal, the great internal friction forces arise in it. Viscosity is dependent on temperature, and the character of this dependence is different for liquids and gases (for liquids with increasing temperature decreases from gases, in contrast, increases), which indicates the difference in their internal friction mechanisms. Especially strongly on temperature is depend the viscosity of oils. For example, the viscosity of castor oil in the range of 18-40° C falls four times. Russian physicist Kapitza (1894-1984; Nobel Prize 1978) discovered that at a temperature of 2.17 K, liquid helium becomes in superfluid state in which its viscosity is equal zero. There are two modes of fluid flow. The flow is called laminar (layered) if along the flow each dedicated thin layer slides relative to the neighboring, not mixing with them, and turbulent (vortex), if there is an intensive flow along the vortex formation and mixing of the fluid (gas). Laminar flow of the fluid is observed at low speeds its movement. The outer layer of the fluid adjacent to the surface of the pipe in which it flows, due to the molecular cohesion forces adhere thereto and remains fixed. Subsequent layers speed the greater the greater their distance from the surface of the pipe, and has the highest rate of layer moving along the tube axis. Laminar flow of the fluid is observed at low speeds of its movement. The external layer of the fluid adjacent to the surface of the pipe in which it flows, due to the molecular cohesion forces adhere to it and remains fixed. Subsequent layers speed the greater the greater their distance from the surface of the pipe, and the layer moving along the tube axis has the highest velocity. In turbulent flow the liquid particles acquire velocity components perpendicular to the flow, so that they can pass from one layer to another. The fluid velocity of the particles increases rapidly with increasing distance from the surface of the pipe, and then varies quite significantly. Since the fluid particles move from one layer to another, 102

their velocities in various layers differ a little. Due to the large velocity gradient at the surface of the pipe there is usually a vortex formation. The profile of average speed in a turbulent flow in the tubes (Fig. 51) differs from the parabolic profile in laminar flow by the rapid increase in the speed of the pipe wall and lesser curvature in the central part of the flow. The nature of the flow depends on the dimensionless quantity called the Reynolds number (O.Reynolde (1842-1912) – British scientist):


vd vd ,   

where v = /p – kinematic viscosity;  – density of the liquid; – average velocity of the fluid by pipe section; d – characteristic linear dimension, such as diameter of the pipe.

Figure 51. Laminar and turbulent flow of the fluid

For small values of the Reynolds number (Re ≲ 1000) there is a laminar flow, the transition from laminar to turbulent flow occurs in the region 1000 ≲ Re ≲ 2000 and at Re = 2300 (for smooth tubes) the flow – turbulent. If Reynolds number is the same, the flow regime of various fluids (gas) in the same sections of different pipes is equal. 103

32. Methods of viscosity determining 1. Stokes method *. This method of viscosity determining is based on speed measurement of moving slowly in a liquid of small bodies of spherical shape. On the ball falling into the liquid vertically downward, three forces act: the force of gravity Р = 4/3R3G ( – density of the ball), the Archimedes force FA= 4/3R3G (' – density of the liquid) and the resistance force, empirically set by J. Stokes:P=6RV, where r – radius of the ball, v – speed. At uniform movement of the ball

4 4 P  FA  F or r 3 g  r 3g  6rv 3 3 2   g r 2 v . 9 By measuring the speed of uniform motion of the ball, we can determine the viscosity of the liquid (gas). 2. Poiseuille method. This method is based on a laminar flow in a thin capillary. Consider capillary with radius R and a length l. In liquid mentally isolate a cylindrical layer of radius r and thickness dr (Fig. 52). The force of internal friction (see. (1.31)) acting on the lateral surface of this layer,

F  

dv dv dS   2rl , dr dr

where dS – the lateral surface of the cylindrical layer; a minus sign means that with increasing radius the speed decreases.

Figure 52


For steady flow of fluid the viscous force acting on the lateral surface of the cylinder is balanced by the force of pressure acting on its base: dv   * 2rl  pr 2 , dv   p rdr . dr 2l After integration, believing that there is a sticking liquids to the wall, i.e. the speed at a distance R from the axis is equal to zero, we get p 2 R  r 2  . v 4l This shows that the liquid particles velocity are distributed according by a parabolic law, the apex of the parabola lies on the axis of the tube (see. also Fig. 53). During the time t from the tube will drained fluid, volume of which R

2pt R 2 2 pt  r 2 R 2 r 4  R 4 pt . V   vt * 2rdr      r R  r dr  4l 0 2l  2 4 0 8l 0 R

From this the viscosity


R 4 pt 8Vl


33. Motion in liquids and gases One of the most important tasks of aerodynamics and hydrodynamics is the study of the motion of solids in gas and liquids, in particular the study of the forces with which the medium acts on the moving body. This problem has become particularly important due to the rapid development of aviation and increase in the speed of ships. On a body moving in a liquid or a gas, two forces (the resultant of their denoted R) act, one of which is directed in the direction oppo105

site to movement of the body (in flow direction), – the drag, and the second is perpendicular to this direction – the lifting force (Fig. 53).

Figure 53. The drag and the lifting force

If the body is symmetric and its symmetry axis coincides with the direction of the velocity, thenon it only drag acts, the lifting force in this case is zero. It can be proved that in a perfect fluid the uniform motion is without drag. If we consider the motion of the cylinder in same liquid (Fig. 54), the current line pattern is symmetrical both with respect to the line passing through points A and B, as well as with respect to the line passing through points C and D, i.e. the resultant pressure force on the surface of cylinder is zero.

Figure 54

The situation is different at bodies motion in a viscous fluid (especially with increasing flow rate). Due to the medium viscosity in the region adjacent to the body surface, the boundary layer of particles is formed moving at slower speeds. As a result of the inhibitory effect of this layer there is the rotation of the particlesand the fluid motion in the boundary layer becomes a vortex. If the body has not a streamlined shape (not smoothly thinning the tail part), the boundary 106

layer of the liquid separates from surface of the body.After the body there is a flow of liquid (gas), the direction of which is opposite to the oncoming flow. Detached boundary layer, following this course, forming vortices rotating in opposite directions (Fig. 55).

Figure 55. Vortices in fluid

Drag depends on the shape of the body and its position relative to the flow that is taken into account by the dimensionless coefficient Cx resistance determined experimentally:

Rx  C x

v 2 2



where  – density of the medium; v – velocity of the body; S – largest cross section of the body. Rx component can be significantly reduced by choosing a body of such shape, which is not conducive to the formation of vortices. The lifting force can be determined by the formula similar to (33.1): v 2 , Ry  C y S 2 where Су – dimensionless coefficient of lifting force. For a wing of the aircraft it is neededlarge lifting force at low drag (this condition is satisfied at small angles of attack  (angle to the flow), see Fig. 55.). Wing better satisfy this condition, the higher 107

the value of K=Cl /Cx called wing quality. Great achievements in the construction of the required wing profile and the study of the influence of the body geometric shape on the coefficient of lifting force belong to the «father of Russian aviation» N.E. Zhukovsky (1847-1921). Control questions by the theme 1. Pressure and unit of its measurement in SI. 2. Pascal's Law. 3. List out-of-system units of pressure. 4. Law of Archimedes. 5. Bernoulli’s law 6. Laminar and turbulent flow 7. Torricelli’s equation 8. What are the vortices in fluid

Problems 6.1. The hollow iron sphere (p =7.87 g/cm3) weighs 5 Nin air and 3 N in water ('=1 g/cm3). Neglecting buoyancy force of the air, determine the volume of the internal cavity of the ball. [139 cm3] 6.2. The cylindrically shaped buck of base area S=1 m2 and the volume of V=3m3 is filled with water. Neglecting the viscosity of water, determine the time (required for emptying the tank, if on the bottom of the buck a circular hole with the squareS=10 cm2have formed.  1 t  S 1 

 2SV  13 min  . g 

6.3. Fountain nozzle, jet vertical FLOW WITH height H=5 M HASfrustoconical tapering upwards. Diameter of lower section d1=6 cm, upper -d2=2 cm. Nozzle height hN = 1 m.NEGLECTING the air resistance IN THE FLOW and the resistance in the jet nozzle, determine: 1) the flow of water in a 1S supplied BY A FOUNTAIN 2) the difference in pressure р IN lower section and atmospheric pressure. Density of water 1 g/cm3.[1) 2 gH d 2 4  3,1  103 м3/с; 2) p  gh  gH 1  d 42 d14   58,3 kPa] SHAPE,

6.4. On a horizontal surface it stands a cylindrical vessel in lateral surface of which there is a hole. The cross section of the hole is much smaller than the cross section of the vessel. The hole is located at a distance h1= 64 cm below the water level in the vessel, which is kept constant, and a distance h2=25 cm from the bottom of the vessel. Neglecting the viscosity of water, determine how far horizontally from the vessel falls on the surface the stream flowing out of the hole. [80 cm]


6.5. In a broadly vessel filled with glycerol (density =1.2 g/cm3) falls by steady velocity of 5 cm/ s glass ball ('=2,7 g/cm3) of 1 mm diameter. Determine the dynamic viscosity of glycerin. [1.6 Pa-s] 6.6. Into the lateral surface of the cylindrical vessel, mounted on a table, is inserted at a height h1 =5 cm from the bottom the capillary with inner diameter d=2 mm and length l=1 cm. In vessel is maintained a constant level of engine oil (density = 0,9g/cm3, dynamic viscosity =0,1 Pаs) at a height h2=80 cm above the capillary. Determine how far horizontally from the end of the capillary falls on the surface of table the oil stream table, arising out of the hole. [ r  d 2 h2 2 gh1 32l   8,9 сm] 6.7. Determine the highest speed that can get freely falling in the air (=1,29g/сm3) steel ball('=9 g/сm3) with the mass m=20 g. The coefficient Cx is taken equal to 0.5. [94 cm/sec]




34. Galileo transformations. Mechanical principle of relativity In classical mechanics, the mechanical principle of relativity is valid (the principle of Galilean relativity): laws of dynamics are the same in all inertial reference systems. To prove this, consider two reference frames: inertial system K (with coordinates x, y, z), which is conventionally assumed to be fixed, and the system K '(with coordinates х, у', z), moving relatively to K uniformly and by straight line with the velocity u (u=const). The time is started from the moment when the origins of both systems coincide. Suppose that at any time t the position of these systems relative to each other has the form shown in Fig. 56. The speed is directed along the OO', the radius vector drawn from O to O', r0=ut

Figure 56

Let’s find the relationship between the coordinates of an arbitrary point A in both systems. From Fig. 58 it is seen that 110

r  r '  r0  r '  ut. The equation (34.1) can be written in projections on the coordinate axes: ' x  x '  u x t y  y '  u y t z  z  u z t. Equations (34.1) and (34.2) are called the Galilean coordinate transformations. In the particular case when the system K'moves with velocity v along the positive direction of the x-axis of system K (at the initial time the axes are the same), the Galilean transformation of coordinates have the form x  x '  vt y  y ' , z  z ' . In classical mechanics is assumed that the time is not depent on the relative motion of reference system, because to the transformations (34.2) we can add else one equation:

t  t '.


These expressions are valid in the case of classical mechanics, when the velocity much less than the light speed and at the velocities closer to the light speed the Galileo transfformations are exchanged by Lorentz transformations. Differentiationg the expression (34.1) by time (taking into account (34.3)), we obtain

v  v '  u,


which is the velocity composition rule in classical mechanics. The acceleration in K system a

dv d v '  u dv '    a'. dt dt dt 111

So, the acceleration of the point A in systems К and К', moving relatively to each other uniformly and by straight line: а = а'.


Therefore, if on body do not act other bodies (a=0), then, according to (34.5), and а'=0, i. e. the system K ' is inertial (point moves relative to it uniformly and in a straight line or at rest). Thus, from the relation (34.5) follows the confirmation of the mechanical principle of relativity: the equation of dynamics in the transition from one frame to another do not change, i.e., they are invariant with respect to coordinate transformations. Galileo noted that any mechanical experiments carried out in the inertial frame of reference, it is impossible to establish whether it is at rest or moves uniformly by a straight line. For example, sitting in the cabin of the ship moving uniformly straight forward, we can not determine if the ship is resting or moving uniformly and bu straight line, not looking out the window.

35. Postulates of special relativity theory Classical Newtonian mechanics perfectly describes the motion of macroscopic bodies moving at low speeds (v«c). However, in the end of XIX century it was turned out that the conclusions of classical mechanics contradict some experimental data, in particular the study of the motion of fast charged particles turned out that their movement is not subject to the laws of mechanics. Then there were difficulties when attempting to apply Newton's mechanics to explain the propagation of light. If the light source and receiver move relative to each other uniformly and by a straight line, according to classical mechanics, the measured speed will depend on the relative speed of their movement. American physicist AA Michelson (1852-1913) in 1881 and then in 1887 with E. Morley (American physicist, 18381923) attempted to detect the motion of the Earth relative to the ether (ether wind) – the experience of Michelson – Morley using an interferometer, later called the Michelson interferometer. There it was nit possible to discover ethereal wind by Michelson, as, indeed, it was 112

failed to detect in numerous other experiments. Experiments "stubbornly" showed that the speed of light in two systems moving relative to each other are the same. This was contrary to the rule of addition of velocities in classical mechanics. At the same time it was shown the contradiction between the classical theory and equations of J. K. Maxwell (English physicist, 1831-1879), underlying in understanding of light as an electromagnetic wave. For an explanation of these and other experimental data, it was necessary to create a new mechanics that explaining these facts, would contain Newtonian mechanics as a limiting case of small velocities. It was able to be done by Einstein, who concluded that the universal ether – a special environment, which could be taken as an absolute system – does not exist. The existence of a constant velocity of propagation of light in a vacuum is in agreement with Maxwell's equations. Thus, Einstein laid the foundations of the special theory of relativity. This theory is a modern physical theory of space and time in which, as in the classical Newtonian mechanics, it is assumed that the time is homogeneous, and the space is homogeneous and isotropic. Special relativity is often referred to as a relativistic theory, and specific phenomena described by this theory – the relativistic effects. The basis of the special theory of relativity, Einstein's postulates are formulated them in 1905 1. The principle of relativity: no experiments (mechanical, electrical, optical), carried out within a given inertial reference system, make it impossible to detect whether the system is at rest or moves uniformly in a straight line; all laws of nature are invariant with respect to the transition from one inertial reference system to another. 2. The principle of invariance of the speed of light: the speed of light in vacuum does not depend on the speed of movement of the light source or the observer, and the same in all inertial reference systems. Einstein's first postulate, as a generalization of the mechanical principle of Galilean relativity to all physical processes, claims, so that the physical laws are invariant with respect to the choice of inertial reference system and equations describing these laws are identi113

cal in form in all inertial reference systems. According to this postulate, all inertial systems of accounts are completely equivalent, i.e. phenomena (mechanical, electromagnetic, optical, etc.) in all inertial frames of reference occur equally. According to Einstein's second postulate, the constancy of the speed of light – a fundamental property of nature, which is stated as an accomplished fact. Special theory of relativity demanded rejection of traditional ideas of space and time, made in classical mechanics, because they contradict the principle of the constancy of the speed of light. It lost the meaning not only the absolute space and absolute time. Einstein's postulates and theory, based on them, have set a new view on the world and new spatiotemporal representations, such as the relativity of lengths and periods of time, the relativity of simultaneity of events. These and other consequences of Einstein's theory are reliable experimental confirmation, being thus justify Einstein's postulates – justification of special theory of relativity. 36. Lorentz transformations Analysis of the phenomena in inertial reference systems, held by Albert Einstein on the basis of the formulated by them postulates showed that the classical Galilean transformations are not compatible with them and therefore must be replaced by the transformations that satisfy the postulates of the theory of relativity. To illustrate this conclusion, let us consider two inertial reference systems: K (with coordinates x, y, z) and K' (with coordinates х, у, z) moving relative to K (along the x axis) with a speed v=const (Fig 57.). Suppose that at the initial time t= t '=0, when the origins of coordinates O and O' coincide, light pulses emit. According to a second Einstein postulate speed of light in both systems is the same and equal to c. Therefore, if during the time t in K system signal reaches a certain point A (Fig. 59), passed a way х =ct,


then in system К' the coordinate of light impulse at the moment when it achieves the point А 114

х' =ct',


where t' – the time of passing of light impulse from the origin to point A in system К'. Deducting (36.1) from (36.2), we obtain х' – х = c(t' – t). Since xx (system К' moves relativelu to К), then

t '  t, i.е. the time count in systems К and К' is different – the count of time is relative (in classical physics the time in all reference systems is equal, i. е. t=t').

Figure 57. The systems К and К'

Einstein have shown that in relativity theory classical Galileo transformations, describing the transition from inertial system to another: K  K'

 x '  x  vt  ' y  y  ' z  z t '  t 

K'  K  x  x '  vt  ' y  y  ' z  z t  t ' 


Are changed by Lorentz transformations, which take into account the Einsteins postulates (formulas are for the case, when К' moves relatively К with the velocity v along х-axis). These transformations are supposed by Lorentz in 19 04 before the occurance of relativity theory, relative to which the Makswells equations are invariant. Lorentz transformations K  K' x  vt x'  1  2

K  K' x  vt x'  1  2

y'  y

y'  y


' '



' '

2 t  vx / c 2 t '  t  vx / c . t  1  2 1  2


Comparing the equations we can conclude that they are symmetrical and differ only in the sign of v. This is obvious, because if speed of K' system relative to K is v, then the velocity of K with respect to K' is – V. From the Lorentz transformation also implies that at low speeds (compared with the velocity c), i.e. when ≪1, they transfer to the classical Galilean transformation (this is the essence of the principle of conformity), which are, therefore, the limiting case of Lorentz transformations. At v>c expressions (36.3) for х, t, x, t' lose the physical sense (become imaginary). That is, in turn, in accordance with the movement with a velocity greater than the velocity of light in vacuum, it is impossible. From the Lorentz transformation it can be concluded that the distance and the time interval between two events change in the transition from one inertial reference system to another, while in the framework of the Galilean transformations, these values were considered absolute, without changing during the transition from system to system. In addition, both spatial and temporal transformations (see 116

(36.3)) are not independent, since coordinate transformation law includes time and time conservation law – spatial coordinates, i.e, set the relationship of space and time. Thus, Einstein's theory operates not with the three-dimensional space, joined by the concept of time, and considers inextricably linked spatial and temporal coordinates, forming a four-dimensional space-time.

37. Consequences from the Lorentz transformations 1. Simultaneity of events in different reference systems. Let in К system in points with coordinates х1 и х2 at the moments of time t1 and t2 two events occur. In К' system their coordinates are х1 and х'2 and times are t'1, and t2. If the events in К system are in one point (x1=x2) and simultaneous (t1 = t2), then, according to Lorentz transformations (36.3),

x1'  x ' t1'  t2' , '2

i.е. these events are simultaneous and coincide in the space for any inertial reference system. If the events in K system are divided in space (х1 x2), but simultaneous (t1 = t2), then in system К', according to Lorentz transformations (36.3), x  vt x  vt ' x1'  1 , x2  2 2 1  2 1 

t1' 

1  vx1 / c 2 1  2

t 2' 

t  vx2 / c 2 1  2

x1'  x2' , t1'  t 2' . Thus, in system K ' these events, while remaining spatially separated, and are not simultaneous. The sign of t2 – t1 is determined by the sign of expression v(x1 - x2), so at various points of K’ system (at different v) the difference t2 – t1 will be different by module and may be differ in sign. Consequently, in one reference systems first 117

event may be earlier than the second, while in other systems of reference, on the contrary, the first precedes the second event. The foregoing, however, does not apply to causal events as we can show that the sequence of causal events is the same in all inertial reference systems. 2. Duration of events in different reference systems. Let in some point (with the coordinate х), in rest relatively to system К, there is an event, duration of which (the difference of clocks counting at the end and beginning of event)  = t2–t1, where the indexes 1 and 2 correspond to the beginning and the end of event. The duration of event in the system К'

 '  t 2'  t1' .


And to the beginning and the end of the event correspond, according (36.3)

t1' 

t1  vx / c 2 1 



t 2' 

t 2  vx / c 2 . 1  2


Putting (37.2) in (37.1), we obtain

 '  t 2  t1  / 1   2 or

 '   / 1  2 .


From (37.3) it follows that , i. e. the duration of the event occurring at some point, the smallest in the inertial reference frame with respect to which this point is fixed. This result can be further interpreted in the following way: the time interval , reckoned by clocks in system K' from the observer's point of view in the system K is longer of interval  , counted at his watch. Consequently, the clock moving relative to the inertial reference system, go slower of resting clocks, i. e. time slows down in the frame, relative to which the clock moves. Based on the concepts of relativity of the "stationary" and 118

«moving» system the ratio for  и ' are reversible. From (37.3) it follows that the delay of the clock becomes greater only at speeds close to the speed of propagation of light in a vacuum. In connection with the discovery of the relativistic effect of slowing down the clocks at the time there was a problem of «clock paradox» (sometimes seen as the «twins paradox»), which caused many discussions. Imagine that performed fantastic space flight to the star at a distance of 500 light years (the distance that the light from the star comes to Earth for 500 years), at a speed close to the speed of light. By earthly clocks flight to the star and back last 1000 years, while for the system of ship and astronaut the same journey takes only 1 year. Thus, the astronaut returns to Earth in times younger than his twin brother, remaining on Earth. This phenomenon, known as the twins paradox, but it is not paradox. The fact that the relativity principle states equivalence is not any reference systems and inertial only. The fallacy of the argument lies in the fact that the frame of reference associated with the twins, are not equivalent: the earth is inertial system and ship – non-inertial, so to them the principle of relativity does not apply. The relativistic effect of clock slowing is very real and received experimental confirmation in the study of unstable, spontaneously breaking up elementary particles in experiments with -mesons. The average lifetime of resting -mesons (by a clock moving with them)   2,210-8 s. Consequently, -mesons produced in the upper atmosphere (at an altitude of 30 kilometers) and moving at speeds close to the speed, would have to pass the distance с  6,6 m , i. e. can not reach the Earth's surface that is contrary to fact. This phenomena can be explained by the relativistic time dilation: for the Earth observer the life time of -meson     1   2 , and the path of these particles in atmoshere v   c   c

1 2 .

Since 1, then v'≫ с. 3. The length of the body in different frames of reference. Consider the rod located along the axis at rest relative to system K'. Rod length in system K' l0=x1- х'2 where х1 and х'2- do not change with time t' coordinates of the beginning and end of the rod, and the index of 0 indicates that the system of reference K ' is at rest. Let’s 119

define the length of the rod in system K relatively to which it moves with a velocity V. For this it is necessary to measure the coordinates of its ends х1 and х2 at the same time t. Their difference l=x2– x1 determines the length of the rod in the system K. Using the Lorentz transformation (36.3), we obtain

l0'  x2'  x1' 

x2  vt 1 


x1  vt 1 


x2  x1 1 




l0' '  l / 1   2 . Thus, the length of the rod, measured in the system relative to which it moves is less than the length, measured in the system relative to which it at rest. If the rod rests in systeme K, then, determining its length in system K', we again come to the expression (37.4). From (37.4) it follows that the linear dimension of the body moving relative to the inertial reference system, decreases in the direction of movement i. e. the so-called Lorentz contraction length is greater, the greater the speed. From the second and third equations of the Lorentz transformation (36.3) it follows that

y 2' '  y1'  y 2  y1 and z 2'  z1'  z 2  z1 , i.e. transverse dimensions of the body do not depend on the speed of its movement and the same in all inertial reference systems. Thus, the linear dimensions of the body most in inertial frame of reference, relative to which the body is at rest. 4. Relativistic law of velocities addition. Let consider the material point motion in system К', moving relatively to system К with the velocity v. Let determine the velocity of this point in system К. If in system К the motion of the point at each moment of time is determined by coordinates х, у, z, and in system К' at moment of time t' – by coordinates х', у', z', then

ux 

dx dz ' dx ' ' dy' ' dz ' dy , uy  ux  ux  ' u y  ' uz  ' dt dt dt dt dt dt 120

are respectively projections on the axes х, у, z and х', у', z' the velocity vector of considering point relatively to systems К and К'. According to Lorentz transformations (36.3),

uy 

dy dy  dy ' ' dz  dz ' , dt '  vdx ' / c 2 . dt  dt 1  2

Doing some transformations, we took relativistic law of velocities addition by special theory of relativity: K  K' ux  uy  uz 

K  K'

u x'  v 1  vu ''x / c 2 u

' y

1 

u x' 



u 'y 

u z' 1   2 y 1  vu x' / c 2

u z' 

1  vu / c ' x


ux  v 1  vu ' x / c 2 uy 1 



1  vu x / c 2 uz 1  2 1  vu x' / c 2



If the material point moves parallel to x-axes, then и relatively to К coincides with их, and u' respectively to К' – with и'х. Then the expression of velocities addition took the form


u v . u'  v ' , u  ' 2 1 v / c2 1  vu / c


It is easily to prove that if the velocities v, u' and u small compared with the velocity c, then formulas (37.5) and (37.6) become to the law of addition of velocities in classical mechanics (see. (34.4)). Thus, the laws of relativistic mechanics in the limiting case of small velocities of the day (compared to the speed of light in vacuum) become the laws of classical physics, which is therefore a special case of Einstein's mechanics for small velocities.


The relativistic velocity addition law obeys Einstein's second cv c postulate. Indeed, if u=c, then (37.6) takes the form u  1  cv c (similarly we can show that for u=c the velocity и' is also equal to c). This result demonstrates that the relativistic velocity addition law is in agreement with the postulates of Einstein. We also prove that if the adding velocities arbitrarily close to the speed c, the resulting speed always less or equal to c. As an example, consider the limiting case u=v = c. After substituting in (37.6) we obtain u=c. Thus, by adding of all the velocities the result can not exceed the speed of light in vacuo. The speed of light in vacuum is a top speed that can not be exceeded. The speed of light in any environment, equal to c/n (n – the absolute refractive index of the medium), is not the limiting value

38. Interval between events Lorentz transformations and their consequences lead to the conclusion about the relativity of lengths and periods of time, the value of which in different frames of reference is different. At the same time, the relative nature of lengths and periods of time in Einstein's theory of relativity is the relativity of individual components of any real physical quantity, independent of the reference system, i.e. which is invariant with respect to coordinate transformations. In four dimensional Einstein space in which each event is characterized by four coordinates (x, y, z, t), a such physical quantity is the interval between two events: S12  c 2 x 2  x1   y 2  y1  z 2  z12 , 2




x2  x12   y2  y12  z 2  z12  l12 – the distance between the

points of three dimensional space in which these events occurred. Introducing the notation t12 = t2 – t1 we obtain


S12  c 2 t12  l122 . 2

We show that the interval between the two events is the same in all inertial reference systems. Designating t = t2 – t1, x = x2–x1, y = y2–y1 and z=z2-z1 the expression (38.1) can be written as

s122  c 2 (t ) 2  (x) 2  (y) 2 (z ) 2 . The interval between the same events in the system K’ is

s 

\ 2 12

 c 2 (t \ ) 2  (x '\ ) 2  (y \ ) 2 (z \ ) 2 .


According to Lorentz transformations (36.3), x  vt y \  y, z \  z, t \  t  vx / c . x  , 1  2 1  2 2


Put this values in (38.2), after elementary transformations we obtain that

s12 2  c2 t 2  x 2  y 2  z 2 , i.е.

s 

\ 2 12

2  s12 .

Summarizing the results, we can conclude that the interval defining spatio-temporal relations between events, is an invariant in the transition from one inertial reference system to another. The invariance of the interval means that, despite the relative lengths and periods of time, the event is an objective and does not depend on the reference system. The theory of relativity thus formulated a new view of space and time. Spatio-temporal relations are not absolute values, as claimed by Galileo's-Newton mechanics, it is relative. Consequently, the concepts of absolute space and time are invalid. In addition, the invari123

ance of the interval between two events suggests that space and time are organically linked and form a single form of existence of matter – space-time. Space and time do not exist outside of matter and independently of it. Further development of the theory of relativity (general theory of relativity, or the theory of gravitation) showed that the space-time properties in this area are defined by the applicable therein gravitational fields. In the transition to a cosmic scale the space-time geometry is not Euclidean (i.e. independent of space-time dimensions of the region), and varies from one region to another, depending on the mass concentration in these areas and their movement.

39. Main law of relativistic dynamics of a material point Mass of moving particles does not depend on their velocities:


m0 1 v2 / c2



where m0 is the rest mass of the particle, i.e., the mass measured in the inertial reference frame with respect to which the particle is at rest; c is the velocity of light in vacuum; m – mass of the particle in the reference frame, with respect to which it moves at a speed v. Consequently, the mass of the same particle is different in different inertial frames. From Einstein's principle of relativity, asserting the invariance of the laws of nature in the transition from one inertial reference system to another, it should be a condition of invariance of physical laws under the Lorentz transformation equations. Newton's fundamental law of dynamics F

dP d  (mv) dt dt

is also invariant with respect to Lorentz transformations, if the time derivative of the relativistic momentum is right. The fundamental law of the relativistic dynamics of a material point is 124


m0 d  dt  1  v 2 / c 2

 v  

or F

dP , dt

where p  mv 

m0 v 1  v 2 /c 2


– relativistic momentum of a material point. Note that equation (39.3) looks the same as the basic equation of Newtonian mechanics (6.7). However, the physical meaning of it is another: to the right is the time derivative of the relativistic momentum, defined by the formula (39.4). Thus, the equation (39.2) is invariant with respect to Lorentz transformations, and hence satisfies the principle of relativity of Einstein. It should be noted that neither the impulse nor force are not invariant quantities. Moreover, in general, the acceleration does not coincide with the direction of force. According to the homogeneity of space in relativistic mechanics the law of conservation of relativistic momentum: the relativistic momentum of a closed system is conserved, i.e., does not change over time. Often it does not stipulate that the relativistic momentum is considered, as if the body move at speeds close to c, only the relativistic expression for momentum can be used. Analysis of the formulas (39.1), (39.4) and (39.2) shows at velocities much lower light speed, the equation (39.2) transits into fundamental law of classical mechanics. Consequently, the condition of the applicability of the laws of classical (Newtonian) mechanics is the condition v