Matveev - Mechanics - and - Theory - of - Relativity

Matveev - Mechanics - and - Theory - of - Relativity

(Parte 1 de 8)

Mechanics and Theory of Relativity Mechanics and Theory of Relativity

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AN.Matveev

Mechanics and Theory of Relativity

Mir Publishers Moscow ,

Translated from Russian by Ram Wadhwa

First published 1989 Revised from the 1986 Russian edition

Ha auz!luiicKoM J13b1Ke Printed in the Union of Soviet Socialist Republics

ISBN 5-03-000267-7

@ fhnaTeJibCTBO «8biC1UaSI lUKOJia», 1986 © English translation, Mir Publishers, 1989

Preface

This is the first volume (second Russian edition) of a course on general physics (the second, third and fourth volumes were published in 1985 (Molecular Physics), 1986 (Electricity and Magnetism) and 1988 (Optics) respectively; the fifth volume (Atomic Physics) is under preparation).

The task of a course on physics can be summed up as follows. Firstly, it should contain the basic principles and laws of physics and their mathematical formulation, introduce the basic physical phenomena, methods of their investigation and experimental studies, a proper form of expression for physical ideas, a quantitative formulation and solution of physical problems, estimates of the order of physical quantities, and a clear idea about the limits of applicability of physical models and theories. Secondly, it should inculcate in the studies a skill in experimental work, indicate the main methods of exact measurement of physical quantities and simple ways of analyzing the experimental results and basic physical instruments. Thirdly, it should provide an insight into the philosophical and methodological problems of modern physics and describe the various stages of evolution .(5f science. Finally, it should point out the true role of physics in the scientific and technical progress and arouse the student's curiosity, interest and ability to solve scientific, engineering and other applied problems.

These problems can be solved only through a proper combination of experimental and theoretical instruction. Experimental skill is acquired in laboratories with the help of appropriate practical guides for laboratory work. This book provides the theoretical background. Of course, it also contains a description and analysis of physical phenomena, measurement of physical quantities, experimental methods of investigation, and other allied problems, but only from the point of view of theoretical understanding.

The curriculum of physics education in colleges at present aims at strengthening the basic· level of knowledge. Physics is a leading discipline among fundamental sciences. Hence this book contains a detailed material on the measurement and determination of physical quantities, the role of abstractions, and the methods of physical investigation. Kinematics is treated not as a mathematical theory, but from a physical

6 Preface point of view. This allows the introduction of relativistic concepts of space and time, as well as Lorentz transformations, right at the beginning of the book. Consequently, the concepts of space and time, motion and material are linked inseparably in kinematics. The physical content of Newton's laws is described in detail, different methods of substantiation of mechanics are reviewed critically, and the connection between the conservation laws and symmetry of space and time is established in a comprehensible form.

A modern specialist should not only acquire the basic skills, but also learn to effectively apply the results of physical studies to accelerate the pace of scientific progress. In this connection, we have also considered in this book problems like motion in noninertial reference frames, inertial navigational systems, gyroscopic phenomena, motion of the artificial Earth's satellites, dynamics of bodies of variable mass, motion in electromagnetic fields, relation between mass and energy.

The same methodological approach has been used in writing all the volumes of this course. Each chapter contains a resume of the basic ideas, and each section contains a formulation of the crux of the problems discussed in it. Examples have been chosen in such a way that they illustrate the methods of solving the most important problems. Problems for independent work-out are included at the end of each chapter and answers are also provided. Brief formulations of the most important statements and formulas are provided throughout the book, and questions fqf testing the level of understanding of the material are also gi.ven in each section. The material is supplemented by a large number of diagrams. The appendices contain the necessary material for reference.

The author is grateful to Prof. S. P. Kapitza and to the staff of the department chaired by him for a careful review of the manuscript and for valuable comments.

A. N. Matveev

1. Introduction Contents

Sec. 1. Problems and Experimental Methods in Physics 13

Problems of physics. Abstraction and limitedness of models. Experimental methods in physics.

Sec. 2. Physical Quantities and Their Measurement 16

Difference and comparison. Comparison and measurement. Measurement. Units of physical quantities. Number of units of physical quantities.

Sec. 3. On the Definition of Concepts and Quantities in Physics 17

Two categories of concepts used in physics. Two ways of defining physical quantities. On general concepts.

Sec. 4. Systems of Units of Physical Quantities 19

Base and derived units. DimeHsions of a physical quantity. Selection of base units. Number of base units. Arbitrariness in the choice of the system of units. The international system of units (SI). Second, the unit of time. Metre, the unit of length.

Kilogram, the unit of mass. Out-of-system units. Prefixes for fractional and multiple units. Dimensional analysis.

2. Kinematics of a Point and a Rigjd Body

Sec. 5. Coordinate Systems 28

Space and geometry. Geometry and experiment. Point mass. Body. Distance between points. Perfectly rigid body. Reference frame. Coordinate systems. Dimensionality of space. Important coordinate systems. Coordinate transformations.

Sec. 6. Vectors 37

Definition of a vector. Addition of vectors and multiplication of a vector by a scalar. Scalar product. Vector product. Representation of vectors in terms of a unit vector. Advantages of vector notation. Radius vector. Vector projections in the

Cartesian coordinate system. Relation between vectors,,,;, and =· Computation of vector projections. Expression of vector operations in coordinate form.

Transformation of Cartesian coordinates. Transformation of vector projections. Physical vector.

Sec. 7. Time 46 Concept of time. Periodic processes. Synchronization of clocks.

Sec. 8. Displacement, Velocity and Acceleration of a Point 52

Methods of describing motion. Coordinate form of motion. Vector notation of motion. Description of motion with the help of trajectory parameters. Displacement vector. Velocity. Acceleration.

Sec. 9. Kinematics of a Rigid Body 63

motion. Rotational motionAngular velocity vector. Fundamental angular

Degrees of freedom. Degrees of freedom of a rigid body. Decomposition of the motion of a rigid body into components. Euler angles. Translational motion. Plane displacement vector. Angular acceleration. Instantaneous axis of rotation. Problems Answers

8 Contents 3. Coordinate Transformations

Sec. 10. Relativity Principle 76

Geometrical coordinate transformations. Physical coordinate transformations. Inertial reference frames and the relativity principle.

Sec. 1. Galilean Transformations 78

Galilean transformations. Invariants of transformations. Invariance of length. Universality of the concept of simultaneity. In variance of time interval. Velocity summation. Invariance of acceleration.

Sec. 12. Constancy of the Velocity of Light 82

Experimental verification of the validity of Galilean transformations. Evolution of views about the velocity of light. Determination of the velocity of light by Roemer. Aberration of light. Various interpretatiqns ofthe velocity of light. Absolute ether and absolute velocity. Measurements of "absolute" velocity. The Michelson-

Morley experiment. Calculation of the path difference between rays. Results of the Michelson-Morley experiment. Interpretation of the Michelson-Morley experi- ment based on the concept of ether. Ballistic hypothesis. Flaw in the ballistic hypothesis. Incompatibility between the constancy of the velocity of light and conventional concepts. The idea behind Fizeau's experiment. Calculation of the path difference between rays. Result of Fizeau's experiment. Constancy of the velocity of light as a postulate.

Sec. 13. Lorentz Transformations 97

Postulates. Linearity of coordinate transformations. Transformations for y and z. Transformations for x and t. Lorentz transformations. Galilean transformations as limiting case of Lorentz transformations. Four-dimensional vectors.

Problems Answers

4. Corollaries of Lorentz Transformat!ons

Sec. 14. Relativity of Simultaneity ,.107

Definition. Relativity of simultaneity and causality. Interval invariance. Spatially similar and time-similar intervals.

Sec. 15. Length of a Moving Body 112

Definition of the length of a moving body. Formula for the reduction in the length of a moving body. Change in the shape ofa moving body. Estimation of the magnitude of contraction. On the reality of contraction of a moving body. On contraction and perfect rigidity of a body.

Sec. 16. Pace of Moving Clocks. Intrinsic Time 118

Slowing down of the pace of moving clocks. Intrinsic time. Experimental confirmation of time dilatation. Pace of clocks moving with an acceleration. Twin paradox.

Sec. 17. Composition of Velocities and Transformation of Accelerations 127

Formula for composition of velocities. Aberration. Interpretation of Fizeau's experiment. Transformation of accelerations. Problems Answers

5. Dynamics of a Point Mass

Sec. 18. Forces 134 Origin of the concept of force. Interactions. Measurement of force.

Contents 9

Sec. 19. Newton's Laws 136

How many of Newton's laws of motion are independent in nature? Mass. On Newton's second law of motion. On Newton's third law of motion.

Sec. 20. Relativistic Equation of Motion 146

Inertia in the direction of velocity and perpendicular to the velocity. Relativistic equation of motion. Nonalignment offort:e and acceleration in the relativistic case.

Sec. 21. Motion of a System of Point Masses 150

6. Conservation Laws

System of point masses. Angular momentum of a point mass. Moment of force acting on a point mass. Momenta! equation for a point mass. Momentum of a system of point masses. Angular momentum of a system of point masses. Force acting on a system of point masses. Moment of force acting on a system of point masses. Equation of motion for a system of point masses. Centre of mass. Inapplicability of the concept of the centre of mass in the relativistic case. Momenta! equation for a system of point masses. Problems Answers

Sec. 2. Significance and Essence of Conservation Laws 161

Essence of conservation laws. Equations of motion and conservation laws. Mathematical meaning of the mechanical conservation laws.

Sec. 23. Momentum Conservation Law 163

Isolated system. Momentum conservation law for an isolated system. Conservation laws for individual projections of momentum. Application of the momentum conservation law.

Sec. 24. Angular Momentum Conservation Law 165

Formulation of the law. Conservation laws for individual projections of angular momentum.

Sec. 25. Energy Conservation Law 167

Work done by a force. Potential forces. Mathematical criterion for the potential nature of a field. Work in a potential ,tield. Normalization of potential energy. Interaction energy. Applications. Total energy and rest energy. Kinetic energy. Mass-energy relation. Experimental verification of the mass-energy relation. Inertial nature of potential energy. Binding energy. Energy conservation law for a system of point masses.

Sec. 26. Conservation Laws and the Symmetry of Space and Time 188

Momentum conservation law and the homogeneity of space. Angular momentum conservation law and the isotropy of space. Energy conservation law and the homogeneity of time. Universality and general nature of conservation laws. Problems Answers

7. Noninertial Reference Frames

Sec. 27. Inertial Forces 195

Definition ofnonincrtial reference frames. Time and space in noninertial reference frames. Inertial forces. On the reality of existence of inertial forces. Determination of inertial forces.

Sec. 28. Noninertial Reference Frames of Translational Motion in a

Straight Line 198 Expression for inertial forces. Pendulum on a cart. A falling pendulum.

Sec. 29. Zero Gravity. The Equivalence Principle 202 Zero gravity. Gravitational and inertial masses. Equivalence principle. Red shift.

10 Contents

Sec. 30. Noninertial Rotating Reference Frames 206

Coriolis acceleration. Expression for Coriolis acceleration. Inertial forces in a rotating reference frame. Equilibrium of a pendulum on a rotating disc. Motion of a body along a rotating rod. Noninertial reference frame fixed to the Earth's surface. Foucault pendulum. Conservation laws in noninertial reference frames.

Problems - Answers

8. Dynamics of a Rigid Body

Sec. 31. Equations of Motion 218

System of equations. Proof of the closure of a system of equations for a rigid body. Choice of a coordinate system.

Sec. 32. Moments of Inertia 220

Inertia tensor. Principal axes of the inertia tensor. Determination of principal axes. Computation of the moment of inertia about an axis. Huygens' theorem.

Sec. 3. Kinetic Energy of a Rotating Rigid Body 226

Expressing the inertia tensor with the help of the Kronecker delta. Kinetic energy of rotation.

Sec. 34. Plane Motion. Pendulums 229

Peculiarities of the dynamics of plane motion. Rolling of a cylinder down an inclined plane. Maxwell's pendulum. Physical pendulum.

Sec. 35. Motion of a Rigid Body Fixed at a Point. Gyroscopes 240

Choice of the coordinate system. Euler's equations. Free axes. Nutation. Gy- roscopes. Precession of a gyroscope. Direction and velocity of precession. Gyroscopic pendulum. Egg-shaped top. Powered gyroscope. Gyroscopic forces.

Sec. 36. Motion under Friction 253 .

Dry friction. Flujd friction. Work of frictional forces. Stagnation. Skidding. Limiting velocity. Approaching the limiting velocity. Free fall of bodies in air.

Rolling friction. Self-propelled means of transport. On nature of frictional forces. Problems Answers

9. Dynamics of Bodies of Variable Mass

10. Collisions

Sec. 37. Nonrelativistic Rockets 268

Reaction propulsion. Equation of motion. Tsiolkovsky formula. Multistage rockets. Characteristic velocity.

Sec. 38. Relativistic Rockets 274

Equation of motion. Dependence of final mass on velocity. Photon rockets. Problems

Answers

Sec. 39. Description of Collision Processes 279

Definition of collision. Diagrammatic representation of collision processes. Conser- vation laws and collisions. Momentum conservation law. Energy conservation law. Angular momentum conservation law. Elastic and inelastic collisions. Centre-of-mass system.

Sec. 40. Elastic Collisions 285

Collision of two particles in a nonrelativistic case. Head-on collision. Moderation of neutrons. Compton effect.

(Parte 1 de 8)

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