Mechanics And Theory Of Relativity | A. N. Matveev

About the book:

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).

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 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 non-inertial 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 for testing the level of understanding of the material are also given in each section. The material is supplemented by a large number of diagrams. The appendices contain the necessary material for reference.

The book was translated from the Russian by Ram Wadhwa and was published first by Mir Publishers in 1989.

Contents

Chapter 1.

Introduction

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.

Dimensions 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.

Chapter 2.

Kinematics of a Point and a Rigid 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 i_x, i_y and i_z·

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

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 motion.

Rotational motion.

Angular velocity vector.

Fundamental angular displacement vector.

Angular acceleration.

Instantaneous axis of rotation.

Problems

Answers

Chapter 3.

Coordinate Transformations

Sec. 10. Relativity Principle 76

Geometrical coordinate transformations.

Physical coordinate transformations.

Inertial reference frames and the relativity principle.

Sec. 11. 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 interpretations of the 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 experiment 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

Chapter 4.

Corollaries of Lorentz Transformations

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 of a 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

Chapter 5.

Dynamics of a Point Mass

Sec. 18. Forces 134

Origin of the concept of force.

Interactions.

Measurement of force.

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 of force and acceleration in the relativistic case.

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

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.

Momental equation for a system of point masses.

Problems

Answers

Chapter 6.

Conservation Laws

Sec. 22. 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 field.

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

Chapter

7. Noninertial Reference Frames

Sec. 27. Inertial Forces 195

Definition of noninertial 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.

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

Chapter 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. 33. 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.

Gyroscopes.

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.

Fluid 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

Chapter 9.

Dynamics of Bodies of Variable Mass

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

Chapter 10.

Collisions

Sec. 39. Description of Collision Processes 279

Definition of collision.

Diagrammatic representation of collision processes.

Conservation 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.

Sec. 41. Inelastic Collisions 292

General properties of inelastic collisions. Inelastic collision of two particles.

AbSorption of a photon. Emission of a photon.

Sec. 42. Reactions of Subatomic Particles 295

Threshold energy.

Activation energy.

Transition to laboratory coordinate system.

Threshold of generation of \pi^{0}-mesons.

Threshold of generation of proton-antiproton pairs.

Role of collisions in physical investigations.

Problems

Answers

Chapter 11.

Motion in a Gravitational Field

Sec. 43. Gravitational Forces 301

Newton's law of gravitation.

Gravitational field near the Earth's surface.

Gravitational energy of a spherical object.

Gravitational radius.

Size of the Universe.

"Black holes".

Sec. 44. Motion of Planets and Comets 306

Equation of motion.

Momental equation.

Plane of motion.

Kepler's second law.

Kepler's first law.

Kepler's third law.

Motion of comets.

Repulsion of light rays in the gravitational field of the Sun.

Interplanetary flights.

Sec. 45. Motion of the Earth's Artificial Satellites 317

Distinction between the laws of motion of the Earth's artificial satellites and

Kepler's laws of planetary motion.

Trajectory of a satellite.

The shape of the Earth.

Atmospheric drag.

Geostationary orbit.

Sec. 46. The Two-Body Problem 323

Reduced mass.

Transition to the centre-of-mass system.

Tides.

Problems

Answers

Chapter 12.

Motion in an Electromagnetic Field

Sec. 47. Properties of Electromagnetic Forces 329

Lorentz force.

Potential of an electrostatic field.

Equation of motion.

Sec. 48. Motion in Stationary Electric and Magnetic Fields 332

Motion in a uniform magnetic field.

Motion in a transverse nonuniform magnetic field.

Motion in a longitudinal electric field.

Motion in a transverse electric field.

Small deviations. Drift in crossed electric and magnetic fields.

Drift in a nonuniform magnetic field.

Drift due to the curvature of the magnetic field line.

Magnetic moment.

Adiabatic invariance of the magnetic moment.

Magnetic mirrors.

Radiation belts of the Earth.

Sec. 49. Motion in Variable Electromagnetic Fields 349

Motion in the field of a plane electromagnetic wave.

Motion in a variable electric field and a constant magnetic field.

Case 1.

Case 2.

Case 3.

Case 4.

Problems

Answers

Chapter 13.

Oscillations

Sec. 50. Harmonic Oscillations 355

Role of harmonic oscillations in nature.

Equation of harmonic oscillations.

Harmonic functions.

Amplitude, phase and frequency.

Complex representation of harmonic oscillations.

Addition of harmonic oscillations with the same frequency.

Addition of harmonic oscillations with nearly equal frequencies.

Beats.

Sec. 51. Natural Oscillations 363

Definition.

Initial conditions.

Energy.

Relation between displacement, velocity and acceleration.

Nonlinear oscillations.

General condition of harmonicity of oscillations.

Example 51.1.

Computation of the period of oscillations.

Example 51.2.

Sec. 52. Damped Oscillations 373

Friction.

Equation of motion.

Frequency and damping decrement.

Logarithmic decrement.

Large friction (\gamma >> \omega_{0}).

Analysis of damping based on energy losses due to friction.

Example 52.1.

Damping in the case of dry friction.

Example 52.2.

Damping in the case of arbitrary frictional forces.

Sec. 53. Forced Oscillations. Resonance 380

External force.

Equation of motion.

Transient conditions.

Steady-state forced oscillations.

Amplitude-frequency characteristic. Case I. Case 2. Case 3.

Q-factor.

Phase characteristic.

Periodic anharmonic force.

Aperiodic force.

Resonance of nonlinear oscillations.

Sec. 54. Self-Excited Oscillations and Parametric Oscillations 391

Definition.

Self-excited oscillations of a pendulum.

Relaxation oscillations.

Parametric excitation of oscillations.

Sec. 55. Oscillations of Coupled Systems 395

Systems with many degrees of freedom.

Coupled systems.

Normal oscillations of coupled systems.

Problems

Answers

Appendix 1. SI Units Used in the Book 402

Appendix 2. Physical Constants Encountered in the Book 403

Conclusion 404

Subject Index 409