Introductory Classical Mechanics, with Problems and Solutions

Introductory Classical Mechanics, with Problems and Solutions
There Once Was a Classical Theory
David Morin
Contents
1 Statics I-1
1.1 Balancing forces . I-1
1.2 Balancing torques . I-5
1.3 Exercises . I-9
1.4 Problems . I-12
1.5 Solutions . I-17
2 Using F = ma II-1
2.1 Newton’s Laws II-1
2.2 Free-body diagrams . II-4
2.3 Solving differential equations II-8
2.4 Projectile motion . II-12
2.5 Motion in a plane, polar coordinates II-15
2.6 Exercises . II-18
2.7 Problems . II-24
2.8 Solutions . II-28
3 Oscillations III-1
3.1 Linear differential equations . III-1
3.2 Simple harmonic motion . III-4
3.3 Damped harmonic motion III-6
3.4 Driven (and damped) harmonic motion III-8
3.5 Coupled oscillators III-13
3.6 Exercises . III-18
3.7 Problems . III-22
3.8 Solutions . III-24
4 Conservation of Energy and Momentum IV-1
4.1 Conservation of energy in 1-D IV-1
4.2 Small Oscillations . IV-6
4.3 Conservation of energy in 3-D IV-8
4.3.1 Conservative forces in 3-D IV-9
4.4 Gravity IV-12
4.4.1 Gravity due to a sphere . IV-12
4.4.2 Tides . IV-14
12 CONTENTS
4.5 Momentum IV-17
4.5.1 Conservation of momentum . IV-17
4.5.2 Rocket motion IV-19
4.6 The CM frame IV-20
4.6.1 Definition . IV-20
4.6.2 Kinetic energy IV-22
4.7 Collisions . IV-23
4.7.1 1-D motion IV-23
4.7.2 2-D motion IV-25
4.8 Inherently inelastic processes IV-26
4.9 Exercises . IV-30
4.10 Problems . IV-41
4.11 Solutions . IV-47
5 The Lagrangian Method V-1
5.1 The Euler-Lagrange equations V-1
5.2 The principle of stationary action V-4
5.3 Forces of constraint V-10
5.4 Change of coordinates V-12
5.5 Conservation Laws V-15
5.5.1 Cyclic coordinates V-15
5.5.2 Energy conservation . V-16
5.6 Noether’s Theorem V-18
5.7 Small oscillations . V-21
5.8 Other applications V-24
5.9 Exercises . V-27
5.10 Problems . V-29
5.11 Solutions . V-34
6 Central Forces VI-1
6.1 Conservation of angular momentum VI-1
6.2 The effective potential VI-3
6.3 Solving the equations of motion . VI-5
6.3.1 Finding r(t) and θ(t) . VI-5
6.3.2 Finding r(θ) . VI-6
6.4 Gravity, Kepler’s Laws VI-6
6.4.1 Calculation of r(θ) VI-6
6.4.2 The orbits . VI-8
6.4.3 Proof of conic orbits . VI-10
6.4.4 Kepler’s Laws . VI-11
6.4.5 Reduced mass . VI-13
6.5 Exercises . VI-16
6.6 Problems . VI-18
6.7 Solutions . VI-20CONTENTS 3
7 Angular Momentum, Part I (Constant Lˆ) VII-1
7.1 Pancake object in x-y plane . VII-2
7.1.1 Rotation about the z-axis VII-3
7.1.2 General motion in x-y plane . VII-4
7.1.3 The parallel-axis theorem VII-5
7.1.4 The perpendicular-axis theorem . VII-6
7.2 Non-planar objects VII-7
7.3 Calculating moments of inertia . VII-9
7.3.1 Lots of examples . VII-9
7.3.2 A neat trick VII-11
7.4 Torque . VII-12
7.4.1 Point mass, fixed origin . VII-13
7.4.2 Extended mass, fixed origin . VII-13
7.4.3 Extended mass, non-fixed origin VII-14
7.5 Collisions . VII-17
7.6 Angular impulse . VII-19
7.7 Exercises . VII-21
7.8 Problems . VII-28
7.9 Solutions . VII-34
8 Angular Momentum, Part II (General Lˆ) VIII-1
8.1 Preliminaries concerning rotations . VIII-1
8.1.1 The form of general motion . VIII-1
8.1.2 The angular velocity vector . VIII-2
8.2 The inertia tensor VIII-5
8.2.1 Rotation about an axis through the origin . VIII-5
8.2.2 General motion VIII-9
8.2.3 The parallel-axis theorem VIII-10
8.3 Principal axes . VIII-11
8.4 Two basic types of problems . VIII-15
8.4.1 Motion after an impulsive blow . VIII-15
8.4.2 Frequency of motion due to a torque VIII-18
8.5 Euler’s equations . VIII-20
8.6 Free symmetric top VIII-22
8.6.1 View from body frame VIII-22
8.6.2 View from fixed frame VIII-24
8.7 Heavy symmetric top . VIII-25
8.7.1 Euler angles VIII-25
8.7.2 Digression on the components of ~! . VIII-26
8.7.3 Torque method VIII-29
8.7.4 Lagrangian method VIII-30
8.7.5 Gyroscope with θ˙ = 0 VIII-31
8.7.6 Nutation . VIII-33
8.8 Exercises . VIII-36
8.9 Problems . VIII-384 CONTENTS
8.10 Solutions . VIII-44
9 Accelerated Frames of Reference IX-1
9.1 Relating the coordinates . IX-2
9.2 The fictitious forces . IX-4
9.2.1 Translation force: ¡md2R=dt2 . IX-5
9.2.2 Centrifugal force: ¡m~! £ (~! £ r) IX-5
9.2.3 Coriolis force: ¡2m~! £ v IX-7
9.2.4 Azimuthal force: ¡m(d!=dt) £ r IX-11
9.3 Exercises . IX-13
9.4 Problems . IX-15
9.5 Solutions . IX-17
10 Relativity (Kinematics) X-1
10.1 The postulates X-2
10.2 The fundamental effects . X-4
10.2.1 Loss of Simultaneity . X-4
10.2.2 Time dilation . X-7
10.2.3 Length contraction X-10
10.3 The Lorentz transformations X-14
10.3.1 The derivation X-14
10.3.2 The fundamental effects . X-18
10.3.3 Velocity addition . X-20
10.4 The invariant interval X-23
10.5 Minkowski diagrams . X-26
10.6 The Doppler effect X-29
10.6.1 Longitudinal Doppler effect . X-29
10.6.2 Transverse Doppler effect X-30
10.7 Rapidity X-32
10.8 Relativity without c . X-35
10.9 Exercises . X-39
10.10Problems . X-46
10.11Solutions . X-52
11 Relativity (Dynamics) XI-1
11.1 Energy and momentum . XI-1
11.1.1 Momentum XI-2
11.1.2 Energy . XI-3
11.2 Transformations of E and ~p . XI-7
11.3 Collisions and decays . XI-10
11.4 Particle-physics units . XI-13
11.5 Force XI-14
11.5.1 Force in one dimension XI-14
11.5.2 Force in two dimensions . XI-16
11.5.3 Transformation of forces . XI-17CONTENTS 5
11.6 Rocket motion XI-19
11.7 Relativistic strings XI-22
11.8 Mass XI-24
11.9 Exercises . XI-26
11.10Problems . XI-30
11.11Solutions . XI-34
12 4-vectors XII-1
12.1 Definition of 4-vectors XII-1
12.2 Examples of 4-vectors XII-2
12.3 Properties of 4-vectors XII-4
12.4 Energy, momentum XII-6
12.4.1 Norm . XII-6
12.4.2 Transformation of E,p XII-6
12.5 Force and acceleration XII-7
12.5.1 Transformation of forces . XII-7
12.5.2 Transformation of accelerations . XII-8
12.6 The form of physical laws XII-10
12.7 Exercises . XII-12
12.8 Problems . XII-13
12.9 Solutions . XII-14
13 General Relativity XIII-1
13.1 The Equivalence Principle XIII-1
13.2 Time dilation . XIII-2
13.3 Uniformly accelerated frame . XIII-4
13.3.1 Uniformly accelerated point particle XIII-5
13.3.2 Uniformly accelerated frame . XIII-6
13.4 Maximal-proper-time principle . XIII-8
13.5 Twin paradox revisited XIII-9
13.6 Exercises . XIII-12
13.7 Problems . XIII-15
13.8 Solutions . XIII-18
14 Appendices XIV-1
14.1 Appendix A: Useful formulas XIV-1
14.1.1 Taylor series . XIV-1
14.1.2 Nice formulas . XIV-2
14.1.3 Integrals XIV-2
14.2 Appendix B: Units, dimensional analysis XIV-4
14.2.1 Exercises . XIV-6
14.2.2 Problems . XIV-7
14.2.3 Solutions . XIV-8
14.3 Appendix C: Approximations, limiting cases XIV-11
14.3.1 Exercise XIV-136 CONTENTS
14.4 Appendix D: Solving differential equations numerically XIV-15
14.5 Appendix E: F = ma vs. F = dp=dt XIV-17
14.6 Appendix F: Existence of principal axes XIV-19
14.7 Appendix G: Diagonalizing matrices XIV-22
14.8 Appendix H: Qualitative relativity questions XIV-24
14.9 Appendix I: Lorentz transformations XIV-29
14.10Appendix J: Resolutions to the twin paradox . XIV-32
14.11Appendix K: Physical constants and data . XIV-34
14.11 Appendix K: Physical constants and data
Earth
Mass ME = 5:98 ¢ 1024 kg
Mean radius RE = 6:37 ¢ 106 m
Mean density 5.52 g=cm3
Surface acceleration g = 9:81 m=s2
Mean distance from sun 1:5 ¢ 1011 m
Orbital speed 29:8 km/s
Period of rotation 23 h 56 min 4 s = 8:6164 ¢ 104 s
Period of orbit 365 days 6 h = 3:16 ¢ 107 s
Moon
Mass ML = 7:35 ¢ 1022 kg
Radius RL = 1:74 ¢ 106 m
Mean density 3.34 g=cm3
Surface acceleration 1:62 m=s2 … g=6
Mean distance from earth 3:84 ¢ 108 m
Orbital speed 1:0 km/s
Period of rotation 27:3 days = 2:36 ¢ 106 s
Period of orbit 27:3 days = 2:36 ¢ 106 s
Sun
Mass MS = 1:99 ¢ 1030 kg
Radius RS = 6:96 ¢ 108 m
Surface acceleration 274 m=s2 … 28g
Fundamental constants
Speed of light c = 2:998 ¢ 108 m/s
Gravitational constant G = 6:673 ¢ 10¡11 N m2=kg2
Planck’s constant h = 6:63 ¢ 10¡34 J s
Electron charge e = 1:602 ¢ 10¡19 C
Electron mass me = 9:11 ¢ 10¡31 kg = 0:511 MeV=c2
Proton mass m
p = 1:673 ¢ 10¡27 kg = 938:3 MeV=c2
Neutron mass mn = 1:675 ¢ 10¡27 kg = 939:6 MeV=c2
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