Essential Dynamics and Relativity

330 Pages·2014·3.268 MB·English

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• Explains the main elements of dynamics and special relativity together
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• Presents the mathematics of special relativity in a straightforward way
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K20382 cvr mech.indd 1 10/21/14 8:52 AM
Essential
Dynamics
&
Relativity
K20382_FM.indd 1 10/30/14 11:34 AM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Essential
Dynamics
&
Relativity
Peter J. O’Donnell
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Fellow of St. Edmund’s College, Cambridge
K20382_FM.indd 3 10/30/14 11:34 AM
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2015 by Taylor & Francis Group, LLC
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Version Date: 20141021
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TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
List of Symbols xi
Preface xiii
About the Author xv
Acknowledgments xvii
I Dynamics 1
1 The Galileo–Newton Formulation of Dynamics 3
1.1 Galilean relativity . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Frames of reference . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Galileo’s law of inertia and inertial frames . . . . . . . 10
1.1.3 Galilean transformations . . . . . . . . . . . . . . . . . 12
1.1.4 General Galilean transformations . . . . . . . . . . . . 15
1.2 Newton’s dynamical laws . . . . . . . . . . . . . . . . . . . . 16
1.3 Gravitational and inertial mass . . . . . . . . . . . . . . . . . 19
1.3.1 The weak equivalence principle . . . . . . . . . . . . . 20
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Particle Dynamics in One Dimension 25
2.1 Motion of a particle under a force . . . . . . . . . . . . . . . 25
2.1.1 Potential energy . . . . . . . . . . . . . . . . . . . . . 27
2.1.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Potential energy diagrams . . . . . . . . . . . . . . . . . . . 30
2.3 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Resistive motion . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.1 Vertical motion in a resistive medium . . . . . . . . . 43
2.6 Escape velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Oscillations 57
3.1 Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . 60
3.3 Period of small oscillations . . . . . . . . . . . . . . . . . . . 64
vii
viii Contents
3.4 Damped simple harmonic motion . . . . . . . . . . . . . . . 67
3.5 Damped simple harmonic motion with a forcing term . . . . 71
3.5.1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 The LCR circuit . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Particle Dynamics in Two and Three Dimensions 83
4.1 Projectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.1 Projectiles without air resistance . . . . . . . . . . . . 84
4.1.2 Projectiles with linear air resistance . . . . . . . . . . 87
4.2 Energy and force . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.1 Work and kinetic energy . . . . . . . . . . . . . . . . . 88
4.2.2 Potential energy and conservative force . . . . . . . . 90
4.3 Charged particles in an electromagnetic field . . . . . . . . . 94
4.3.1 Particle in a magnetic field . . . . . . . . . . . . . . . 94
4.3.2 Crossed electric and magnetic fields . . . . . . . . . . 96
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Central Forces and Orbits 101
5.1 Central forces and angular momentum . . . . . . . . . . . . 101
5.2 Circular motion . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.1 Plane polar coordinates . . . . . . . . . . . . . . . . . 104
5.2.2 Motion of a particle in a circular orbit . . . . . . . . . 106
5.2.3 Motion of a particle in a vertical circle . . . . . . . . . 107
5.3 Orbital motion . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4 The inverse square law . . . . . . . . . . . . . . . . . . . . . 111
5.5 The orbital equation . . . . . . . . . . . . . . . . . . . . . . . 114
5.5.1 Orbital paths . . . . . . . . . . . . . . . . . . . . . . . 115
5.5.2 Orbital energy . . . . . . . . . . . . . . . . . . . . . . 118
5.6 Perturbed orbits . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Kepler’s laws of planetary motion . . . . . . . . . . . . . . . 125
5.8 The perihelion precession of Mercury . . . . . . . . . . . . . 127
5.9 Rutherford scattering . . . . . . . . . . . . . . . . . . . . . . 129
5.9.1 Scattering angle and impact parameter . . . . . . . . 130
5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Multi-Particle Systems 137
6.1 Conservation of linear momentum . . . . . . . . . . . . . . . 138
6.2 Conservation of angular momentum . . . . . . . . . . . . . . 140
6.3 The centre of mass frame . . . . . . . . . . . . . . . . . . . . 141
6.3.1 Linear momentum . . . . . . . . . . . . . . . . . . . . 143
6.3.2 Angular momentum . . . . . . . . . . . . . . . . . . . 143
6.3.3 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3.4 A caveat to the torque equation . . . . . . . . . . . . 145
6.3.5 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Contents ix
6.4 The two-body problem . . . . . . . . . . . . . . . . . . . . . 150
6.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.5.1 Elastic collisions in the laboratory (LAB) frame. . . . 155
6.5.2 Elastic collisions in the centre of mass (CM) frame . . 158
6.5.3 Transformations between CM and LAB frames . . . . 160
6.6 Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . 166
6.6.1 The coefficient of restitution. . . . . . . . . . . . . . . 169
6.7 Variable mass problems . . . . . . . . . . . . . . . . . . . . . 171
6.7.1 Rocket motion . . . . . . . . . . . . . . . . . . . . . . 171
6.7.2 A falling raindrop . . . . . . . . . . . . . . . . . . . . 175
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7 Rigid Bodies 181
7.1 Rotation of a rigid body about a fixed axis . . . . . . . . . . 181
7.1.1 Angular velocity and kinetic energy . . . . . . . . . . 181
7.1.2 Moment of inertia . . . . . . . . . . . . . . . . . . . . 183
7.1.3 Angular momentum . . . . . . . . . . . . . . . . . . . 184
7.1.4 Calculating the centre of mass and the moment of iner-
tia of some uniform rigid bodies . . . . . . . . . . . . 185
7.1.5 Parallel axis theorem . . . . . . . . . . . . . . . . . . . 194
7.1.6 Perpendicular axis theorem . . . . . . . . . . . . . . . 196
7.2 Planar motion of a rigid body . . . . . . . . . . . . . . . . . 198
7.2.1 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . 199
7.2.2 Instantaneous centre . . . . . . . . . . . . . . . . . . . 199
7.2.3 The compound pendulum . . . . . . . . . . . . . . . . 202
7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8 Rotating Reference Frames 211
8.1 Rates of change in a rotating frame . . . . . . . . . . . . . . 211
8.2 Newton’s second law in a rotating frame . . . . . . . . . . . 214
8.3 The centrifugal force . . . . . . . . . . . . . . . . . . . . . . 216
8.3.1 Actual and apparent gravitational acceleration . . . . 217
8.4 The Coriolis force . . . . . . . . . . . . . . . . . . . . . . . . 222
8.4.1 Motion of a freely falling body . . . . . . . . . . . . . 224
8.4.2 Foucault’s pendulum . . . . . . . . . . . . . . . . . . . 228
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
II Relativity 235
9 Special Relativity 237
9.1 Inception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.1.1 The Michelson–Morley experiment (a pr´ecis) . . . . . 238
9.2 Einstein’s postulates of special relativity . . . . . . . . . . . 240
9.3 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . 241
9.4 Minkowski diagrams (space-time diagrams) . . . . . . . . . . 245
9.4.1 Calibration hyperbolae and the s2 invariant . . . . . . 246