Figures ...................................................... хiii
Preface ...................................................... xvii
Acknowledgments ............................................... xix
1 Energy, Mass, Momentum ..................................... 1
1.1 Energy ..................................................... 1
1.1.1 Car driving ......................................... 3
1.2 Mass ....................................................... 4
1.3 Momentum ................................................... 7
2 Kinematics, Newton's Laws of Motion ....................... 10
2.1 Whether to Stop or Run Through? ........................... 10
2.2 Vertical Jump ............................................. 13
2.2.1 Height equation: from conservation law ............. 14
2.2.2 Jumps of animals ................................... 15
2.3 Hourglass ................................................. 16
2.4 Motion of a Chain in a Tube ............................... 17
2.5 Forces .................................................... 19
2.5.1 Gravitation ........................................ 19
2.5.2 Electro-weak interaction ........................... 21
2.5.3 Strong interaction ................................. 21
3 Circular Motion ........................................... 23
3.1 Cartesian vs Polar Coordinates ............................ 23
3.2 Coriolis Force ............................................ 27
3.3 Rotation Group ............................................ 28
4 The Principle of Least Action ............................. 29
4.1 Action of the Principle ................................... 29
4.1.1 Mechanics .......................................... 29
4.1.2 Electric circuits .................................. 30
4.1.3 Optics ............................................. 31
4.2 The Principle of Least Action ............................. 32
4.3 More Thoughts on Why "(T - V)"? ........................... 34
5 Work and Energy ........................................... 36
5.1 At the T-junction ......................................... 36
5.2 Motion of a Heavy Particle on a Smooth Curve in
a Vertical Plane .......................................... 39
5.3 Motion of a Heavy Particle, Placed on the Outside of
a Smooth Circle in a Vertical Plane and Allowed to Slide
Down ...................................................... 40
5.4 Motion in a Vertical Plane of a Heavy Particle Attached
by a Fine String to a Fixed Point ......................... 41
5.5 Conservative Force ........................................ 42
5.5.1 Interpretation of grad V ........................... 43
5.5.2 Relation with curl of the force .................... 44
5.6 Work-energy Theorem and Galilean Invariance ............... 44
5.6.1 Galilean invariant ................................. 44
5.6.2 Example ............................................ 46
6 Mechanics of a System of Particles ........................ 48
6.1 Analysing the Leaky Pendulum .............................. 49
6.1.1 Simple 'usual' pendulum ............................ 49
6.1.2 Leaking bob ........................................ 50
6.2 Work-energy Theorem Revisited ............................. 53
6.3 Displacement .............................................. 55
6.4 Rotation .................................................. 56
6.5 Rigid Body Motion: Basic Ideas ............................ 57
6.6 Rotation of a Rigid Body about an Arbitrary Axis .......... 60
6.6.1 Special cases ...................................... 61
6.7 Moments of Inertia of Simple Bodies ....................... 62
6.8 Principal Axes - Stationary Points of Kinetic Energy ...... 64
6.9 Euler's Equations ......................................... 66
7 Friction .................................................. 70
7.1 Non-conservative Forces and Energy Loss ................... 70
7.1.1 Energy loss ......................................... 71
7.2 Bowling - Physics of the Rolling Ball ..................... 72
7.3 Squealing and Squeaking ................................... 77
8 Impulse and Collisions .................................... 78
8.1 Impact of Smooth Spheres .................................. 78
8.1.1 Direct impact ...................................... 78
8.1.2 Poisson's hypothesis ............................... 79
8.1.3 Kinetic energy lost by impact ...................... 80
8.1.4 Generalization of Newton's rule .................... 81
8.2 Forward Karate Punch ...................................... 82
8.2.1 Deformation energy ................................. 82
8.2.2 Impact force ....................................... 83
8.3 Falling Pencil on a Table ................................. 86
9 Central Forces ............................................ 91
9.1 The Two-body Problem ...................................... 91
9.1.1 Bounded orbits ...................................... 94
9.2 Two Bodies Under their Own Gravitational Interaction ...... 96
9.2.1 Case I ............................................. 96
9.2.2 Case II ............................................ 97
9.2.3 Case III ........................................... 98
9.3 Satellite Paradox ......................................... 98
9.3.1 Descending path on a near-circular orbit ........... 99
9.4 Rotation Curves: an Anomaly .............................. 101
9.5 The Rosetta-Philae Comet Mission ......................... 102
10 Dimensional Analysis ..................................... 104
10.1 Black Holes at LHC ....................................... 104
10.2 Nuclear Explosion ........................................ 105
10.3 Insect Right ............................................. 106
11 Oscillations ............................................. 108
11.1 Free Oscillations ........................................ 108
11.2 Transverse Oscillations in Mass-spring System ............ 109
11.3 Compound Pendulum ........................................ 109
11.4 Damped Harmonic Oscillator ............................... 110
11.5 Driven Damped Simple Harmonic Oscillator ................. 111
11.6 Beats .................................................... 114
11.7 Another Instance of Simple Harmonic Motion ............... 114
11.8 Two Coupled Oscillators .................................. 116
11.9 Three Coupled Oscillators ................................ 119
11.10 Many Coupled Oscillators ................................ 120
11.10.1 Phase velocity ................................... 122
11.10.2 Three-dimensional long-range order ............... 122
11.11 Dissipation by a Rapidly Oscillating Potential .......... 122
12 Waves .................................................... 125
12.1 Transverse Modes of a String ............................. 125
12.2 Standing Waves in One Dimension .......................... 127
12.2.1 Reflection and transmission of waves on a string .. 127
12.3 Standing Waves on Planar Membranes ....................... 128
12.4 Speed of Sound in Air .................................... 131
12.4.1 Newton's derivation ............................... 131
12.4.2 Correct derivation (Laplace) ...................... 133
13 Sound of Music ........................................... 136
13.1 Physics of Music ......................................... 136
13.2 Western Classical Music .................................. 139
13.3 Transposition, Musical Scales, and Algebraic Groups ...... 140
14 Fluid Mechanics .......................................... 142
14.1 Equation of Continuity ................................... 144
14.2 Euler's Equation ......................................... 144
14.2.1 Applications ........................................... 145
14.3 Bernoulli's Equation ..................................... 146
14.4 Streamlines .............................................. 146
14.4.1 Applications ...................................... 147
14.5 Speed of Sound Inside a Fluid ............................ 147
14.5.1 Effect of bubbles ................................. 148
14.6 Sound of a Brook ......................................... 150
14.7 Why is Water Watery? ..................................... 151
15 Water Waves .............................................. 153
15.1 Gravity Waves in Liquid .................................. 153
15.1.1 Deep water waves .................................. 155
15.1.2 Shallow water waves (Tsunami) ..................... 155
15.2 Capillary Waves .......................................... 156
16 The Kinetic Theory of Gases .............................. 159
16.1 Equipartition of Kinetic Energy, Ideal Gas Law ........... 160
16.2 Football Game: Kinetic Theory Perspective ................ 162
16.3 Adiabatic Reversible Compression ......................... 164
16.4 Adiabatic Reversible Compression (from Mechanics and
Kinetic Theory) .......................................... 165
16.5 Maxwellian Distribution of Velocities of Gas Molecules ... 167
17 Concepts and Laws of Thermodynamics ...................... 168
17.1 Adiabatic Transitions and Accessibility of States of
a System - Empirical Entropy, First and Second Laws ...... 169
17.2 Sears' Illustration of Caratheodory's Treatment .......... 170
17.2.1 Temperature as a property ......................... 174
17.3 Reversible and Irreversible Adiabatic Processes .......... 174
17.3.1 Reversible process ................................ 174
17.3.2 Carnot cycle ...................................... 175
17.3.3 Irreversibility ................................... 175
17.4 Order or Disorder ........................................ 176
17.5 How Does Entropy Look Like? .............................. 177
18 Some Applications of Thermodynamics ...................... 181
18.1 Thermodynamic Potentials ................................. 181
18.2 Van der Waals Equation for Real Gases .................... 184
18.2.1 Liquefaction of gases ............................. 185
18.3 The Third Law of Thermodynamics .......................... 186
18.4 Gaseous Mixture .......................................... 187
18.4.1 Diffusion ......................................... 187
18.4.2 Law of mass action ................................ 189
18.5 Chemical Potential ....................................... 190
18.6 Van't Hoff Equation of State for Dilute Solutions ........ 192
19 Basic Ideas of Statistical Mechanics ..................... 194
19.1 Gibbs and Boltzmann Entropies ............................ 194
19.1.1 Entropy and "energy-spreading" .................... 197
19.2 Boltzmann Factor: Application to "Phases of Matter" ...... 197
19.2.1 Gases and solids .................................. 198
19.2.2 Liquids ........................................... 200
19.3 Failure of Classical Physics ............................. 201
Bibliography .................................................. 203
Index ......................................................... 209
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