ISBN: 354067652X
TITLE: Elements of Newtonian Mechanics
AUTHOR: Knudsen, Hjorth
TOC:

1. The Foundation of Classical Mechanics 1
1.1 Principia 1
1.2 Prerequisites for Newton 1
1.3 The Masterpiece 5
The Acceleration of Gravity 11
Circular Motion 12
Communication Satellite 13
Horizontal Throw 14
The Gravitational Constant 17
String Force 18
Forces and Tension 19
Dimensional Analysis 20
1.4 Concluding Remarks 21
1.5 Problems 22
2. Newton's Five Laws 27
2.1 Newton's Laws of Motion 27
2.2 Integration of the Equation of Motion 31
Constant Force 31
The Harmonic Oscillator 33
Mass on a Spring in the Gravitational Field of Earth 36
Sphere Falling Through a Liquid 38
Solid Against Solid 40
The Atwood Machine 41
Force in Harmonic Motion 42
Charged Particle in a Uniform Magnetic Field 44
Thomson's Experiment 48
Work and Energy in Linear Motion of a Particle 51
Free Fall Towards the Sun from a Great Distance 53
Momentum Conservation 54
Inelastic Collisions 56
Rocket Propulsion 57
Some Qualitative Remarks on Rocket Propulsion 59
Ball Against a Wall 61
2.3 Problems 62
3. Gravitational and Inertial Mass 73
3.1 Gravitational Mass 73
3.2 Inertial Mass 74
3.3 Proportionality Between Inertial and Gravitational Mass 75
3.4 Newton's Experiment 76
The Satellite 78
An Elevator in Free Fall 79
Three Balls 80
3.5 Problem 80
4. The Galilei Transformation 83
4.1 The Galilei Transformation 84
4.2 Galileo Speaks 90
Velocity Transformation 91
4.3 Problems 93
5. The Motion of the Earth 95
5.1 Examples 95
Vectors and the Rotation of a Rigid Body 95
Angular Velocities in the Solar System 97
5.2 Problems 99
6. Motion in Accelerated Reference Frames 101
6.1 Newton's 2nd Law Within Accelerated Reference Frames 101
6.2 The Equivalence Principle of Mechanics 108
6.3 The Einstein Box 111
Balloon in Accelerated Frame 114Mass on an Oscillating Plate 115Pendulum in an Elevator 115
6.4 The Centrifugal Force 116Earth's Orbit Around the Sun 117Grass on a Rotating Disk 117The Variation of g with Latitude 118
6.5 Tidal Fields 121The Roche Limit 126
6.6 The Coriolis Force 128Coriolis Force on a Train 133Particle on a Frictionless Disc 133The Vertical Throw 134
6.7 Tidal Forces and Local Inertial Frames 135Global and Local Inertial Frames 137
6.8 The Foucault Pendulum 137
6.9 Newton's Bucket 142
6.10 Review: Fictitious Forces 145
6.11 Problems 146
7. The Problem of Motion 153
7.1 Kinematic and Dynamic Views of the Problem of Motion 153
7.2 Einstein Speaks 155
7.3 Symmetry 157
7.4 The Symmetry (Invariance) of Newton's 2nd Law 157
7.5 Limited Absolute Space 159
7.6 The Asymmetry (Variance) of Newton's 2nd Law 159
7.7 Critique of the Newtonian View 162
7.8 Concluding Remarks 163
8. Energy 167
8.1 Work and Kinetic Energy 167
8.2 Conservative Force Fields 169
8.3 Central Force Fields 170
8.4 Potential Energy and Conservation of Energy 171
8.5 Calculation of Potential Energy 174
Constant Gravitational Field 175
Spring Force 175
Gravity Outside a Homogeneous Sphere 175
8.6 The Gravitational Field Around a Homogeneous Sphere 176
8.6.1 The Field Around a Spherical Shell 176
8.6.2 A Solid Sphere 179
8.7 Examples 181
Particle on a Frictionless Curve 181
String Force in the Pendulum 182
The Gravitational Potential Outside the Earth 183
Potential Energy Due to Electric Forces 183
A Tunnel Through the Earth 184
The Asymmetry of Nature 186
8.8 Review: Conservative Forces and Potential Energy 188
8.9 Problems 189
9. The Center-of-Mass Theorem 193
9.1 The Center of Mass 193
9.2 The Center-of-Mass Frame 197
9.3 Examples 199
Two Masses Connected with a Spring 199
Inelastic Collisions 202
The Collision Approximation 204
Freely Falling Spring 207
The Wedge 209
9.4 Review: Center of Mass and Center-of-Mass Theorems 212
9.5 Comments on the Conservation Theorems 212
9.6 Problems 213
10. The Angular Momentum Theorem 219
10.1 The Angular Momentum Theorem for a Particle 219
10.2 Conservation of Angular Momentum 221
10.3 Torque and Angular Momentum Around an Axis 223
10.4 The Angular Momentum Theorem for a System of Particles 224
10.5 Center of Gravity 227
10.6 Angular Momentum Around the Center of Mass 228
10.7 Review: Equations of Motion for a System of Particles 230
10.8 Examples of Conservation of Angular Momentum 230
Particle in Circular Motion 230
Rotation of Galaxies, Solar Systems, etc 231
11. Rotation of a Rigid Body 237
11.1 Equations of Motion 237
11.2 The Rotation Vector 238
11.3 Kinetic Energy of a Rotating Disk 23911.3.1 The Parallel Axis Theorem 24011.3.2 The Perpendicular Axis Theorem 242
11.4 Angular Momentum of an Arbitrary Rigid Body in Rotation Around a Fixed Axis 24311.4.1 The Parallel Axis Theorem in General Form 244
11.5 Calculation of the Moment of Inertia for Simple Bodies 24511.5.1 Homogenous Thin Rod 24511.5.2 Circular Disk 24611.5.3 Thin Spherical Shell 24711.5.4 Homogenous (Solid) Sphere, Mass M and Radius R 24811.5.5 Rectangular Plate 249
11.6 Equation of Motion for a Rigid Body Rotating Around a Fixed Axis 25011.6.1 Conservation of Angular Momentum 253
11.7 Work and Power in the Rotation of a Rigid Body Around a Fixed Axis 25411.7.1 Torsion Pendulum 255
11.8 The Angular Momentum Theorem Referred to Various Points 256
11.9 Examples 258Rotating Cylinder 258Falling Cylinder 260The Atwood Machine 262The Physical Pendulum 263The Rod 264
11.10 Review: Linear Motion and Rotation About a Fixed Axis 266
11.11 Problems 267
12. The Laws of Motion 279
12.1 Review: Classical Mechanics 279
12.2 Remarks on the Three Conservation Theorems 280
12.3 Examples 280
Conservation of Angular Momentum 280
Rotating Rod 284
Man on Disk 286
The Sprinkler 287
Rolling 289
YO-YO on the Floor 295
Rolling Over an Edge 297
Determinism and Predictability 299
12.4 Problems 300
13. The General Motion of a Rigid Body 313
13.1 Inertia in Rotational Motion 313The Dumbbell 314Flywheel on an Axis 316Precession of a Gyroscope 318
13.2 The Inertia Tensor 321The Dumbbell Revisited 324
13.3 Euler's Equations 32613.3.1 Derivation of Euler's Equations 328
13.4 Kinetic Energy 330
13.5 Determination of the Principal Coordinate System 331Rotating Dumbbell 333Flywheel 334The Gyroscope : 335Gyroscope Supported at the Center of Mass 340The Earth as a Gyroscope 341
13.6 Problems 342
14. The Motion of the Planets 345
14.1 Tycho Brahe 345
14.2 Kepler and the Orbit of Mars 346
14.2.1 The Length of a Martian Year 347
14.2.2 The Orbit of the Planet Mars 349
14.2.3 Determination of Absolute Distance in the Solar System 351
14.3 Conic Sections 352
14.4 Newton's Law of Gravity Derived from Kepler's Laws 355
14.5 The Kepler Problem 359
14.5.1 Derivation of Kepler's 3rd Law from Newton's Law of Gravity 364
14.6 The Effective Potential 366
14.7 The Two-Body Problem 36714.7.1 The Two-Body Problem and Kepler's 3rd Law 370
14.8 Double Stars: The Motion of the Heliocentric Reference Frame 370
14.9 Review: Kepler Motion 372
14.10 Examples 373Planetary Orbits and Initial Conditions 373Shape and Size of Planetary Orbits 374Motion Near the Surface of the Earth 376Velocities in an Elliptical Orbit 377Hohman Orbit to Mars 378The Face of the Moon (Spin-Orbit Locking) 381
14.11 Problems 385
15. Harmonic Oscillators 389
15.1 Small Oscillations 389
15.2 Energy in Harmonic Oscillators 391
15.3 Free Damped Oscillations 39215.3.1 Weakly Damped Oscillations 39315.3.2 Strongly Damped Oscillations 39415.3.3 Critical Damping 394
15.4 Energy in Free, Weakly Damped Oscillations 395
15.5 Forced Oscillations 396
15.6 The Forced Damped Harmonic Oscillator 398
15.7 Frequency Characteristics 40015.7.1 w << omega_0 A Low Driving Frequency 40015.7.2 w >> omega_0: A High Driving Frequency 40015.7.3 omega =~ omega_0: Resonance 401
15.8 Power Absorption 402
15.9 The Q-Value of a Weakly Damped Harmonic Oscillator 403
15.10 The Lorentz Curve 405
15.11 Complex Numbers 406
15.12 Problems 408
16. Remarks on Nonlinearity and Chaos 411
16.1 Determinism vs Predictability 411
16.2 Linear and Nonlinear Differential Equations 412
Superposition 413
16.3 Phase Space 414
The Simple Harmonic Oscillator 415
Phase Space of the Pendulum 416
Bifurcation in a Nonlinear Model 421
16.4 A Forced, Damped Nonlinear Oscillator 424
16.5 Liapunov Exponents 427
16.6 Chaos in the Solar System 429
16.7 Problems 431
Appendix. Vectors and Vector Calculus 433
Selected References 439
Answers to Problems 441
Index 449
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