Contents 1Fundamental Concepts: Vectors 1 1.1Introduction 1 1.2Measure of Space and Time: Units and Dimensions 2 1.3Vectors 9 1.4The Scalar Product 15 1.5The Vector Product 19 1.6An Example of the Cross Product: Moment of a Force 22 1.7Triple Products 23 1.8Change of Coordinate System: The Transformation Matrix 25 1.9Derivative of a Vector 30 1.10 Position Vector of a Particle: Velocity and Acceleration in Rectangular Coordinates 31 1.11 Velocity and Acceleration in Plane Polar Coordinates 36 1.12 Velocity and Acceleration in Cylindrical and Spherical Coordinates 39 2 Newtonian Mechanics: Rectilinear Motion of a Particle 47 2.1Newton's Law of Motion: Historical Introduction 47 2.2Rectilinear Motion: Uniform Acceleration Under a Constant Force 60 2.3Forces that Depend on Position: The Concepts of Kinetic and Potential Energy 63 2.4Velocity-Dependent Forces: Fluid Resistance and Terminal Velocity 69 *2.5Vertical Fall Through a Fluid: Numerical Solution 75 3 Oscillations 82 3.1- Introduction 82 3.2 Linear Restoring Force: Harmonic Motion 84 3.3 Energy Considerations in Harmonic Motion 93 3.4 Damped Harmonic Motion 96 *3.5 Phase Space . 106 3.6 Forced Harmonic Motion: Resonance 113 viii Contents *3.7 The Nonlinear Oscillator: Method of Successive Approximations 125 *3.8 The Nonlinear Oscillator: Chaotic Motion 129 3.9 Nonsinusoidal Driving Force: Fourier Series 135 4General Motion of a Particle in Three Dimensions 144 4.1 Introduction: General Principles 144 4.2The Potential Energy Function in Three-Dimensional Motion: The Del Operator 151 4.3Forces of the Separable Type: Projectile Motion 156 4.4The Harmonic Oscillator in Two and Three Dimensions 167 4.5Motion of Charged Particles in Electric and Magnetic Fields 173 4.6Constrained Motion of a Particle 176 5Noninertial Reference Systems 184 5.1 Accelerated Coordinate Systems and Inertial Forces 184 5.2Rotating Coordinate Systems 189 5.3 Dynamics of a Particle in a Rotating Coordinate System 196 5.4Effects of Earth's Rotation 201 5.5 Motion of a Projectile in a Rotating Cylinder 207 5.6The Foucault Pendulum 212 6Gravitation and Central Forces 218 6.1 Introduction 218 6.2 Gravitational Force between a Uniform Sphere and a Particle 223 6.3 Kepler's Laws of Planetary Motion 225 6.4 Kepler's Second Law: Equal Areas 226 6.5Kepler's First Law: The Law of Ellipses 229 6.6Kepler's Third Law: The Harmonic Law 238 6.7Potential Energy in a Gravitational Field: Gravitational Potential 244 6.8Potential Energy in a General Central Field 250 6.9Energy Equation of an Orbit in a Central Field 251 6.10 Orbital Energies in an Inverse-Square Field 251 6.11 Limits of the Radial Motion: Effective Potential 257 6.12 Nearlv Circular Orbits in Central Fields: Stability 260 j J 6.13 Apsides and Apsidal Angles for Nearly Circular Orbits 262 6.14 Motion in an Inverse-Square Repulsive Field: Scattering of Alpha Particles . 264 7Dynamics of Systems of Particles 275 7.1 Introduction: Center of Mass and Linear Momentum of a System 275 7.2Angular Momentum and Kinetic Energy of a System 278 7.3 Motion of Two Interacting Bodies: The Reduced Mass 283 Contents ix *7.4The Restricted Three-Body Problem 288 7.5Collisions 303 7.6Oblique Collisions and Scattering: Comparison of Laboratory and Center of Mass Coordinates 306 7.7Motion of a Body with Variable Mass: Rocket Motion 312 8 Mechanics of Rigid Bodies: Planar Motion 323 8.1Center of Mass of a Rigid Body 323 8.2Rotation of a Rigid Body about a Fixed Axis: Moment of Inertia 328 8.3Calculation of the Moment of Inertia 330 8.4The Physical Pendulum 338 8.5The Angular Momentum of a Rigid Body in Laminar Motion 344 8.6Examples of the Laminar Motion of a Rigid Body 347 8.7Impulse and Collisions Involving Rigid Bodies 354 9 Motion of Rigid Bodies in Three Dimensions 361 9.1Rotation of a Rigid Body about an Arbitrary Axis: Moments and Products of Inertia-Angular Momentum and Kinetic Energy 361 9.2Principal Axes of a Rigid Body 371 9.3Euler's Equations of Motion of a Rigid Body 381 9.4Free Rotation of a Rigid Body: Geometric Description of the Motion 383 9.5Free Rotation of a Rigid Body with an Axis of Symmetry: Analytical Treatment 384 9.6Description of the Rotation of a Rigid Body Relative to a Fixed Coordinate System: The Eulerian Angles 391 9.7 Motion of a Top 397 9.8The Energy Equation and Nutation 401 9.9 The Gyrocompass 407 9.10 Why Lance Doesn't Fall Over (Mostly) 409 10 Lagrangian Mechanics 417 10.1 Hamilton's Variational Principle: An Example 419 10.2 Generalized Coordinates 423 10.3 Calculating Kinetic and Potential Energies in Terms of Generalized Coordinates: An Example 426 10.4 Lagrange's Equations of Motion for Conservative Systems 430 10.5 Some Applications of Lagrange's Equations 431 10.6 Generalized Momenta: Ignorable Coordinates 438 10.7 Forces of Constraint: Lagrange Multipliers 444 10.8 D'Alembert's Principle: Generalized Forces 449 10.9 The Hamiltonian Function: Hamilton's Equations 455 11Dynamics of Oscillating Systems 11.1 Potential Energy and Equilibrium: Stability 11.2 Oscillation of a System with One Degree of Freedom about a Position of Stable Equilibrium 11.3 Coupled Harmonic Oscillators: Normal Coordinates 11.4 General Theory of Vibrating Systems 11.5 Vibration of a Loaded String or Linear Array of Coupled Harmonic Oscillators 11.6 Vibration of a Continuous System: The Wave Equation Appendix AUnits A-1 Appendix BComplex Numbers and Identities A-4 Appendix CConic Sections A-7 Appendix DService Expansions A-11 Appendix E Special Functions A-13 Appendix F Curvilinear Coordinates A-15 Appendix GFourier Series A-17 Appendix HMatrices A-19 Appendix I Software Tools: Mathcad and Mathematica A-24 Answers to Selected Odd-Numbered Problems ANS-1 Selected References R-1 Index 1-1