Classical Mechanics

Description

Classical Mechanics: A professor-student collaboration is a textbook tailored for undergraduate physics students embarking on a first-year module in Newtonian mechanics. This book is a collaboration between Mario Campanelli and students that attended his course in classical mechanics at University College London, based on the course lecture notes w Title Copyright Contents Preface Acknowledgments Editor biography Short biographies of contributors 1 Mathematical preliminaries 1.0 Introduction 1.1 Vectors 1.1.1 Linear operations 1.1.2 Scalar product 1.1.3 Matrices 1.1.4 Vector product 1.2 Complex numbers 1.2.1 Arithmetic in C 1.2.2 Polar coordinates 1.2.3 Complex exponential and Euler’s formula 1.3 Calculus 1.3.1 Differentiation 1.3.2 Taylor series 1.3.3 Integration 1.3.4 Integration by parts 1.3.5 Integration with vectors 1.4 Differential equations 1.4.1 Separable first order ordinary differential equations 1.4.2 First order ordinary differential equation. Integrating factor 1.4.3 Second order homogeneous linear differential equations 1.4.4 Second order inhomogeneous linear differential equations 2 Newton’s laws 2.0 Introduction 2.1 Newton’s laws of motion 2.2 The concept of force 2.2.1 The vector nature of force 2.2.2 Types of forces 2.2.3 Analysis of a physical system using Newton’s second law 2.3 Motion under a constant force 2.3.1 General theory 2.3.2 One dimension 2.3.3 Free fall under gravity—one dimension 2.3.4 Particle motion with air friction 2.4 Projectiles 2.4.1 Projectiles without air resistance 2.5 Momentum and impulse 2.6 Conservation of momentum for isolated systems 3 Kinematic relations 3.0 Introduction: what is energy? 3.1 Work and energy 3.2 Relationship between work and kinetic energy 3.3 Power 3.4 Potential energy and conservative forces 3.4.1 Potential energy in 1D 3.4.2 Potential energy in 3D 3.4.3 Conservation of mechanical energy 3.4.4 Work–energy theorem for systems of many particles 4 Oscillatory motion 4.0 Introduction 4.1 Simple harmonic motion 4.1.1 Energy of simple harmonic motion 4.2 Damped harmonic motion 4.3 Driven and damped harmonic motion 4.3.1 Driven oscillator curves 4.3.2 Quality factor 4.3.3 Work done by driving force 4.4 Coupled oscillators 5 Angular momentum and central forces 5.0 Introduction 5.1 Polar coordinates 5.2 Circular motion 5.3 Angular momentum 5.4 Central forces 5.4.1 Potential energy for central forces 6 Centre of mass and collisions 6.0 Introduction 6.1 The centre of mass 6.1.1 The centre of mass frame 6.1.2 Relative displacement and reduced mass 6.1.3 Kinetic energy in centre of mass frame 6.2 Collisions 6.2.1 Elastic and inelastic collisions 6.2.2 Coefficient of restitution 6.2.3 Single-body collision with a rigid wall 6.2.4 Collision between two bodies of finite mass 7 Orbits 7.0 Introduction: a historical note 7.1 Orbital forces 7.1.1 Potentials 7.2 Circular motion approximation 7.3 Motion under the inverse square law of force 7.3.1 Trajectories 7.4 Orbits under an attractive force: elliptical orbits and Kepler’s laws 7.4.1 Eccentricity 7.4.2 Understanding orbits from the effective potential 7.4.3 Elliptical orbit 7.4.4 Kepler’s laws 7.5 Orbits with positive energy: unbound orbits 7.6 Reduced mass and the two-body problem 7.7 Variable mass problems 8 Rigid bodies 8.0 Introduction 8.1 Preliminaries 8.2 Centre of mass 8.3 Flat object in x–y plane 8.3.1 Rotation about the z-axis 8.3.2 General motion in the x–y plane 8.3.3 Theorem of parallel axis 8.3.4 Theorem of perpendicular axis 8.3.5 Non-planar objects moving around an axis 8.3.6 Calculating moments of inertia 8.4 General motion of a non-planar object in 3D space 8.4.1 Angular momentum and the inertia matrix 8.4.2 General motion in 3D 8.4.3 Correspondences between translation and rotation 8.4.4 Compound pendulum 8.4.5 Effect of external force 8.4.6 Simple theory of the gyroscope 9 Accelerating frames of reference 9.0 Accelerating reference frames 9.0.1 Inertial and non-inertial frames 9.0.2 Accelerating and rotating frames of reference 9.0.3 Derivation of fictitious forces 9.1 Fictitious forces 9.1.1 Translation force 9.1.2 Centrifugal force 9.1.3 Coriolis force 9.1.4 Azimuthal force 10 Fluid mechanics 10.0 Introduction 10.1 Hydrostatics 10.1.1 Variation of pressure with depth 10.1.2 Buoyancy 10.1.3 Archimedes’ principle 10.2 Hydrodynamics—fluids in motion 10.2.1 The ideal fluid flow 10.2.2 Streamlines 10.2.3 The equation of continuity 10.2.4 Bernoulli’s equation 11 Solutions to Chapter 1: Mathematical preliminaries 12 Solutions to Chapter 2: Newton’s laws 13 Selected solutions to Chapter 3: Kinematic relations 14 Selected solutions to Chapter 4: Oscillatory motion 15 Selected solutions to Chapter 5: Angular momentum and central forces 16 Solutions to Chapter 6: Centre of mass and collisions 17 Solutions to Chapter 7: Orbits 18 Selected solutions to Chapter 8: Rigid bodies 19 Selected solutions to Chapter 9: Accelerating frames of reference 20 Solutions to Chapter 10: Fluid mechanics Appendices Appendix A Appendix B Appendix C Further reading