THERE ONCE WAS A CLASSICAL THEORY… Introductory Classical Mechanics, with Problems and Solutions David Morin …Of which quantum disciples were leery. They said, “Why spend so long On a theory that’s wrong?” Well, it works for your everyday query! Copyright © 2004 by David Morin All rights reserved Contents 1 Statics I-1 1.1 Balancing forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1 1.2 Balancing torques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-5 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-9 1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-12 1.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-17 2 Using F = ma II-1 2.1 Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-1 2.2 Free-body diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . II-4 2.3 Solving differential equations . . . . . . . . . . . . . . . . . . . . . . II-8 2.4 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-12 2.5 Motion in a plane, polar coordinates . . . . . . . . . . . . . . . . . . II-15 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-18 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-24 2.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-28 3 Oscillations III-1 3.1 Linear differential equations . . . . . . . . . . . . . . . . . . . . . . . III-1 3.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . III-4 3.3 Damped harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . III-6 3.4 Driven (and damped) harmonic motion . . . . . . . . . . . . . . . . III-8 3.5 Coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-13 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-18 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-22 3.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-24 4 Conservation of Energy and Momentum IV-1 4.1 Conservation of energy in 1-D . . . . . . . . . . . . . . . . . . . . . . IV-1 4.2 Small Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-6 4.3 Conservation of energy in 3-D . . . . . . . . . . . . . . . . . . . . . . IV-8 4.3.1 Conservative forces in 3-D . . . . . . . . . . . . . . . . . . . . IV-9 4.4 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-12 4.4.1 Gravity due to a sphere . . . . . . . . . . . . . . . . . . . . . IV-12 4.4.2 Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-14 1 2 CONTENTS 4.5 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-17 4.5.1 Conservation of momentum . . . . . . . . . . . . . . . . . . . IV-17 4.5.2 Rocket motion . . . . . . . . . . . . . . . . . . . . . . . . . . IV-19 4.6 The CM frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-20 4.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-20 4.6.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . IV-22 4.7 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-23 4.7.1 1-D motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-23 4.7.2 2-D motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-25 4.8 Inherently inelastic processes . . . . . . . . . . . . . . . . . . . . . . IV-26 4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-30 4.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-41 4.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV-47 5 The Lagrangian Method V-1 5.1 The Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . . V-1 5.2 The principle of stationary action . . . . . . . . . . . . . . . . . . . . V-4 5.3 Forces of constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-10 5.4 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . V-12 5.5 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-15 5.5.1 Cyclic coordinates . . . . . . . . . . . . . . . . . . . . . . . . V-15 5.5.2 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . V-16 5.6 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-18 5.7 Small oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-21 5.8 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-24 5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-27 5.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-29 5.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-34 6 Central Forces VI-1 6.1 Conservation of angular momentum . . . . . . . . . . . . . . . . . . VI-1 6.2 The effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . VI-3 6.3 Solving the equations of motion . . . . . . . . . . . . . . . . . . . . . VI-5 6.3.1 Finding r(t) and θ(t) . . . . . . . . . . . . . . . . . . . . . . . VI-5 6.3.2 Finding r(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-6 6.4 Gravity, Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . VI-6 6.4.1 Calculation of r(θ) . . . . . . . . . . . . . . . . . . . . . . . . VI-6 6.4.2 The orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-8 6.4.3 Proof of conic orbits . . . . . . . . . . . . . . . . . . . . . . . VI-10 6.4.4 Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-11 6.4.5 Reduced mass . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-13 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-16 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-18 6.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-20 CONTENTS 3 7 Angular Momentum, Part I (Constant Lˆ) VII-1 7.1 Pancake object in x-y plane . . . . . . . . . . . . . . . . . . . . . . . VII-2 7.1.1 Rotation about the z-axis . . . . . . . . . . . . . . . . . . . . VII-3 7.1.2 General motion in x-y plane . . . . . . . . . . . . . . . . . . . VII-4 7.1.3 The parallel-axis theorem . . . . . . . . . . . . . . . . . . . . VII-5 7.1.4 The perpendicular-axis theorem. . . . . . . . . . . . . . . . . VII-6 7.2 Non-planar objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-7 7.3 Calculating moments of inertia . . . . . . . . . . . . . . . . . . . . . VII-9 7.3.1 Lots of examples . . . . . . . . . . . . . . . . . . . . . . . . . VII-9 7.3.2 A neat trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-11 7.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-12 7.4.1 Point mass, fixed origin . . . . . . . . . . . . . . . . . . . . . VII-13 7.4.2 Extended mass, fixed origin . . . . . . . . . . . . . . . . . . . VII-13 7.4.3 Extended mass, non-fixed origin . . . . . . . . . . . . . . . . VII-14 7.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-17 7.6 Angular impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-19 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-21 7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-28 7.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-34 8 Angular Momentum, Part II (General Lˆ) VIII-1 8.1 Preliminaries concerning rotations . . . . . . . . . . . . . . . . . . . VIII-1 8.1.1 The form of general motion . . . . . . . . . . . . . . . . . . . VIII-1 8.1.2 The angular velocity vector . . . . . . . . . . . . . . . . . . . VIII-2 8.2 The inertia tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-5 8.2.1 Rotation about an axis through the origin . . . . . . . . . . . VIII-5 8.2.2 General motion . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-9 8.2.3 The parallel-axis theorem . . . . . . . . . . . . . . . . . . . . VIII-10 8.3 Principal axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-11 8.4 Two basic types of problems . . . . . . . . . . . . . . . . . . . . . . . VIII-15 8.4.1 Motion after an impulsive blow . . . . . . . . . . . . . . . . . VIII-15 8.4.2 Frequency of motion due to a torque . . . . . . . . . . . . . . VIII-18 8.5 Euler’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-20 8.6 Free symmetric top . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-22 8.6.1 View from body frame . . . . . . . . . . . . . . . . . . . . . . VIII-22 8.6.2 View from fixed frame . . . . . . . . . . . . . . . . . . . . . . VIII-24 8.7 Heavy symmetric top . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-25 8.7.1 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-25 8.7.2 Digression on the components of ω(cid:126) . . . . . . . . . . . . . . . VIII-26 8.7.3 Torque method . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-29 8.7.4 Lagrangian method. . . . . . . . . . . . . . . . . . . . . . . . VIII-30 8.7.5 Gyroscope with θ˙ = 0 . . . . . . . . . . . . . . . . . . . . . . VIII-31 8.7.6 Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-33 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-36 8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-38 4 CONTENTS 8.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-44 9 Accelerated Frames of Reference IX-1 9.1 Relating the coordinates . . . . . . . . . . . . . . . . . . . . . . . . . IX-2 9.2 The fictitious forces . . . . . . . . . . . . . . . . . . . . . . . . . . . IX-4 9.2.1 Translation force: −md2R/dt2 . . . . . . . . . . . . . . . . . IX-5 9.2.2 Centrifugal force: −mω(cid:126) ×(ω(cid:126) ×r) . . . . . . . . . . . . . . . . IX-5 9.2.3 Coriolis force: −2mω(cid:126) ×v . . . . . . . . . . . . . . . . . . . . IX-7 9.2.4 Azimuthal force: −m(dω/dt)×r . . . . . . . . . . . . . . . . IX-11 9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX-13 9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX-15 9.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX-17 10 Relativity (Kinematics) X-1 10.1 The postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-2 10.2 The fundamental effects . . . . . . . . . . . . . . . . . . . . . . . . . X-4 10.2.1 Loss of Simultaneity . . . . . . . . . . . . . . . . . . . . . . . X-4 10.2.2 Time dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . X-7 10.2.3 Length contraction . . . . . . . . . . . . . . . . . . . . . . . . X-10 10.3 The Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . X-14 10.3.1 The derivation . . . . . . . . . . . . . . . . . . . . . . . . . . X-14 10.3.2 The fundamental effects . . . . . . . . . . . . . . . . . . . . . X-18 10.3.3 Velocity addition . . . . . . . . . . . . . . . . . . . . . . . . . X-20 10.4 The invariant interval . . . . . . . . . . . . . . . . . . . . . . . . . . X-23 10.5 Minkowski diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . X-26 10.6 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-29 10.6.1 Longitudinal Doppler effect . . . . . . . . . . . . . . . . . . . X-29 10.6.2 Transverse Doppler effect . . . . . . . . . . . . . . . . . . . . X-30 10.7 Rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-32 10.8 Relativity without c . . . . . . . . . . . . . . . . . . . . . . . . . . . X-35 10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-39 10.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-46 10.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-52 11 Relativity (Dynamics) XI-1 11.1 Energy and momentum . . . . . . . . . . . . . . . . . . . . . . . . . XI-1 11.1.1 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-2 11.1.2 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-3 11.2 Transformations of E and p(cid:126) . . . . . . . . . . . . . . . . . . . . . . . XI-7 11.3 Collisions and decays . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-10 11.4 Particle-physics units . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-13 11.5 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-14 11.5.1 Force in one dimension . . . . . . . . . . . . . . . . . . . . . . XI-14 11.5.2 Force in two dimensions . . . . . . . . . . . . . . . . . . . . . XI-16 11.5.3 Transformation of forces . . . . . . . . . . . . . . . . . . . . . XI-17 CONTENTS 5 11.6 Rocket motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-19 11.7 Relativistic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-22 11.8 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-24 11.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-26 11.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-30 11.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI-34 12 4-vectors XII-1 12.1 Definition of 4-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . XII-1 12.2 Examples of 4-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . XII-2 12.3 Properties of 4-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . XII-4 12.4 Energy, momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII-6 12.4.1 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII-6 12.4.2 Transformation of E,p . . . . . . . . . . . . . . . . . . . . . . XII-6 12.5 Force and acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . XII-7 12.5.1 Transformation of forces . . . . . . . . . . . . . . . . . . . . . XII-7 12.5.2 Transformation of accelerations . . . . . . . . . . . . . . . . . XII-8 12.6 The form of physical laws . . . . . . . . . . . . . . . . . . . . . . . . XII-10 12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII-12 12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII-13 12.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII-14 13 General Relativity XIII-1 13.1 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . XIII-1 13.2 Time dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII-2 13.3 Uniformly accelerated frame . . . . . . . . . . . . . . . . . . . . . . . XIII-4 13.3.1 Uniformly accelerated point particle . . . . . . . . . . . . . . XIII-5 13.3.2 Uniformly accelerated frame . . . . . . . . . . . . . . . . . . . XIII-6 13.4 Maximal-proper-time principle . . . . . . . . . . . . . . . . . . . . . XIII-8 13.5 Twin paradox revisited . . . . . . . . . . . . . . . . . . . . . . . . . . XIII-9 13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII-12 13.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII-15 13.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII-18 14 Appendices XIV-1 14.1 Appendix A: Useful formulas . . . . . . . . . . . . . . . . . . . . . . XIV-1 14.1.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-1 14.1.2 Nice formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-2 14.1.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-2 14.2 Appendix B: Units, dimensional analysis . . . . . . . . . . . . . . . . XIV-4 14.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-6 14.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-7 14.2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-8 14.3 Appendix C: Approximations, limiting cases . . . . . . . . . . . . . . XIV-11 14.3.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV-13 6 CONTENTS 14.4 Appendix D: Solving differential equations numerically . . . . . . . . XIV-15 14.5 Appendix E: F = ma vs. F = dp/dt . . . . . . . . . . . . . . . . . . XIV-17 14.6 Appendix F: Existence of principal axes . . . . . . . . . . . . . . . . XIV-19 14.7 Appendix G: Diagonalizing matrices . . . . . . . . . . . . . . . . . . XIV-22 14.8 Appendix H: Qualitative relativity questions. . . . . . . . . . . . . . XIV-24 14.9 Appendix I: Lorentz transformations . . . . . . . . . . . . . . . . . . XIV-29 14.10Appendix J: Resolutions to the twin paradox . . . . . . . . . . . . . XIV-32 14.11Appendix K: Physical constants and data . . . . . . . . . . . . . . . XIV-34 Preface This textbook has grown out of the first-semester honors freshman physics course that has been taught at Harvard University during recent years. The book is essen- tially two books in one. Roughly half of it follows the form of a normal textbook, consisting of text, along with exercises suitable for homework assignments. The other half takes the form of a problem book, with all sorts of problems (with so- lutions) of varying degrees of difficulty. If you’ve been searching for a supply of practice problems to work on, this should keep you busy for a while. A brief outline of the book is as follows. Chapter 1 covers statics. Most of this will probably look familiar, but you’ll find some fun problems. In Chapter 2, we learn about forces and how to apply F = ma. There’s a bit of math here needed for solving some simple differential equations. Chapter 3 deals with oscillations and coupled oscillators. Again, there’s a fair amount of math needed for solving linear differential equations, but there’s no way to avoid it. Chapter 4 deals with conservation of energy and momentum. You’ve probably seen much of this before, but again, it has lots of neat problems. In Chapter 5, we introduce the Lagrangian method, which will undoubtedly be new to you. It looks rather formidable at first, but it’s really not all that rough. There are difficult concepts at the heart of the subject, but the nice thing is that the technique is easy to apply. The situation here analogous to taking a derivative in calculus; there are substantive concepts on which the theory rests, but the act of taking a derivative is fairly straightforward. Chapter 6 deals with central forces, Kepler’s Laws, and such things. Chapter 7 covers the easier type of angular momentum situations, ones where the direction of theangularmomentumisfixed. Chapter8coversthemoredifficulttype,oneswhere the direction changes. Gyroscopes, spinning tops, and other fun and perplexing objects fall into this category. Chapter 9 deals with accelerated frames of reference and fictitious forces. Chapters 10 through 13 cover relativity. Chapter 10 deals with relativistic kine- matics – abstract particles flying through space and time. Chapter 11 covers rel- ativistic dynamics – energy, momentum, force, etc. Chapter 12 introduces the im- portant concept of “4-vectors.” The material in this chapter could alternatively be put in the previous two, but for various reasons I thought it best to create a separate chapter for it. Chapter 13 covers a few topics from general relativity. It’s not possible for one chapter to do this subject justice, of course, so we’ll just look at some basic (but still very interesting) examples. 1 2 CONTENTS The appendices contain various useful things. Indeed, Appendices B and C, which cover dimensional analysis and limiting cases, are the first parts of this book you should read. Throughout the book, I have included many “remarks.” These are written in a slightly smaller font than the surrounding text. They begin with a small-capital “Remark” and end with a shamrock (♣). The purpose of these remarks is to say somethingthatneedstobesaid,withoutdisruptingtheoverallflowoftheargument. In some sense these are “extra” thoughts, although they are invariably useful in understanding what is going on. They are usually more informal than the rest of the text, and I reserve the right to occasionally use them to babble about things I find interesting, but which you may find a bit tangential. For the most part, however, the remarks address issues and questions that arise naturally in the course of the discussion. At the end of the solutions to many problems, the obvious thing to do is to check limiting cases.1 I have written these in a smaller font, but I have not always bothered to start them with a “Remark” and end them with a “♣”, because they are not “extra” thoughts. Checking limiting cases of your answer is something you should always do. For your reading pleasure (I hope), I have included many limericks scattered throughout the text. I suppose that they might be viewed as educational, but they certainly don’t represent any deep insight I have on the teaching of physics. I have written them solely for the purpose of lightening things up. Some are funny. Some are stupid. But at least they’re all physically accurate (give or take). A word on the problems. Some are easy, but many are very difficult. I think you’ll find them quite interesting, but don’t get discouraged if you have trouble solving them. Some are designed to be brooded over for hours. Or days, or weeks, or months (as I can attest to). I have chosen to write them up for two reasons: (1) Students invariably want extra practice problems, with solutions, to work on, and (2) I find them rather fun. The problems are marked with a number of asterisks. Harder problems earn more asterisks, on a scale from zero to four. You may, of course, disagree with my judgment of difficulty, but I think that an arbitrary weighting scheme is better than none at all. As a rough idea of what I mean by the number of stars: one-star problems are solid problems that require some thought, and four-star problems are really really really difficult. Try a few and you’ll see what I mean. Just to warn you, even if you understand the material in the text backwards and forwards, the four-star (and many of the three-star) problems will still be extremely challenging. But that’s how it should be. My goal was to create an unreachable upper bound on the number (and difficulty) of problems, because it would be an unfortunate circumstance, indeed, if you were left twiddling your thumbs, having run out of problems to solve. I hope I have succeeded. For the problems you choose to work on, be careful not to look at the solution too soon. There is nothing wrong with putting a problem aside for a while and 1This topic is discussed in Appendix C.