Why do the homework?
Since you're sophomores and juniors, you've all figured this out by now. But, I'll say it anyway...
I can't emphasize enough the importance of working the problems. In some of your classes homework is primarily evaluative; the point is for you to demonstrate what you've learned from the readings and lectures. In physics the homeworks are primarily instructional; you learn physics primarily by doing working problems. You must work the problems, think about the results, and understand any mistakes you've made if you wish to attain the type of understanding of the subject required of a working physicist. In at nutshell: If you can't work problems you don't know physics. I (or a grader) will grade the problems, and I'll hand out solutions. I encourage you to read the solutions and understand any mistakes immediately. If something doesn't make sense, ask me about it right away---don't wait until right before an exam.
If you've got a good reason why you need an extension, come talk to me in advance. I'll usually grant the extension for some additional reasonable amount of time that we agree upon. However, I will not grant a homework extension without penalty if you ask for it on the day the homework is due, so don't ask for one. In general, life will be easier for both of us if you do your best to finish the problem set on time and hand in as much as you've been able to complete by the deadline. [If you need such a last-minute or post-facto extension due to extenuating circumstances (e.g. death in the family, sudden illness, travel problem), consult the Dean of Students or your Class Dean formally make such a request to me and suggest a rescheduled due date. You should also take this route if you need an extension but you don't want to tell me why (say, it's for personal or legal reasons). If you explain your reason to a Dean and the Dean tells me it's OK, that's good enough for me.]
The College requires that all written work for a course except for a final be submitted by 5 pm on the last day of classes. The physics department takes this deadline seriously. After that day/time, no homework will be accepted.
The roles of lectures and textbooks
Lecture will not be a regurgitation of the text, a summary of all you need to know for the course, or a how-to guide for the homework. Rather, I'll try go deeper into selected points. In lecture I'll cover material and do demonstrations related to the readings, but I won't feel obliged to be comprehensive in those places where I feel the text is adequate and I may focus only on a few points that I feel are particularly interesting or subtle. You shouldn't expect to understand what's going on without close study of the readings, and you should come to class with questions you have on the readings. Further, after we settle into the semester a bit, I expect the classes will become less lecture-oriented and more participatory; it will be difficult to reap the maximum benefit from that format if you're not sufficiently prepared to fully participate.
For the problems you can't solve, talk to classmates, attend the problem sessions, or ask me. When you ask me, either try to give you just enough of a hint to get you through, or I'll guide you through the problem with a series of leading questions. I'll never just tell you how to do it. If you run out of time and don't finish the set, start earlier next week. When the solutions come out, look over them right away, before you've forgotten all of the points you were confused about. You think you'll just get clear on it before the next exam, but there's never as much time as you think.
On the other hand, if you find the class too slow for your liking, if you have questions that you aren't getting answers to, if you'd like more detail, if you are frustrated that we aren't digging deeply enough, if you crave more applications, come talk to me. I'm very happy to provide you with additional materials or explanations that will will stimulate you and challenge you at whatever level you can handle.
One word of warning: Amherst College students tend to have lots of extracurriculars of all types. I support this (enthusiastically), and I am occasionally willing to be flexible to facilitate your participation in range of activities, but don't let your extracurriculars overshadow your academics. If you become concerned that your courses are getting in the way of your extracurriculars, you definitely have the wrong mindset. Remember why you're here.
Key derivations / chains of logic / results to commit to memory
|1. September 6||Newton's Laws
Sept 7: Introduction to the course / Space, time, vectors
General course information. Time, space. Vectors and vector operations.
[Handouts: Course information sheet]
Sept 9: More on vectors: algebra and calculus
Expressing vectors in terms of orthonormal basis vectors. Cartesian and polar coordinate systems. Writing polar coords and basis vectors in terms of cartesian coords and basis vectors, and vice-versa. Scalar ("dot") products of vectors: define, geom. interp, observe it's scalar under rotations. Vector ("cross") product: define in terms of cartesian coords, geometric interp., express in terms of Levi-Civita symbol. Differentiation of vectors (def, chain rule). Diff. of polar unit vectors (both algebraic and geometrical).
Read: TM, Chap. 1 (review of scalars, vectors, and rotation matrices),
start reading Chap. 2
Problems: TM: 1.13, 1.22, 1.23, 1.25, 1.27, 1.37, 2.3, 2.5
|2. September 12||Newton's Laws
Sept 13: Vectors / Newton's Laws
Differentiating vectors in polar coordinates. Finite rotations are not vectors, inf. ones are. Definitions: reference frames, mass, force. Newton I: defines inertial frames, describes "natural state" of matter. Newton II: forces change momentum. Newton III: for every action there's an equal and oppose reaction. Strong form and weak form. Newton III fails: in relativity (except for contact forces), in E&M (for magnetic forces). Consequence? Conservation of momentum for composite systems hinges on Newton III.
Sept 15: Projectile motion: particles in drag
Consider only drag effects (neglect lift), keep terms linear and quadratic in speed. Use dimensional analysis to guess the physics that contributes to linear and quadratic drag terms. Reynolds number is approx. f_quad/f_lin. Some numbers and rule of thumb: if you can see it move, it's quadratic drag. Write Newton II for 2D linear drag with gravity. Solve horiz. velocity eqn. Define terminal velocity from vert. velocity eqn.
Read: TM, Chap. 2
Problems: TM: 2.12, 2.13, 2.20, 2.23, 2.33, 2.36, 2.39, 2.50
|3. September 19||Motion of particles / Conservation laws|
Sept 20: Particles in drag
Linear drag: solve vertical motion. Range formula with light linear drag using perturbation expansion. Quadratic drag: solve 1D motion exactly with and without gravity. Define terminal velocity in this case. Note qualitative distinctions between quadratic and linear drag cases. Quadratic in 2D: nonlinear coupled equations, solve numerically.
Sept 22: Charges in B-fields / Conservation laws
Solve motion of charged particle in constant B-field using complex numbers. Conservation laws: linear momentum, angular momentum. Work-energy theorem.
Read: TM, finish Chap. 2, start Chap. 3
Problems: TM: 2.22, 2.42, 2.43, 2.47, 2.49, 2.52, 2.53
|4. September 26||Energy
Sept 27: Conservative forces and potential energy
Finish work-KE thm. Define PE as energy of position: U[r]= -(work[r0 ->r]). Conservative forces are forces with a PE. Path independence: Two necessary and sufficient conditions--(1) Force has position dependence only (F=F(r)), (2) work is path-independent. Example: gravitational PE from grav. force. Change in U is independent of reference point. E=T+U (total mechanical energy) conserved when only conservative forces act. More generally, change in E = nonconservative work. F is negative gradient of U. Path ind. of F implies circulation of F=0. Local formulation of path independence: Line int. of F is path-ind. IFF curlF=0 everywhere. Forward implication is easy. Backward implication by stokes thm. Sketch proof of stokes thm.
Sept 29: Conservative forces / Solving 1D motion using energy
Finish proof that curlF=0 iff path integrals of F are path independent. Calculate form of curlF explicitly in cartesian coords. curlF=0 iff F=-grad(U) for some scalar function U. Energy bookkeeping when F=F(r,t). Energy in 1D systems: (1) F is a function of position only implies path independence of work. (2) Graphs of potential energy vs. x are useful in understanding qualitatively the particle motion.
Read: TM, Chap. 3
Problems: TM: 3.6, 3.7
|5. October 3||Linear harmonic oscillators I: free and damped SHOs
Oct 4: Undamped harmonic oscillators
SHO behavior is generic near an equilibrium point, for small enough disturbance. Newton II gives 1D harmonic oscillator equation. Various equivalent forms of solution to equation of motion, including sine and cos, shifted cos, and complex exponential. Show conservation of energy. 2D oscillator (springs in small displacement limit). Isotropic case has general motions which are ellipses. Non-isotropic case gives Lissajous figures or space-filling curves.
Oct 6: Damped SHOs
Phase diagram for 1D undamped SHO. Lines don't cross on phase plots, by uniqueness. Damped oscillator with linear damping: general solution in terms of exponentials. Three cases: underdamped, overdamped, critical damping. Qualitative features and phase diagram of underdamped and overdamped cases. Form of critically damped solution is slightly different. Brief intro to sinusoidal driving forces.
Read: TM, Finish Chap. 3 / Start Chap. 5
Problems: TM: 3.21, 3.24, 3.26, 3.31
|6. October 10||Linear harmonic oscillators II: Driven damped SHOs
Oct 11: Fall break
Oct 13: Driven damped SHOs
Newton II for driven damped SHO. General form of solution for 2nd order linear ODEs: complementary and particlar solutions. Sinusoidal driving force: solve with (complex) exponentials. Express particular solution in terms of amplitude and phase shift. Complementary functions give transients, particular solutions give steady-state behavior. Amplitude resonance: find peak of amplitude as a function of driving frequency. Examine shape of amplitude and phase curves for large and small damping: amplitude is peaked, phase shifts through 90 degrees passing through resonance.
Read: TM, Finish Chap. 3 / Start Chap. 5
Problems: TM: 3.28, 3.29, 3.38, 5.7, 5.9, 5.10
|Exam practice: P43F04 exam 1|
|7. October 17||Linear harmonic oscillators / Matters of gravity
Oct 18: Linear harmonic oscillators III: Superposition gone wild
Linear superposition. Fourier's theorem. Using Fourier's theorem to obtain response of oscillator to arbitrary driving force. Orthogonality relations. Fourier integrals. Fourier transform pairs. Green's functions with causal boundary conditions. Using Green's functions to obtain response of oscillator to arbitrary driving force.
Oct 20: Matters of gravity
Newton's law of gravitation for point particles. Newtonian gravity obeys linear superposition. Force due to arbitrary mass distribution. Gravitational field and potential. Shell theorem. Gauss's law for gravitation (integral and differential forms). Poisson's equation for gravitation.
Read: TM, Chap. 6, start Chap. 7
Problems: TM: 6.4, 6.6, 6.8, 6.9, 6.12, 6.15
|8. October 24||Variational principles
Oct 25: Extremum problems and Euler's equation
General intro. Canonical examples: (1) Shortest distance between two points on a plane, (2) Fermat's principle. General variational problem: given a functional, which path extremizes it (assuming fixed endpoints for the path)? Ans: the path described by the Euler equations. Solve shortest distance problem using Euler equations.
Oct 27: Euler's equation applied
Recap of last time. Show that Euler equations must be zero only if first derivative of the functional vanishes for ALL variations. Shortest path between two points is a line. Brachistochrone problem.
Read: TM, Chap. 7
Problems: TM: 7.4, 7.7, 7.9, 7.15, 7.18, 7.21, 7.25, 7.40
|9. October 31||Euler's equation
Nov 1: Euler in several variables: with and without constraints
Euler's equation with several independent variables. Euler with several variables and a constraint: method of Lagrange multipliers.
Applications to mechanics: Define the Lagrangian. Lagrange's equations are equivalent to Newton's second law. They have the form of Euler's equation and can be deduced by applying a variational principle to the action (Hamilton's principle). Lagrange's equations are often simple than Newton, especially in curvilinear coordinates.
Nov 3: Lagrange's equation
Lagrange's equations have form (generalized force) = d/dt (generalized momentum). Application: 2D motion in central force field, in cartesian and polar coordinates, comparing solutions using Lagrange equations and Newton II. Natural choice of coords gives an analog of the second law which involves the natural analogs of force and momentum. Finding conserved quantities facilitated by Lagrange equation: if L is no explicitly dependent on one of the generalized coords, the corresponding gen. force is zero and the gen. mom. is conserved This is tied to a symmetry of the system related to the generalized coord: L is invariant under translations of the gen. coord. that is missing (Noether's thm). Systems of multiple unconstrained particles are similar: 3N Lagrange eqns corresponding to 3N coords needed to specify the positions of the particles in the system completely. Constrained systems: plane pendulum can be formulated in terms of single variable (polar coords) or two variables with constraint (cartesian coords). Degree of freedom: number of coords that can be independently varied in a small displacement. Holonomic system: # of DOF = # of generalized coordinate. Nonholonomic system requires more generalized coords than there are DOF to specify state of system (e.g. ball rolling on table has 2 DOF, requires 5 gen. coords).
Read: TM, Chap. 8
Problems: TM: 7.23, 7.26, 7.37
|10. November 7||Conservation laws / Hamiltonian mechanics / Two-body central force problems
Nov 8: Conservation laws / Hamiltonian mechanics
Nov 10: Two-body central force problem
Read: TM, Chap. 9
Problems: TM: 8.8, 8.11, 8.22, 8.32, 9.4, 9.13, 9.56, 9.61
Solutions (.pdf files):
|11. November 14||Two-body central force problem / Dynamics of systems of particles
Nov 15: Two-body central force problem
Nov 17: Dynamics of systems of particles
Read: TM, Chap. 10, start Chap. 11
Problems: TM: 10.6, 10.8, 10.12, 10.18
Solutions (.pdf files):
P43F04 exam 2
|12. November 28||Non-inertial reference frames
Nov 29 (evening): Exam 2
Read: TM, Chap. 11, start Chap. 12
Problems: TM: 11.3, 11.13, 11.17, 11.24, 11.27, 11.28
P43F05 exam 2
P43F05 exam 2 problem 2 solution
P43F05 exam 2 problem 3 solution
P43F05 exam 2 problem 5 solution
|13. December 5||Rigid body rotation
Read: TM, Chap.
|14. December 12||Coupled oscillations
Dec 15: Reading period
Read: TM, Chap.
Area Seminars and colloquia