Homework Assignment 6 Physics 302, Classical Mechanics

Fall 2010 A. V. Kotwal

Handed out:  Friday, October 15, 2010
Due in class on: Friday, October 22, 2010

Problems

1. “Classical Dynamics”, José & Saletan, 6.3 [particle on a vertical cylinder];

2. “Classical Mechanics”, Goldstein, et al., 12.6 [the relativistic correction of a 1D harmonic oscillator using canonical perturbation];

3. “Classical Dynamics”, José & Saletan, 6.25 [a bouncing object in an accelerating elevator];

4. A particle with mass m moves in one dimension under the influence of a potential V (x) = −\frac{k}{|x|}, where k > 0. For energies that are negative, the motion is bounded and oscillatory.

(a) Write down the Hamiltonian for this system, draw the potential energy function, and then explain why the orbit is bounded.

(b) If k is a constant, by the method of action-angle variables, find the period of the motion as a function of the particle’s energy.

(c) If k is slowly varied from an initial value, k = k_{0} + αt, where k_{0} and α are positive constants. Will the energy of the system decrease or increase with time? How about the period of the oscillation? How about the maximum amplitude of each oscillation?

5. “Classical Mechanics”, Goldstein, et al., 3.19 [a particle’s motion in the Yukowa potential V (r) = −\frac{k}{r} exp (−\frac{r}{a})].