Physics 422 -- Fall 2012

Homework #3, due Friday Sep. 21 at beginning of class

1. [5 pts] (a) Find the equation of motion for the apparatus in Marion & Thornton problem 7.18 using the Lagrangian method. Use the angle theta of the massless string with respect to vertical as your coordinate variable. You will find it convenient to define a constant A by A=(length of string)/R - Pi/2. Do not assume a small angle approximation, and ignore the stuff in Thornton about theta1 and theta2. (Hint: You need to compute the kinetic and potential energies. One way to do that is to first calculate the Cartesian coordinates (x,y) of the mass as a function of theta, and then take derivatives to get the velocity.)

(b) Find the Hamiltonian and use it to obtain Hamilton's equations of motion.

(c) Now make the approximation that the angle is small and find the frequency for small oscillations.

2. [5 pts] Consider a pendulum that consists of a mass M hanging from a massless string of length r. The string is being pulled upward at constant velocity through a tiny hole in the ceiling, so the length of the pendulum is given by r = r0 - alpha*t, where alpha is a constant. Let theta be the angle of the string with respect to vertical. Assume that the motion is in a vertical plane, but do not make small angle approximations.
(a) Find the Lagrange equation of motion.

(b) Find the Hamilton equations of motion.

(c) Find the equation of motion in the approximation that the angle theta is small. (You do not have to solve the equation of motion -- its solution will be discussed in lecture.)