Assignment #6

Question #1

You shouldn't need a hint for this one.

Question #2

Consider introducing a fourth generalized coordinate that represents the position of the pulley attached to the string of length l1. Then you will have four generalized coordinates and two equations of constraints relating them to the lengths of the strings. The solution is a straight forward solution of a system of equations but it gets quite involved. Just setting up the system is good enough.

Question #3

First work out Lagrange's equations. Solve for the equilibrium angle, theta0, by setting d^2 theta/dt^2 = 0. Then expand theta about theta0 to first order in some small angle. It should end up looking like a typical harmonic oscillator problem.

Question #4

Actually, the translation and rotation modes decouple so you won't need to work out any determinants in this case. This wouldn't necessarily be the case if the springs had different spring constants or if the springs weren't equally spaced on either side of the center of mass.