Introduction to Classical Mechanics · Problem 6.46
The Lagrangian method — Morin
Physics — Mechanics · small-oscillations
Problem
Mass sliding on a rim, with a spring ***
Consider the setup in the previous exercise, but now let the mass m be attached to a spring with spring constant k and relaxed length zero, the other end of which is attached to a point on the rim. Assume that the spring is constrained to run along the rim, and assume that the mass can pass freely over the point where the spring is attached to the rim. To keep things from getting too messy here, you can set M=m.
(a) Find the frequencies of the normal modes for small oscillations. Check the g=0 limit, and (if you've done the previous exercise) the k=0 limit.
(b) For the special case where g/R=k/m, show that the frequencies can be written as k/m(5±1)/2. This numerical factor is the golden ratio (and its inverse). Describe what the normal modes look like.