A particle of mass m moves in a potential given by V (r) = rk. Let the

Question:

A particle of mass m moves in a potential given by V (r) = βrk. Let the angular momentum be L.
(a) Find the radius, r0, of a circular orbit.
(b) If the particle is given a tiny kick so that the radius oscillates around r0, find the frequency, ωr, of these small oscillations in r.
(c) What is the ratio of the frequency ωr to the frequency of the (nearly) circular motion, ωθ ≡ θ? Give a few values of k for which the ratio is rational, that is, for which the path of the nearly circular motion closes back on itself.