Introduction to Classical Mechanics · Problem 7.4

Angular momentum, Part I (Constant L) — Morin

Physics — Mechanics · gravity-kepler-laws

Problem

r^k

potential *** A particle of mass

m

moves in a potential given by

V(r) = \beta r^k

. Let the angular momentum be

L

. ((a)) Find the radius,

r_0

, of the circular orbit. ((b)) If the particle is given a tiny kick so that the radius oscillates around

r_0

, find the frequency,

\omega_r

, of these small oscillations in

r

. ((c)) What is the ratio of the frequency

\omega_r

to the frequency of the (nearly) circular motion,

\omega_\theta \equiv \dot{\theta}

? 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.