Particle oscillating around equilibrium radius

[AbigailM]

Homework Statement

A particle of mass m moving in three dimensions is attracted to the origin by the gravitational force of a much heavier object. It can be shown that the radial motion is governed by the following equation
$m\ddot{r}=-\frac{k}{r^{2}}+\frac{l^{2}}{mr^{3}}$

where k is a constant and l is the angular momentum. Determine an equilibrium radius $r_{0}$ in terms of k, l, and m. If the particle is put near that equilibrium radius, $r=r_{0}+\epsilon$(where $\epsilon << r_{0}$), it will have an oscillatory radial motion about $r_{0}$. What will be the frequency of that oscillation?

The Attempt at a Solution

Attached to thread as I'm horribly slow at typing latex.

 

[gabbagabbahey]

Your final answer looks correct, but I'm not quite sure what it is you've done to get it. Specifically, why do you assert that $\omega=\sqrt{\frac{1}{m}\frac{d^2U_{eff}}{dr^2}}$? Is the RHS of this equation even a constant?

The method I would suggest is to just plug $r=r_0+\epsilon$ into your equation of motion and Taylor expand the RHS of it in powers of $\frac{\epsilon}{r_0}$ (since you know that it is much smaller than one).