Solving the Rocking Motion of a Dumbbell

[Math Jeans]

Homework Statement

A dumbbell consisting of two spheres of mass $$\frac{m}{2}$$, and connected by a massless rod of length $$2a$$ is in circular orbit. The dumbbell is at radius $$r_0$$ from the planet, and orbits with frequency $$\omega_0$$. The angle of the dumbbell to the downward gravitational force is given by $$\phi$$.

The position of stable equilibrium for the dumbbell is when $$\phi=\pi$$, and the position of unstable equilibrium is at $$\phi=0$$.

The dumbbell is rocking back and forth. Show that the angular frequency of the rocking motion about the stable equilibrium is equal to $$\omega_0*\sqrt{3}$$.

Homework Equations

The Attempt at a Solution

I keep attempting to set up a lagrangian equation to describe the motion of $$\phi$$, however, my $$\omega_0$$ term keeps dropping out at the beginning of my calculations, so I know right there that the answer will turn out wrong.

What method am I supposed to use?

 

[Math Jeans]

Bump. Please please please, this is really important.