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.