# Lagrangian Problem. Two masses on a massless circle

## Homework Statement

Two equal masses are glued to a massless hoop of radius R that is free to rotate about its center in a vertical plane. The angle between the masses is 2*theta. Find the frequency of small oscillations.

## Homework Equations

$$\frac{d}{dt} \frac{∂L}{∂\dot{q}}=\frac{∂L}{∂q}$$

## The Attempt at a Solution

So since the masses are glued to the hoop, the radius is constant. So I am assuming my professor is referring to the oscillations in the angle? Meaning the hoop will just turn back and fourth?

So going by that (r=constant) My lagrangian is..
$$L=\frac{1}{2}mr^2 \dot{\theta}^2+2mr^2 \dot{\theta}^2+mgr(cos\theta +cos2\theta)$$
Which can be simplified obviously. I left it this way so it is easier to see my logic in the formulation of the Lagrangian? Is this correct? For the second mass, I used 2θ.

Now when I take the respective derivatives I get..
$$\ddot{\theta}-\frac{5g}{9r^2}\theta=0$$

The problem is, this is not an oscillator! If that was a plus sign I would be good to go. Could anyone help me out in figuring out this problem? Where was my logic flawed? :\??

Thanks!!