Finding Angular Frequency of Small Oscillations about an Equilibrium

[Oijl]

Homework Statement

Consider a system of one generalized coordinate theta, having the following Lagrangian equation of motion:

r and b are constants
m is mass

(1/3)mb^{2}\ddot{\theta} = r(r+b)\theta + r^{2}\theta^{3} + gr\theta

And this potential energy (if it matters):

U = mg(r+b) - mgr\theta^{2}


There is an equilibrium point where theta is equal to zero.

Find the angular frequency of small oscillations about \theta = 0.

Homework Equations

The Attempt at a Solution

Using the potential energy, can't I just say

U = (1/2)k\theta^{2}
where
k = 2mgr
so that I can write
\omega = (k/m)^(1/2)
\omega = (2gr)^(1/2)
and call that the angular frequency?

But the problem asks me to do it the Lagrangian way.

So
\omega = (2\pi)/\tau

How can I find tau?

 

[ideasrule]

Using the potential energy, can't I just say

U = (1/2)k\theta^{2}
where
k = 2mgr
so that I can write
\omega = (k/m)^(1/2)
\omega = (2gr)^(1/2)
and call that the angular frequency?

Clever, but it's not right. You can see this by comparing the units of U=(1/2)kx^2 with those of U = (1/2)k\theta^{2}: both U's have the same unit, both x doesn't have the same unit as theta, so the two k's must have different units. That means the equation omega=(k/m)^1/2 is not correct.

To start, do you know the characteristic differential equation for simple harmonic motion? Try to get the Lagrangian equation of motion into that form, remembering that theta^3 is much smaller than theta for small values of theta.