Why does the small oscillation formula give the wrong frequency for a pendulum?

[asdf60]

For small oscillations, the oscillation behaves like a spring, because the potential energy function can be approximated by a parabola at the equilibrium point. Now, the effective spring constant in these situations is equal to the second derivative of the potential energy function, and so the frequency w = sqrt(k/m), where k is the second derivative of the potential energy function.

I'm confused by this. In particular, I don't understand when this actually works. For example, for a pendulum, the potential energy function is U(t) = mgL(1-cos(t)), where t is theta. In this case the effective spring constant is mgL, so w = sqrt(gL). Obviously this doesn't agree with the accepted formula (which is also for small angles only). So what's going on here?

 

[asdf60]

Hmm, it occurs to me now that if insead of m, i use the moment of inertia I = ml^2, i get the right formula. Is this a coincidence?

 

[asdf60]

Yes, if you write it as a function of x, horizontal displacement, rather than theta, it comes out all right.