Small oscillations

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

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