Balancing a circle on a triangle?

[anotherghost]

Homework Statement

You're given a hoop with mass m and radius R balanced on top of a knife blade. (The diagram looks like a triangle with a circle balanced on the tip.) Find the period of small oscillations.

(Yes, that is all the problem says.)

Homework Equations

Moment of inertia of a hoop: I = mR²
U = mgh
K = 1/2 I w² (writing omega as w)

The Attempt at a Solution

OK, here's the thing - I can't figure out how the hoop could be oscillating because I don't know what the restoring force is supposed to be. When questioned about it the professor said that has to do with the center of mass, but that we should use energy methods to solve the problem. So...

U = mgh (where h is the height of the center of mass)
h = R cos O (writing theta as O. theta is the angle the hoop rotates from the point of the triangle)
Then U = mgRcosO
K = 1/2 (I_hoop + mR²)w² (by parallel axis theorem)
K = 1/2 2mRw² = mRw²

E = K + U, dE/dt = 0

d/dt (MRw² + mgRcosO ) = 0
2R d²O/dt² - g O = 0 after small angle approximation and dividing through by m R w

The solution of that is just O(t) = c₁ e^sqrt(g/2R)t + c₂ e^-sqrt(g/2R)t

That isn't oscillating. Of course that makes sense to me, because from what I understand there can be no restoring force...

What am I missing here?

 

[Dadface]

From your description I suspect you are not seeing the set up clearly.The blade is inside the hoop so that when at rest the centre of the hoop is vertically below the blade.when the hoop is displaced there will be a restoring force.