A uniform hoop of mass M and radius R is hanging from a pivot on its rim. The axis through this pivot is perpendicular to the plane of the hoop (and perpendicular to the plane of the paper).
Since the mass is all at distance R from the center of the hoop, the moment of inertia of the hoop is just MR^2 about an axis through its center.
What is the moment of inertia of this hoop about the axis through its rim? (10 points)
The hoop is now pulled aside and released so that it swings back and forth about this axis. If the angle of oscillation is small so that sin θ ≈ θ, show that the motion is simple harmonic, and derive an expression for the period of the oscillation. (15 points)