How Do Rocking Bowl Oscillations Relate to Torque and Moment of Inertia?

[Fuzzy Static]

Homework Statement

A hemispherical shell with its curved surface resting on a table will rock back and forth. To derive the period, use $$\sum$$T=I$$\alpha$$$$_{}z$$. Sum the torques about the instantaneous point of contact and make the small angle approximation sin $$\theta$$$$\approx$$$$\theta$$.
You will need to know the location of the center of mass. It can be shown that the center of mass of the shell is halfway between the center of curvature and surface. (You don't have to show it.)
You will also need the moment of inertia. Since we will restrict ourselves to small angle oscillations, just use the moment of inertia about the equilibrium point of contact.

Homework Equations

$$\sum$$T=I$$\alpha$$$$_{}z$$
sin $$\theta$$$$\approx$$$$\theta$$

The Attempt at a Solution

Um... Using the sum of Torque = the moment of Inertia times alpha. No clue.

 

[AvroArrow]

Since the center of mass is halfway between the center of curvature and surface, the moment of inertia would be half of the total moment of inertia. The small angle approximation will help us simplify the expression for the period. We can use the formula T=2\pi\sqrt{\frac{I}{\sumT}} to find the period. Using the given information, we can solve for T as follows:T=2\pi\sqrt{\frac{I/2}{\sumT}}