Physical Pendulum

A physical pendulum is constructed from a thin rod of negligible mass inserted into a uniform ball of mass $M$ and radius $R$. The rod has length $L$ from the pivot point to the center of the ball. At time $t = 0$ the ball is released from rest when the rod is at an initial small angle $\theta_0$ with respect to its vertical equilibrium position.

Answer all the questions below in terms of $M,R,L,g,\theta_0$. You may make the small angle approximation where appropriate.

a) Determine the equation of motion for the system, solving for $\alpha = \frac{d^2\theta}{dt^2}$.

b) Determine the angular frequency of oscillation $\omega$ and write down $\theta(t)$ for the ball.

c) Find the maximum speed $v$ of the ball. Is this larger or smaller than it would have been if the ball had been a point mass $M$ at the end of the rod? Why?