A simple pendulum consists of a bob of mass m attached to a rigid rod, with R being the distance from the pivot to the bob center. The rod mass and bob diameter are each small (relative to m and R, respectively), the pivoting is frictionless, and the effects of the surrounding air are negligible. The bob is initially at rest, with a small angle θ₀ between the rod and a vertical line. It is desired to predict the period t_p of the oscillations, once the bob is released.

(a) What can you conclude about t_p from dimensional analysis alone?

(b) Derive from basic mechanics the expression for t_p. When put in dimensionless form, how does it compare with what you found in part (a)?