A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M and length L that is pivoted freely about one end, with a solid sphere of the same mass, M, and a radius of L/2 centered about the free end of the rod.
a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of M and L.
b) Obtain an expression for the period of the pendulum for small oscillations.
c) Determine the length L that gives a period of T = 2.0 s.
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