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\includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-3_698_505_275_801} A uniform solid cylinder has radius \(a\), height \(3 a\), and mass \(M\). The line \(A B\) is a diameter of one of the end faces of the cylinder (see diagram).

1. Show by integration that the moment of inertia of the cylinder about \(A B\) is \(\frac { 13 } { 4 } M a ^ { 2 }\). (You may assume that the moment of inertia of a uniform disc of mass \(m\) and radius \(a\) about a diameter is \(\frac { 1 } { 4 } m a ^ { 2 }\).) The line \(A B\) is now fixed in a horizontal position and the cylinder rotates freely about \(A B\), making small oscillations as a compound pendulum.
2. Find the approximate period of these small oscillations, in terms of \(a\) and \(g\).