5
\includegraphics[max width=\textwidth, alt={}, center]{0d4a352c-4eda-45b4-9284-60c6fc680f02-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find

1. the period of small oscillations,
2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.