5 A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The point \(C\) on the rod is such that \(A C = a\). The rod can rotate freely in a vertical plane about a fixed horizontal axis passing through \(C\) and perpendicular to the rod.

1. Show by integration that the moment of inertia of the rod about this axis is \(7 m a ^ { 2 }\). The rod starts at rest with \(B\) vertically below \(C\). A couple of constant moment \(\frac { 6 m g a } { \pi }\) is then applied to the rod.
2. Find, in terms of \(a\) and \(g\), the angular speed of the rod when it has turned through one and a half revolutions.
   \includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-3_721_621_872_762} A light pulley of radius \(a\) is free to rotate in a vertical plane about a fixed horizontal axis passing through its centre \(O\). Two particles, \(P\) of mass \(5 m\) and \(Q\) of mass \(3 m\), are connected by a light inextensible string. The particle \(P\) is attached to the circumference of the pulley, the string passes over the top of the pulley, and \(Q\) hangs below the pulley on the opposite side to \(P\). The section of string not in contact with the pulley is vertical. The fixed line \(O X\) makes an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 4 } { 5 }\), and \(O P\) makes an angle \(\theta\) with \(O X\) (see diagram). You are given that the total potential energy of the system (using a suitable reference level) is \(V\), where $$V = m g a ( 3 \sin \theta - 4 \cos \theta - 3 \theta ) .$$