7. \begin{figure}[h] \end{figure} A body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), with a uniform rod. The centres of the discs are fixed to the ends \(A\) and \(B\) of the rod, which has mass \(3 m\) and length 8a. The discs and the rod are coplanar, as shown in Fig. 2. The body is free to rotate in a vertical plane about a smooth fixed horizontal axis. The axis is perpendicular to the plane of the discs and passes through the point \(O\) of the rod, where \(A O = 3 a\).

1. Show that the moment of inertia of the body about the axis is \(54 m a ^ { 2 }\). The body is held at rest with \(A B\) horizontal and is then released. When the body has turned through an angle of \(30 ^ { \circ }\), the rod \(A B\) strikes a small fixed smooth peg \(P\) where \(O P = 3 a\). Given that the body rebounds from the peg with its angular speed halved by the impact,
2. show that the magnitude of the impulse exerted on the body by the peg at the impact is $$9 m \sqrt { \left( \frac { 5 g a } { 6 } \right) } .$$ END