6. (a) Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through the centre of the disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
[0pt]
[You may assume without proof that the moment of inertia of a uniform hoop of mass \(m\) and radius \(r\) about an axis through its centre and perpendicular to its plane is \(m r ^ { 2 }\).]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1b010d7-14fb-4d23-88eb-d8183e9da3c7-09_528_528_635_708}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A uniform plane shape \(S\) of mass \(M\) is formed by removing a uniform circular disc with centre \(O\) and radius \(a\) from a uniform circular disc with centre \(O\) and radius \(2 a\), as shown in Figure 1. The shape \(S\) is free to rotate about a fixed smooth axis \(L\), which passes through \(O\) and lies in the plane of the shape.
(b) Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 5 } { 4 } M a ^ { 2 }\).
The shape \(S\) is at rest in a horizontal plane and is free to rotate about the axis \(L\). A particle of mass \(M\) falls vertically and strikes \(S\) at the point \(A\), where \(O A = \frac { 3 } { 2 } a\) and \(O A\) is perpendicular to \(L\). The particle adheres to \(S\) at \(A\). Immediately before the particle strikes \(S\) the speed of the particle is \(u\).
(c) Find, in terms of \(M\) and \(u\), the loss in kinetic energy due to the impact.