6. (a) Show by integration that the moment of inertia of a uniform disc, of mass \(m\) and radius \(a\), about an axis through the centre of disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
(3 marks)
\begin{figure}[h]
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\caption{Figure 1}
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\end{figure}
A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). A uniform disc, of mass \(4 m\) and radius \(\frac { 1 } { 2 } a\), is attached to the rod with the centre of the disc lying on the rod a distance \(\frac { 3 } { 2 } a\) from \(A\). The rod lies in the plane of the disc, as shown in Fig. 1. The disc and rod together form a pendulum which is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the pendulum.
(b) Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 27 } { 2 } m a ^ { 2 }\).
The pendulum makes small oscillations about its position of stable equilibrium.
(c) Show that the motion of the pendulum is approximately simple harmonic, and find the period of the oscillations.
(6 marks)