6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5180a4e0-dafe-4595-a517-e3501f7aed40-4_419_773_196_664}
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\caption{Figure 2}
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A lamina \(S\) is formed from a uniform disc, centre \(O\) and radius \(2 a\), by removing the disc of centre \(O\) and radius \(a\), as shown in Figure 2. The mass of \(S\) is \(M\).
- Show that the moment of inertia of \(S\) about an axis through \(O\) and perpendicular to its plane is \(\frac { 5 } { 2 } M a ^ { 2 }\).
(3)
The lamina is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) lies in the plane of \(S\) and is a tangent to its outer circumference, as shown in Figure 2. - Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 21 } { 4 } M a ^ { 2 }\).
(4)
\(S\) is displaced through a small angle from its position of stable equilibrium and, at time \(t = 0\), it is released from rest. Using the equation of motion of \(S\), with a suitable approximation, - find the time when \(S\) first passes through its position of stable equilibrium.
(6)