5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d3e55cec-05f7-4db3-8eb5-5d0adca38d4c-09_723_707_214_621}
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\caption{Figure 1}
\end{figure}
A uniform circular lamina has radius \(2 a\) and centre \(C\). The points \(P , Q , R\) and \(S\) on the lamina are the vertices of a square with centre \(C\) and \(C P = a\). Four circular discs, each of radius \(\frac { a } { 2 }\), with centres \(P , Q , R\) and \(S\), are removed from the lamina. The remaining lamina forms a template \(T\), as shown in Figure 1.
The radius of gyration of \(T\) about an axis through \(C\), perpendicular to \(T\), is \(k\).
- Show that \(k ^ { 2 } = \frac { 55 a ^ { 2 } } { 24 }\)
The template \(T\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(T\) and passes through a point on its outer rim.
- Write down an equation of rotational motion for \(T\) and deduce that the period of small oscillations of \(T\) about its stable equilibrium position is
$$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$