4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-06_796_789_276_566}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A uniform circular disc has centre \(O\) and radius 4a. The lines \(P Q\) and \(S T\) are perpendicular diameters of the disc. A circular hole of radius \(2 a\) is made in the disc, with the centre of the hole at the point \(R\) on \(O P\) where \(O R = 2 a\), to form the lamina \(L\), shown shaded in Figure 2.
- Show that the distance of the centre of mass of \(L\) from \(P\) is \(\frac { 14 a } { 3 }\).
The mass of \(L\) is \(m\) and a particle of mass \(k m\) is now fixed to \(L\) at the point \(P\). The system is now suspended from the point \(S\) and hangs freely in equilibrium. The diameter \(S T\) makes an angle \(\alpha\) with the downward vertical through \(S\), where \(\tan \alpha = \frac { 5 } { 6 }\).
- Find the value of \(k\).