6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be16c17a-c4db-4f0c-9f32-8d5614b4f2f3-16_1031_915_116_513}
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\caption{Figure 2}
\end{figure}
The uniform lamina \(L\) shown shaded in Figure 2 is formed by removing two circular discs, \(C _ { 1 }\) and \(C _ { 2 }\), from a circular disc with centre \(O\) and radius \(8 a\). Disc \(C _ { 1 }\) has centre \(A\) and radius \(a\). Disc \(C _ { 2 }\) has centre \(B\) and radius \(2 a\). The diameters \(P R\) and \(Q S\) are perpendicular. The midpoint of \(P O\) is \(A\) and the midpoint of \(O R\) is \(B\).
- Show that the centre of mass of \(L\) is \(\frac { 484 } { 59 } a\) from \(R\).
The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(S\). The lamina with the attached particle is suspended from \(R\) and hangs freely in equilibrium with the diameter \(P R\) at an angle of arctan \(\left( \frac { 1 } { 4 } \right)\) to the downward vertical through \(R\).
- Find the value of \(k\).