4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-10_787_814_246_566}
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\caption{Figure 1}
\end{figure}
The uniform lamina \(L\), shown shaded in Figure 1, is formed by removing a circular disc of radius \(2 a\) from a uniform circular disc of radius \(4 a\). The larger disc has centre \(O\) and diameter \(A B\). The radius \(O D\) is perpendicular to \(A B\). The smaller disc has centre \(C\), where \(C\) is on \(A B\) and \(B C = 3 a\)
- Show that the centre of mass of \(L\) is \(\frac { 13 } { 3 } a\) from \(B\).
The mass of \(L\) is \(M\) and a particle of mass \(k M\) is attached to \(L\) at \(B\). When \(L\), with the particle attached, is freely suspended from point \(D\), it hangs in equilibrium with \(A\) higher than \(B\) and \(A B\) at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\)
- Find the value of \(k\).