- (a) Use algebraic integration to show that the centre of mass of a uniform semicircular disc of radius \(r\) and centre \(O\) is at a distance \(\frac { 4 r } { 3 \pi }\) from the diameter through \(O\) [You may assume, without proof, that the area of a circle of radius \(r\) is \(\pi r ^ { 2 }\) ]
A uniform lamina L is in the shape of a semicircle with centre \(B\) and diameter \(A C = 8 a\). The semicircle with diameter \(A B\) is removed from \(L\) and attached to the straight edge \(B C\) to form the template \(T\), shown shaded in Figure 4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-16_419_1273_680_397}
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\caption{Figure 4}
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The distance of the centre of mass of \(T\) from \(A C\) is \(d\).
(b) Show that \(d = \frac { 4 a } { \pi }\)
The template \(T\) is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at an angle \(\theta\) to the downward vertical.
(c) Find the exact value of \(\tan \theta\)