3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-06_645_684_260_639}
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\caption{Figure 1}
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The uniform lamina \(A B C D\) is a square of side \(6 a\). The template \(T\), shown shaded in Figure 1, is formed by removing the right-angled triangle \(E F G\) and the circle, centre \(H\) and radius \(a\), from the square lamina.
Triangle \(E F G\) has \(E F = E G = 4 a\), with \(E F\) parallel to \(A B\) and \(E G\) parallel to \(A D\). The distance between \(A B\) and \(E F\) is \(a\) and the distance between \(A D\) and \(E G\) is \(a\).
The point \(H\) lies on \(A C\) and the distance of \(H\) from \(B C\) is \(2 a\).
- Show that the centre of mass of \(T\) is a distance \(\frac { 4 ( 67 - 3 \pi ) } { 3 ( 28 - \pi ) } a\) from \(A D\).
The template \(T\) is suspended from the ceiling by two light inextensible vertical strings. One string is attached to \(T\) at \(A\) and the other string is attached to \(T\) at \(B\) so that \(T\) hangs in equilibrium with \(A B\) horizontal.
The weight of \(T\) is \(W\). The tension in the string attached to \(T\) at \(B\) is \(k W\), where \(k\) is a constant.
- Find the value of \(k\), giving your answer to 2 decimal places.