4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_351_597_246_735}
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\caption{Figure 3}
\end{figure}
The uniform triangular lamina \(A B C\) has \(A B\) perpendicular to \(A C\), \(A B = 9 a\) and \(A C = 6 a\). The point \(D\) on \(A B\) is such that \(A D = a\).
The rectangle \(D E F G\), with \(D E = 2 a\) and \(E F = 3 a\), is removed from the lamina to form the template shown shaded in Figure 3.
The distance of the centre of mass of the template from \(A C\) is \(d\).
- Show that \(d = \frac { 23 } { 7 } a\)
The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical through \(A\).
- Find the value of \(\theta\)
A new piece, of exactly the same size and shape as the template, is cut from a lamina of a different uniform material. The template and the new piece are joined together to form the model shown in Figure 4. Both parts of the model lie in the same plane.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_369_1185_1667_440}
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\caption{Figure 4}
\end{figure}
The weight of \(C P Q R S T A\) is \(W\)
The weight of \(A D G F E B C\) is \(4 W\)
The model is freely suspended from \(A\).
A horizontal force of magnitude \(X\), acting in the same vertical plane as the model, is now applied to the model at \(T\) so that \(A C\) is vertical, as shown in Figure 4. - Find \(X\) in terms of \(W\).