7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-20_679_695_260_628}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The template shown in Figure 3 is formed by joining together three separate laminas. All three laminas lie in the same plane.
- PQUV is a uniform square lamina with sides of length \(3 a\)
- URST is a uniform square lamina with sides of length \(6 a\)
- \(Q R U\) is a uniform triangular lamina with \(U Q = 3 a , U R = 6 a\) and angle \(Q U R = 90 ^ { \circ }\)
The mass per unit area of \(P Q U V\) is \(k\), where \(k\) is a constant.
The mass per unit area of URST is \(k\).
The mass per unit area of \(Q R U\) is \(2 k\).
The distance of the centre of mass of the template from \(Q T\) is \(d\).
- Show that \(d = \frac { 29 } { 14 } a\)
The template is freely suspended from the point \(Q\) and hangs in equilibrium with \(Q R\) at \(\theta ^ { \circ }\) to the downward vertical.
- Find the value of \(\theta\)