7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{47420c50-c232-41e9-8c4d-a890d86ea933-12_837_565_226_694}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A uniform rod \(A B\) of weight \(W\) has its end \(A\) freely hinged to a point on a fixed vertical wall. The rod is held in equilibrium, at angle \(\theta\) to the horizontal, by a force of magnitude \(P\). The force acts perpendicular to the rod at \(B\) and in the same vertical plane as the rod, as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall. The magnitude of the vertical component of the force exerted on the rod by the wall at \(A\) is \(Y\).
- Show that \(Y = \frac { W } { 2 } \left( 2 - \cos ^ { 2 } \theta \right)\).
Given that \(\theta = 45 ^ { \circ }\)
- find the magnitude of the force exerted on the rod by the wall at \(A\), giving your answer in terms of \(W\).