7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{f70e9177-fbda-409d-8f80-d900a33a6481-5_622_506_395_803}
\end{figure}
A uniform rod \(A B\), of mass \(m\) and length \(2 a\), can rotate freely in a vertical plane about a fixed smooth horizontal axis through \(A\). The fixed point \(C\) is vertically above \(A\) and \(A C = 4 a\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), joins \(B\) to \(C\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 3.
- Show that the potential energy of the system is
$$- m g a [ \cos \theta + \sqrt { } ( 5 - 4 \cos \theta ) ] + \text { constant. }$$
- Hence determine the values of \(\theta\) for which the system is in equilibrium.
END