7.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-4_558_1180_845_440}
\end{figure}
A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is smoothly hinged at a fixed point on a fixed straight horizontal wire. A smooth light ring \(R\) is threaded on the wire. The ring \(R\) is attached by a light elastic string, of natural length \(a\) and modulus of elasticity \(m g\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle R A B = \theta\), as shown in Fig. 3.
- Show that the potential energy of the system is
$$m g a \left( 2 \sin ^ { 2 } \theta - 3 \sin \theta \right) + \text { constant }$$
(6)
- Hence determine the value of \(\theta , \theta < \frac { \pi } { 2 }\), for which the system is in equilibrium.
- Determine whether this position of equilibrium is stable or unstable.
END