5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{656fb620-e80b-4598-a8cd-0f5b8a11e487-08_581_784_221_589}
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\caption{Figure 1}
\end{figure}
A uniform rod \(A B\), of length \(2 l\) and mass \(12 m\), has its end \(A\) smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end \(B\) of the rod. The string passes over a small smooth pulley which is fixed at the point \(C\), where \(A C\) is horizontal and \(A C = 2 l\). A particle of mass \(m\) is attached to the other end of the string and the particle hangs vertically below \(C\).
The angle \(B A C\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1.
- Show that the potential energy of the system is
$$4 m g l \left( \sin \frac { \theta } { 2 } - 3 \sin \theta \right) + \mathrm { constant }$$
- Find the value of \(\theta\) when the system is in equilibrium and determine the stability of this equilibrium position.