2 A uniform rigid rod AB of mass \(m\) and length \(4 a\) is freely hinged at the end A to a horizontal rail. The end B is attached to a light elastic string BC of modulus \(\frac { 1 } { 2 } m g\) and natural length \(a\). The end C of the string is attached to a ring which is small, light and smooth. The ring can slide along the rail and is always vertically above B . The angle that AB makes below the rail is \(\theta\). The system is shown in Fig. 2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9763e6c4-e372-46ef-a666-3ccb185aa5d2-2_277_707_1398_717}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure}
- Find the potential energy, \(V\), of the system when the string is stretched and show that
$$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 4 m g a \cos \theta ( 2 \sin \theta - 1 )$$
- Hence find any positions of equilibrium of the system and investigate their stability.