2 Fig. 2 shows a system in a vertical plane. A uniform rod AB of length \(2 a\) and mass \(m\) is freely hinged at A . The angle that AB makes with the horizontal is \(\theta\), where \(- \frac { 2 } { 3 } \pi < \theta < \frac { 2 } { 3 } \pi\). Attached at B is a light spring BC of natural length \(a\) and stiffness \(\frac { m g } { a }\). The other end of the spring is attached to a small light smooth ring C which can slide freely along a vertical rail. The rail is at a distance of \(a\) from A and the spring is always horizontal.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-2_737_703_1356_680}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure}
- Find the potential energy, \(V\), of the system and hence show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \cos \theta ( 1 - 4 \sin \theta )\).
- Hence find the positions of equilibrium of the system and investigate their stability.