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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{9763e6c4-e372-46ef-a666-3ccb185aa5d2-2_277_707_1398_717}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

(i) Find the potential energy, $V$, of the system when the string is stretched and show that

$$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 4 m g a \cos \theta ( 2 \sin \theta - 1 )$$

(ii) Hence find any positions of equilibrium of the system and investigate their stability.

\hfill \mbox{\textit{OCR MEI M4 2009 Q2 [12]}}