3 A uniform rigid rod AB of length $2 a$ and mass $m$ is smoothly hinged to a fixed point at A so that it can rotate freely in a vertical plane. A light elastic string of modulus $\lambda$ and natural length $a$ connects the midpoint of AB to a fixed point C which is vertically above A with $\mathrm { AC } = a$. The rod makes an angle $2 \theta$ with the upward vertical, where $\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \pi$. This is shown in Fig. 3.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c4d3b482-5d09-4128-891d-4499fa49670c-3_339_563_534_737}
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\caption{Fig. 3}
\end{center}
\end{figure}

(i) Find the potential energy, $V$, of the system relative to A in terms of $m , \lambda , a$ and $\theta$. Show that

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

Assume now that the system is set up so that the result (*) continues to hold when $\pi < 2 \theta \leqslant \frac { 5 } { 3 } \pi$.\\
(ii) In the case $\lambda < 2 m g$, show that there is a stable position of equilibrium at $\theta = \frac { 1 } { 2 } \pi$. Show that there are no other positions of equilibrium in this case.\\
(iii) In the case $\lambda > 2 m g$, find the positions of equilibrium for $\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \frac { 5 } { 3 } \pi$ and determine for each whether the equilibrium is stable or unstable, justifying your conclusions.

\hfill \mbox{\textit{OCR MEI M4 2012 Q3 [23]}}