2 A small ring of mass $m$ can slide freely along a fixed smooth horizontal rod. A light elastic string of natural length $a$ and stiffness $k$ has one end attached to a point A on the rod and the other end attached to the ring. An identical elastic string has one end attached to the ring and the other end attached to a point B which is a distance $a$ vertically above the rod and a horizontal distance $2 a$ from the point A . The displacement of the ring from the vertical line through B is $x$, as shown in Fig. 2.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{0166dd50-5069-47f4-a015-d01a9c54faf4-2_405_1063_1270_539}
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\caption{Fig. 2}
\end{center}
\end{figure}

(i) Find an expression for $V$, the potential energy of the system when $0 < x < a$, and show that

$$\frac { \mathrm { d } V } { \mathrm {~d} x } = 2 k x - k a - \frac { k a x } { \sqrt { a ^ { 2 } + x ^ { 2 } } }$$

(ii) Show that $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$ is always positive.\\
(iii) Show that there is a position of equilibrium with $\frac { 1 } { 2 } a < x < a$. State, with a reason, whether it is stable or unstable.

\hfill \mbox{\textit{OCR MEI M4 2011 Q2 [12]}}