4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a4cdf2b0-8dd0-4c12-9259-95b78875c6cb-07_768_666_123_609}
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\caption{Figure 3}
\end{figure}
A small smooth peg \(P\) is fixed at a distance \(d\) from a fixed smooth vertical wire. A particle of mass \(3 m\) is attached to one end of a light inextensible string which passes over \(P\). The particle hangs vertically below \(P\). The other end of the string is attached to a small ring \(R\) of mass \(m\), which is threaded on the wire, as shown in Figure 3.
- Show that when \(R\) is at a distance \(x\) below the level of \(P\) the potential energy of the system is
$$3 m g \sqrt { } \left( x ^ { 2 } + d ^ { 2 } \right) - m g x + \text { constant }$$
- Hence find \(x\), in terms of \(d\), when the system is in equilibrium.
- Determine the stability of the position of equilibrium.