7.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-5_955_855_349_573}
\end{figure}
A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.
- Show that the potential energy of the system is given by
$$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
- Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
- Determine the stability of each position of equilibrium.
END