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\includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-4_640_608_267_715} A smooth wire forms a circle with centre \(O\) and radius \(a\), and is fixed in a vertical plane. The highest point on the wire is \(A\). A small ring \(R\) of mass \(m\) moves along the wire. A light elastic string, with natural length \(\frac { 1 } { 2 } a\) and modulus of elasticity \(2 m g\), has one end attached to \(A\) and the other end attached to \(R\). The string \(A R\) makes an angle \(\theta\) (measured anticlockwise) with the downward vertical (see diagram), and you may assume that the string does not become slack.

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is \(m g a \left( 6 \cos ^ { 2 } \theta - 4 \cos \theta + \frac { 1 } { 2 } \right)\).
2. Show that there are two positions of equilibrium for which \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
3. For each of these positions of equilibrium, determine whether it is stable or unstable.