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\includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_493_748_1393_708} The diagram shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The string joins \(B\) to a light ring \(R\) which slides along a smooth horizontal wire fixed at a height \(a\) above \(A\) and in the same vertical plane as \(A B\). The string \(B R\) remains vertical. The angle between \(A B\) and the horizontal is denoted by \(\theta\), where \(0 < \theta < \pi\).

1. Taking the reference level for gravitational potential energy to be the horizontal through \(A\), show that the total potential energy of the system is $$m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) .$$
2. Find the three values of \(\theta\) for which the system is in equilibrium.
3. For each position of equilibrium, determine whether it is stable or unstable.