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\includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-3_585_801_991_647} A light rod \(A B\) of length \(2 a\) can rotate freely in a vertical plane about a fixed horizontal axis through \(A\). A particle of mass \(m\) is attached to the rod at \(B\). A fixed smooth ring \(R\) lies in the same vertical plane as the rod, where \(A R = a\) and \(A R\) makes an angle \(\frac { 1 } { 4 } \pi\) above the horizontal. A light elastic string, of natural length \(a\) and modulus of elasticity \(m g \sqrt { } 2\), passes through the ring \(R\); one end is fixed to \(A\) and the other end is fixed to \(B\). The rod makes an angle \(\theta\) below the horizontal, where \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\) (see diagram).

1. Use the cosine rule to show that \(R B ^ { 2 } = a ^ { 2 } ( 5 - ( 2 \sqrt { } 2 ) \cos \theta + ( 2 \sqrt { } 2 ) \sin \theta )\).
2. Show that \(\theta = 0\) is a position of stable equilibrium.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - k \sin \theta\), expressing the constant \(k\) in terms of \(a\) and \(g\), and hence write down the approximate period of small oscillations about the equilibrium position \(\theta = 0\).