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\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-2_439_1045_1512_550} Two small smooth pegs \(A\) and \(B\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(C\) is the mid-point of \(A B\). A uniform rod \(C D\), of mass \(m\) and length \(a\), is freely pivoted at \(C\) and can rotate in the vertical plane containing \(A B\), with \(D\) below the level of \(A B\). A light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), passes round the peg \(A\) and its ends are attached to \(C\) and \(D\). Another light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), passes round the peg \(B\) and its ends are also attached to \(C\) and \(D\). The angle \(C A D\) is \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), so that the angle \(B C D\) is \(2 \theta\) (see diagram).

1. Taking \(A B\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$\frac { 1 } { 2 } m g a ( 14 - 2 \cos 2 \theta - \sin 2 \theta )$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.