A non-uniform rod \(B C\) has mass \(m\) and length \(3 l\). The centre of mass of the rod is at distance \(l\) from \(B\). The rod can turn freely about a fixed smooth horizontal axis through \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\frac { m g } { 6 }\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3 l\) vertically above \(B\).
Show that, while the string is stretched, the potential energy of the system is
$$m g l \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant, }$$
where \(\theta\) is the angle between the string and the downward vertical and \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\).
Find the values of \(\theta\) for which the system is in equilibrium with the string stretched.