6. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(4 m\) and length \(4 l\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). A particle of mass \(k m\), where \(k < 7\), is attached to the rod at \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity 4 mg , is attached to the point \(D\) of the rod, where \(A D = 3 l\). The other end of the string is attached to a fixed point \(E\) which is vertically above \(A\), where \(A E = 3 l\), as shown in Figure 2. The angle between the rod and the upward vertical is \(2 \theta\), where \(\arcsin \left( \frac { 1 } { 6 } \right) < \theta \leqslant \frac { \pi } { 2 }\).

1. Show that, while the string is stretched, the potential energy of the system is $$8 m g l \left\{ ( 7 - k ) \sin ^ { 2 } \theta - 3 \sin \theta \right\} + \text { constant }$$ There is a position of equilibrium with \(\theta \leqslant \frac { \pi } { 6 }\).
2. Show that \(k \leqslant 4\) Given that \(k = 4\),
3. show that this position of equilibrium is stable.