4. \begin{figure}[h] \end{figure} A uniform rod \(P Q\) has mass \(m\) and length \(2 l\). A small smooth light ring is fixed to the end \(P\) of the rod. This ring is threaded on to a fixed horizontal smooth straight wire. A second small smooth light ring \(R\) is threaded on to the wire and is attached by a light elastic string, of natural length \(l\) and modulus of elasticity \(k m g\), to the end \(Q\) of the rod, where \(k\) is a constant.

1. Show that, when the rod \(P Q\) makes an angle \(\theta\) with the vertical, where \(0 < \theta \leq \frac { \pi } { 3 }\), and \(Q\) is vertically below \(R\), as shown in Figure 1, the potential energy of the system is $$m g l \left[ 2 k \cos ^ { 2 } \theta - ( 2 k + 1 ) \cos \theta \right] + \text { constant. }$$ Given that there is a position of equilibrium with \(\theta > 0\),
2. show that \(k > \frac { 1 } { 2 }\).