Show that, when \(\angle B F O = \theta\), the potential energy of the system is
$$\frac { 1 } { 10 } m g a ( 8 \cos \theta - 5 ) ^ { 2 } - 2 m g a \cos ^ { 2 } \theta + \text { constant } .$$
Hence find the values of \(\theta\) for which the system is in equilibrium.
Determine the nature of the equilibrium at each of these positions.