A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end is attached to a fixed point \(O\).

When the particle is hanging vertically below \(O\), it is projected horizontally with speed \(u\), so that it begins to move along a circular path. When \(P\) is at the lowest point of its motion, the tension in the string is \(T\). When \(OP\) makes an angle \(\theta\) with the upward vertical, the tension in the string is \(S\).

(a) Show that \(S=T-3mg(1+\cos\theta)\).

(b) Given that \(u=\sqrt{4ag}\), find \(\cos\theta\) when the string goes slack.