Question 11 EITHER alternative.

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held so that the string is taut, with \(OP\) horizontal. The particle is projected downwards with speed \(\sqrt{\frac{2}{5}ag}\) and begins to move in a vertical circle. The string breaks when its tension is equal to \(\frac{11}{5}mg\).

(i) Show that the string breaks when \(OP\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos\theta=\frac35\). Find the speed of \(P\) at this instant.

(ii) For the subsequent motion after the string breaks, find the distance \(OP\) when the particle \(P\) is vertically below \(O\).