- A particle \(P\) is attached to one end of a light inextensible string of length \(l \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u \mathrm {~ms} ^ { - 1 }\) and starts to move in a vertical circle.
Given that the string becomes slack when it makes an angle of \(120 ^ { \circ }\) with the downward vertical through \(O\),
- show that \(u ^ { 2 } = \frac { 7 g l } { 2 }\).
- Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion.
(7 marks)