5. A particle \(P\), of mass \(m \mathrm {~kg}\), 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 \(O\). Initially, \(P\) is held at rest with the string just taut and making an angle of \(60 ^ { \circ }\) with the downward vertical. It is then given a velocity \(u \mathrm {~ms} ^ { - 1 }\) perpendicular to the string in a downward direction.

1. 1. When the string makes an angle \(\theta\) with the downward vertical, the velocity of the particle is \(v\) and the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m , l , u ^ { 2 }\) and \(\theta\).
   2. Given that \(P\) describes complete circles in the subsequent motion, show that \(u ^ { 2 } > 4 l g\).
2. Given that now \(u ^ { 2 } = 3 l g\), find the position of the string when circular motion ceases. Briefly describe the motion of \(P\) after circular motion has ceased.
3. The string is replaced by a light rigid rod. Given that \(P\) describes complete circles in the subsequent motion, show that \(u ^ { 2 } > k l g\), where \(k\) is to be determined.