4 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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\).

1. Show that the least possible value of \(u\) is \(\sqrt { } ( a g )\).
   It is now given that \(u = \sqrt { } ( a g )\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac { 1 } { 4 } m\).
2. Find the speed of the combined particle immediately after the collision.
   In the subsequent motion, when \(O P\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\).
3. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
4. Find the value of \(\cos \theta\) when the string becomes slack.