3 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 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 \(O P\) makes an angle \(\theta\) with the upward vertical, the tension in the string is \(S\).

1. Show that \(S = T - 3 m g ( 1 + \cos \theta )\).
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-04_2718_38_141_2010}
2. Given that \(u = \sqrt { 4 a g }\), find the value of \(\cos \theta\) when the string goes slack.
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-06_348_707_285_680} A light spring of natural length \(a\) and modulus of elasticity \(k m g\) is attached to a fixed point \(O\) on a smooth plane inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = \frac { 3 } { 4 }\). A particle of mass \(m\) is attached to the lower end of the spring and is held at the point \(A\) on the plane, where \(O A = 2 a\) and \(O A\) is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed \(V\) when it passes through the point \(B\) on the plane, where \(O B = \frac { 3 } { 2 } a\). The speed of the particle is \(\frac { 1 } { 2 } V\) when it passes through the point \(C\) on the plane, where \(O C = \frac { 3 } { 4 } a\). Find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-06_2718_35_141_2012}