1 A particle of mass \(m\) is attached to one end \(A\) of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of \(120 ^ { \circ }\). Show that the speed of the particle is then \(\sqrt { } \left( \frac { 1 } { 2 } g a \right)\) and find the initial speed of projection.