3 A particle \(P\), of mass \(m\), is placed at the highest point of a fixed solid smooth sphere with centre \(O\) and radius \(a\). The particle \(P\) is given a horizontal speed \(u\) and it moves in part of a vertical circle, with centre \(O\), on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the surface of the sphere, the speed of \(P\) is \(v\) and the reaction of the sphere on \(P\) has magnitude \(R\). Show that \(R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the sphere at the instant when \(v = 2 u\). Find \(u\) in terms of \(a\) and \(g\).