A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt{\frac{7}{2} g a}\).

The particle \(P\) loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
(a) Show that \(\theta=60^{\circ}\).

(b) Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\).