8 A bead, of mass \(m\), moves on a smooth circular ring, of radius \(a\) and centre \(O\), which is fixed in a vertical plane. At \(P\), the highest point on the ring, the speed of the bead is \(2 u\); at \(Q\), the lowest point on the ring, the speed of the bead is \(5 u\).
- Show that \(u = \sqrt { \frac { 4 a g } { 21 } }\).
(4 marks) - \(\quad S\) is a point on the ring so that angle \(P O S\) is \(60 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-4_600_540_657_760}
Find, in terms of \(m\) and \(g\), the magnitude of the reaction of the ring on the bead when the bead is at \(S\).