7
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-4_517_677_267_733} A particle \(P\) of mass \(m \mathrm {~kg}\) is slightly disturbed from rest at the highest point on the surface of a smooth fixed sphere of radius \(a\) m and centre \(O\). The particle starts to move downwards on the surface. While \(P\) remains on the surface \(O P\) makes an angle of \(\theta\) radians with the upward vertical and has angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The sphere exerts a force of magnitude \(R \mathrm {~N}\) on \(P\).

1. Show that \(a \omega ^ { 2 } = 2 g ( 1 - \cos \theta )\).
2. Find an expression for \(R\) in terms of \(m , g\) and \(\theta\). At the instant that \(P\) loses contact with the surface of the sphere, find
3. the transverse component of the acceleration of \(P\),
4. the rate of change of \(R\) with respect to time \(t\), in terms of \(m , g\) and \(a\).