- A fixed smooth sphere has centre \(O\) and radius \(a\). A particle \(P\) is placed on the surface of the sphere at the point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(O P\) makes an angle \(\theta\) to the upward vertical through \(O , P\) is on the surface of the sphere and the speed of \(P\) is \(v\).
Given that \(\cos \alpha = \frac { 3 } { 5 }\)
- show that
$$v ^ { 2 } = \frac { 2 g a } { 5 } ( 3 - 5 \cos \theta )$$
- find the speed of \(P\) at the instant when it loses contact with the sphere.