Challenging +1.2 This is a standard Further Maths mechanics problem on motion on a sphere with projectile motion afterward. It requires energy conservation, circular motion equations (N=0 at loss of contact), and projectile trajectory analysis, but follows a well-established template with straightforward algebra. The multi-part structure and specific numerical verification elevate it slightly above average difficulty.
5 A particle \(P\) of mass \(m\) is placed at the point \(Q\) on the outer surface of a fixed smooth sphere with centre \(O\) and radius \(a\). The acute angle between \(O Q\) and the upward vertical is \(\alpha\), where \(\cos \alpha = \frac { 9 } { 10 }\). The particle is released from rest and begins to move in a vertical circle on the surface of the sphere. Show that \(P\) loses contact with the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical, where \(\cos \theta = \frac { 3 } { 5 }\), and find the speed of \(P\) at this instant.
Show that, in the subsequent motion, when \(P\) is at a distance \(\frac { 7 } { 5 } a\) from the vertical diameter through \(O\), its distance below the horizontal through \(O\) is \(\frac { 31 } { 30 } a\).
5 A particle $P$ of mass $m$ is placed at the point $Q$ on the outer surface of a fixed smooth sphere with centre $O$ and radius $a$. The acute angle between $O Q$ and the upward vertical is $\alpha$, where $\cos \alpha = \frac { 9 } { 10 }$. The particle is released from rest and begins to move in a vertical circle on the surface of the sphere. Show that $P$ loses contact with the sphere when $O P$ makes an angle $\theta$ with the upward vertical, where $\cos \theta = \frac { 3 } { 5 }$, and find the speed of $P$ at this instant.
Show that, in the subsequent motion, when $P$ is at a distance $\frac { 7 } { 5 } a$ from the vertical diameter through $O$, its distance below the horizontal through $O$ is $\frac { 31 } { 30 } a$.
\hfill \mbox{\textit{CAIE FP2 2011 Q5 [12]}}