Standard +0.8 This is a classic vertical circular motion problem requiring energy conservation and Newton's second law in the radial direction. While the derivation of the reaction force formula is standard Further Maths fare, the second part requires students to combine the loss-of-contact condition (R=0) with the constraint v=2u and the energy equation, involving algebraic manipulation of multiple equations. This is moderately challenging but follows established patterns for FP2/FM mechanics.
3 A particle \(P\), of mass \(m\), is placed at the highest point of a fixed solid smooth sphere with centre \(O\) and radius \(a\). The particle \(P\) is given a horizontal speed \(u\) and it moves in part of a vertical circle, with centre \(O\), on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the surface of the sphere, the speed of \(P\) is \(v\) and the reaction of the sphere on \(P\) has magnitude \(R\). Show that \(R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\).
The particle loses contact with the sphere at the instant when \(v = 2 u\). Find \(u\) in terms of \(a\) and \(g\).
3 A particle $P$, of mass $m$, is placed at the highest point of a fixed solid smooth sphere with centre $O$ and radius $a$. The particle $P$ is given a horizontal speed $u$ and it moves in part of a vertical circle, with centre $O$, on the surface of the sphere. When $O P$ makes an angle $\theta$ with the upward vertical, and $P$ is still in contact with the surface of the sphere, the speed of $P$ is $v$ and the reaction of the sphere on $P$ has magnitude $R$. Show that $R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }$.
The particle loses contact with the sphere at the instant when $v = 2 u$. Find $u$ in terms of $a$ and $g$.
\hfill \mbox{\textit{CAIE FP2 2015 Q3 [9]}}