Challenging +1.2 This is a standard two-part circular motion problem requiring energy conservation and the condition for losing contact (N=0 giving v²=g·r·cosθ). While it involves multiple steps (finding θ, then u, then projectile motion), each step follows well-established methods taught in Further Maths mechanics. The algebraic manipulation is moderate but routine for this level.
4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\).
Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)
4 A particle $P$ is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre $O$ and radius $a$. The particle is projected horizontally with speed $u$ and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point $A$, where $O A$ makes an angle $\theta$ with the upward vertical through $O$. Given that the speed of $P$ at $A$ is $\sqrt { } \left( \frac { 3 } { 5 } a g \right)$, find $u$ in terms of $a$ and $g$.
Find, in terms of $a$, the greatest height above the level of $O$ achieved by $P$ in its subsequent motion. (You may assume that $P$ achieves its greatest height before it makes any further contact with the sphere.)
\hfill \mbox{\textit{CAIE FP2 2016 Q4 [10]}}