| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Particle on inner surface of sphere/bowl |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring energy conservation, circular motion dynamics, and projectile motion analysis. Part (a) involves finding the loss-of-contact condition (normal force = 0) combined with energy methods, requiring careful angle work. Part (b) requires setting up and solving projectile equations to show the particle returns to point A, involving trigonometry and algebraic manipulation. While the techniques are standard for FM students, the multi-stage reasoning and the need to work with the specific angle result makes this significantly harder than typical A-level questions. |
| Spec | 6.05e Radial/tangential acceleration6.05f Vertical circle: motion including free fall |
A hollow cylinder of radius $a$ is fixed with its axis horizontal. A particle $P$, of mass $m$, moves in part of a vertical circle of radius $a$ and centre $O$ on the smooth inner surface of the cylinder. The speed of $P$ when it is at the lowest point $A$ of its motion is $\sqrt{\frac{7}{2}ga}$.
The particle $P$ loses contact with the surface of the cylinder when $OP$ makes an angle $\theta$ with the upward vertical through $O$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\theta = 60°$. [5]
\item Show that in its subsequent motion $P$ strikes the cylinder at the point $A$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q7 [10]}}