| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Vertical circle: string becomes slack |
| Difficulty | Challenging +1.2 This is a standard Further Maths vertical circle problem requiring energy conservation and circular motion equations (T - mg cos θ = mv²/a), followed by projectile motion. The multi-step nature and combination of techniques elevates it above average, but the methods are well-practiced in FM courses with no novel insight required. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings6.05d Variable speed circles: energy methods |
A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is held so that the string is taut, with $O P$ horizontal. The particle is projected downwards with speed $\sqrt { } \left( \frac { 2 } { 5 } a g \right)$ and begins to move in a vertical circle. The string breaks when its tension is equal to $\frac { 11 } { 5 } m g$.\\
\begin{enumerate}[label=(\roman*)]
\item Show that the string breaks when $O P$ makes an angle $\theta$ with the downward vertical through $O$, where $\cos \theta = \frac { 3 } { 5 }$. Find the speed of $P$ at this instant.
\item For the subsequent motion after the string breaks, find the distance $O P$ when the particle $P$ is vertically below $O$.
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2018 Q11 EITHER}}