| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Circular Motion 2 |
| Type | Vertical circle: string becomes slack |
| Difficulty | Standard +0.8 This is a substantial vertical circle problem requiring energy conservation, circular motion dynamics, and finding the slack condition. Part (ii) requires deriving the standard complete circles condition, while part (iii)(b) involves solving a transcendental equation. More demanding than typical A-level mechanics but standard for Further Maths, with multiple connected steps requiring careful application of principles. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05f Vertical circle: motion including free fall |
10 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. A particle of mass $m$ is attached to the free end of the string and the particle hangs at rest vertically below $O$. The particle is projected horizontally with speed $u$.\\
(i) Find the tension in the string when it makes an angle $\theta$ with the downward vertical, whilst the string remains taut.\\
(ii) Deduce that the particle will perform complete circles provided that $u ^ { 2 } \geqslant 5 a g$.\\
(iii) It is given that $u ^ { 2 } = 4 a g$. Find
\begin{enumerate}[label=(\alph*)]
\item the tension in the string when $\theta = 60 ^ { \circ }$,
\item the value of $\theta$, to the nearest degree, at the instant when the string becomes slack.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q10}}