| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Advanced work-energy problems |
| Type | Particle on sphere or circular surface |
| Difficulty | Standard +0.3 This is a standard M2 particle-on-sphere problem requiring energy conservation and circular motion conditions. Part (a) is straightforward application of energy conservation (given as 'show that'), and part (b) requires setting normal reaction to zero, leading to a routine equation to solve. The setup is familiar and the techniques are well-practiced in M2, making it slightly easier than average. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05e Radial/tangential acceleration |
7 A particle $P$, of mass $m \mathrm {~kg}$, is placed at the point $Q$ on the top of a smooth upturned hemisphere of radius 3 metres and centre $O$. The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of $2 \mathrm {~ms} ^ { - 1 }$. When the particle is on the surface of the hemisphere, the angle between $O P$ and $O Q$ is $\theta$ and the particle has speed $v \mathrm {~ms} ^ { - 1 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-4_415_1007_1573_513}
\begin{enumerate}[label=(\alph*)]
\item Show that $v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )$.
\item Find the value of $\theta$ when the particle leaves the hemisphere.
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2006 Q7 [9]}}