| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Work done and energy |
| Type | Particle on smooth curved surface |
| Difficulty | Standard +0.3 This is a standard M2 particle-on-curved-surface problem requiring energy conservation and circular motion equations. Part (a) is straightforward application of energy conservation (given as 'show that'), and part (b) requires setting normal reaction to zero. While it involves multiple steps, it follows a well-established method taught in all M2 courses with no novel insight required, making it slightly easier than average. |
| Spec | 6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Driving force \(= \frac{P}{v} = \frac{91100}{20} = 4555\) N | M1 A1 | |
| Newton's second law up slope: \(F - R - Mg\sin\theta = Ma\) | M1 | |
| \(4555 - 4000 - 1400 \times 9.8 \times \sin\theta = 1400 \times 0.2\) | A1 | |
| \(555 - 13720\sin\theta = 280\) | A1 | |
| \(\sin\theta = \frac{275}{13720}\) | M1 | |
| \(\theta = 1.15°\) | A1 A1 | Final answer |
# Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Driving force $= \frac{P}{v} = \frac{91100}{20} = 4555$ N | M1 A1 | |
| Newton's second law up slope: $F - R - Mg\sin\theta = Ma$ | M1 | |
| $4555 - 4000 - 1400 \times 9.8 \times \sin\theta = 1400 \times 0.2$ | A1 | |
| $555 - 13720\sin\theta = 280$ | A1 | |
| $\sin\theta = \frac{275}{13720}$ | M1 | |
| $\theta = 1.15°$ | A1 A1 | Final answer |6 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]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}
\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 Q6}}