| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2015 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle in hemispherical bowl |
| Difficulty | Standard +0.3 This is a standard circular motion problem requiring resolution of forces (weight and normal reaction) and application of F=mrω². Part (i) involves straightforward use of v=rω to find the radius, then geometry. Part (ii) requires setting up two equations (vertical equilibrium and centripetal force) and solving simultaneously. While multi-step, it follows a well-established method taught in M2 with no novel insight required, making it slightly easier than average. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(4 = 10r\) | B1 | \(v = \omega r\), \(r < 0.5\) |
| \(\theta = 53.1°\) | B1 | From \(\sin\theta = 0.4/0.5\) [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(0.4g = 6\cos\theta\) | M1 | |
| \(\theta = 48.2°\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(6\sin\theta = 0.4\omega^2 \times 0.5\sin\theta\) | M1 A1\(\checkmark\) | \(\text{Accn} = \omega^2 \times 0.5\sin\theta\); Using \(\theta = 48.2°\), allow any acute \(\theta\) |
| \(\omega = 5.48 \text{ rad s}^{-1}\) | A1 | [3] |
## Question 4(i):
| Working | Mark | Guidance |
|---------|------|----------|
| $4 = 10r$ | B1 | $v = \omega r$, $r < 0.5$ |
| $\theta = 53.1°$ | B1 | From $\sin\theta = 0.4/0.5$ **[2]** |
## Question 4(ii)(a):
| Working | Mark | Guidance |
|---------|------|----------|
| $0.4g = 6\cos\theta$ | M1 | |
| $\theta = 48.2°$ | A1 | **[2]** |
## Question 4(ii)(b):
| Working | Mark | Guidance |
|---------|------|----------|
| $6\sin\theta = 0.4\omega^2 \times 0.5\sin\theta$ | M1 A1$\checkmark$ | $\text{Accn} = \omega^2 \times 0.5\sin\theta$; Using $\theta = 48.2°$, allow any acute $\theta$ |
| $\omega = 5.48 \text{ rad s}^{-1}$ | A1 | **[3]** |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-2_387_613_1073_767}
A particle $P$ of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre $O$ and radius 0.5 m (see diagram).\\
(i) Given that the speed of the particle is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its angular speed is $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$, calculate the angle between $O P$ and the vertical.\\
(ii) Given instead that the magnitude of the force exerted on $P$ by the hemisphere is 6 N , calculate
\begin{enumerate}[label=(\alph*)]
\item the angle between $O P$ and the vertical,
\item the angular speed of $P$.
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2015 Q4 [7]}}