| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Rotating lamina about fixed axis |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring recall of a standard formula for the centre of mass of a semicircular lamina (4r/3π from the diameter) and basic circular motion (v = rω). Both parts are direct applications with no problem-solving or novel insight required, making it easier than average. |
| Spec | 6.04b Find centre of mass: using symmetry6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\bar{r} = \frac{2 \times 5 \sin 90°}{3 \times \pi/2}\) | M1 | For using \(\bar{r} = \frac{2r\sin\alpha}{3\alpha}\) |
| Distance is 2.12 m | A1 | Accept \(20/3\pi\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For using \(v = r\omega\) | |
| Speed is \(8.49 \text{ ms}^{-1}\) | A1ft | 2 |
## Question 1:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{r} = \frac{2 \times 5 \sin 90°}{3 \times \pi/2}$ | M1 | For using $\bar{r} = \frac{2r\sin\alpha}{3\alpha}$ |
| Distance is 2.12 m | A1 | Accept $20/3\pi$ |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For using $v = r\omega$ |
| Speed is $8.49 \text{ ms}^{-1}$ | A1ft | 2 | ft their answer to part (i) |
**Total: 4 marks**
---1\\
\includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-2_467_645_274_749}
A uniform semicircular lamina has radius 5 m . The lamina rotates in a horizontal plane about a vertical axis through $O$, the mid-point of its diameter. The angular speed of the lamina is $4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ (see diagram). Find\\
(i) the distance of the centre of mass of the lamina from $O$,\\
(ii) the speed with which the centre of mass of the lamina is moving.
\hfill \mbox{\textit{CAIE M2 2007 Q1 [4]}}