| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Lamina suspended in equilibrium |
| Difficulty | Standard +0.3 This is a standard centre of mass problem using the known formula for a circular quadrant (distance 4r/3π from centre) followed by a straightforward equilibrium calculation with trigonometry. Both parts require routine application of memorized results with minimal problem-solving, making it slightly easier than average. |
| Spec | 6.04b Find centre of mass: using symmetry6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 2 \times 0.6\sin(\pi/4)/(3\pi/4)\ [= 0.36(0)]\) | B1 | Centre of mass from O |
| \(d^2 = 0.6^2 + 0.36^2 - 2 \times 0.6 \times 0.36\cos(\pi/4)\) | M1 | |
| \(d = 0.429\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sin\alpha / 0.36 = \sin(\pi/4)/0.429\) | M1 | |
| \(\alpha = 36.4°\) or \(0.635^c\) | A1, A1 [3] |
## Question 2:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 2 \times 0.6\sin(\pi/4)/(3\pi/4)\ [= 0.36(0)]$ | B1 | Centre of mass from O |
| $d^2 = 0.6^2 + 0.36^2 - 2 \times 0.6 \times 0.36\cos(\pi/4)$ | M1 | |
| $d = 0.429$ | A1 [3] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin\alpha / 0.36 = \sin(\pi/4)/0.429$ | M1 | |
| $\alpha = 36.4°$ or $0.635^c$ | A1, A1 [3] | |
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\includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_406_483_431_829}\\
$A O B$ is a uniform lamina in the shape of a quadrant of a circle with centre $O$ and radius 0.6 m (see diagram).\\
(i) Calculate the distance of the centre of mass of the lamina from $A$.
The lamina is freely suspended at $A$ and hangs in equilibrium.\\
(ii) Find the angle between the vertical and the side $A O$ of the lamina.
\hfill \mbox{\textit{CAIE M2 2011 Q2 [6]}}