| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| 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 two-part centre of mass problem requiring (i) calculation using known formulae for semicircular lamina (4r/3π) and arc (2r/π) centres of mass, then combining using weighted averages, and (ii) applying equilibrium condition tan(θ) = x̄/ȳ. While it involves multiple steps and careful coordinate work, it follows a well-established template with no novel insight required, making it slightly easier than average. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((9+11)OG = \pm[9 \times 0.7/(\pi/2) - 11 \times (2 \times 0.7)/3\pi/2)]\) | M1, A1 | Table of value idea with signs either way round |
| \(OG = 0.0371\text{ m}\) AG | A1 [3] | Accept –ve answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\tan\theta = 0.0371(36..)/0.7\) | M1 | |
| \(\theta = 3.0°\) | A1 | |
| Lamina | B1 [3] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(9+11)OG = \pm[9 \times 0.7/(\pi/2) - 11 \times (2 \times 0.7)/3\pi/2)]$ | M1, A1 | Table of value idea with signs either way round |
| $OG = 0.0371\text{ m}$ AG | A1 [3] | Accept –ve answer |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\tan\theta = 0.0371(36..)/0.7$ | M1 | |
| $\theta = 3.0°$ | A1 | |
| Lamina | B1 [3] | |
---2\\
\includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_481_412_440_865}
The diagram shows a circular object formed from a uniform semicircular lamina of weight 11 N and a uniform semicircular arc of weight 9 N . The lamina and the arc both have centre $O$ and radius 0.7 m and are joined at the ends of their common diameter $A B$.\\
(i) Show that the distance of the centre of mass of the object from $O$ is 0.0371 m , correct to 3 significant figures.
The object hangs in equilibrium, freely suspended at $A$.\\
(ii) Find the angle between $A B$ and the vertical and state whether the lowest point of the object is on the lamina or on the arc.
\hfill \mbox{\textit{CAIE M2 2012 Q2 [6]}}