| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Frame with circular arc or semicircular arc components |
| Difficulty | Standard +0.3 This is a standard centre of mass problem for a composite frame requiring the formula for the centroid of a semicircular arc (2r/π) and basic equilibrium geometry. While it involves multiple steps, both parts use well-known results and straightforward calculations, making it slightly easier than average for A-level mechanics. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(OG(arc) = 0.6\sin(\pi/2)/(\pi/2)\) | B1 | \(0.38197...\) |
| \((0.6\pi + 2 \times 0.6)d\) | M1 | Moment equation |
| \(= 2 \times 0.6 \times 0 + 0.6\pi \times 0.382\) | A1 | |
| \(d = 0.233\) m | A1 [4] | \(0.2333...\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\tan\theta = 0.233/0.6\) | M1 | |
| \(\theta = 21.2/21.3°\) or \(0.371\) radians | A1ft [2] | \(\tan^{-1}(cv(i)/0.6)\) |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $OG(arc) = 0.6\sin(\pi/2)/(\pi/2)$ | B1 | $0.38197...$ |
| $(0.6\pi + 2 \times 0.6)d$ | M1 | Moment equation |
| $= 2 \times 0.6 \times 0 + 0.6\pi \times 0.382$ | A1 | |
| $d = 0.233$ m | A1 [4] | $0.2333...$ |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\tan\theta = 0.233/0.6$ | M1 | |
| $\theta = 21.2/21.3°$ or $0.371$ radians | A1ft [2] | $\tan^{-1}(cv(i)/0.6)$ |
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\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-2_359_686_484_731}
A uniform frame consists of a semicircular arc $A B C$ of radius 0.6 m together with its diameter $A O C$, where $O$ is the centre of the semicircle (see diagram).\\
(i) Calculate the distance of the centre of mass of the frame from $O$.
The frame is freely suspended at $A$ and hangs in equilibrium.\\
(ii) Calculate the angle between $A C$ and the vertical.
\hfill \mbox{\textit{CAIE M2 2013 Q2 [6]}}