| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Non-uniform rod or wire suspended in equilibrium |
| Difficulty | Standard +0.8 This question requires finding the center of mass of a circular arc using integration (or the standard formula), then applying moment equilibrium about the suspension point. While the center of mass formula for an arc is standard, students must correctly set up the geometry and apply equilibrium conditions with careful angle work. The multi-step nature and geometric reasoning place it above average difficulty. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([OG = 1.5\sin45°/(\pi/4)]\) | M1 | For using \(OG = r\sin\alpha/\alpha\) where G is the centre of mass |
| Distance is 1.35m | A1 | AG. Subtotal: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For using \(\tan\theta\) in triangle AMG where M is the midpoint of AB | |
| \(\tan\theta = (1.35 - 1.5\cos45°)/1.5\cos45°\) \((= 4/\pi - 1)\) | A1 | |
| \(\theta = 15.3°\) | A1 | Subtotal: 3. Total: 5 |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[OG = 1.5\sin45°/(\pi/4)]$ | M1 | For using $OG = r\sin\alpha/\alpha$ where G is the centre of mass |
| Distance is 1.35m | A1 | AG. **Subtotal: 2** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For using $\tan\theta$ in triangle AMG where M is the midpoint of AB |
| $\tan\theta = (1.35 - 1.5\cos45°)/1.5\cos45°$ $(= 4/\pi - 1)$ | A1 | |
| $\theta = 15.3°$ | A1 | **Subtotal: 3. Total: 5** |
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\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_686_495_1238_826}
A uniform rigid wire $A B$ is in the form of a circular arc of radius 1.5 m with centre $O$. The angle $A O B$ is a right angle. The wire is in equilibrium, freely suspended from the end $A$. The chord $A B$ makes an angle of $\theta ^ { \circ }$ with the vertical (see diagram).\\
(i) Show that the distance of the centre of mass of the arc from $O$ is 1.35 m , correct to 3 significant figures.\\
(ii) Find the value of $\theta$.
\hfill \mbox{\textit{CAIE M2 2008 Q2 [5]}}