| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Lamina hinged at point with string support |
| Difficulty | Standard +0.3 This is a standard two-part mechanics question requiring center of mass calculation for a composite lamina (routine application of moments formula) and then equilibrium of moments about a hinge. Both parts follow textbook methods with no novel insight required, though the multi-step nature and need to handle composite shapes makes it slightly above average difficulty. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04b Find centre of mass: using symmetry6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | M1 | For using \(M\vec{r} = \sum\) (or difference) of moments of components |
| Answer | Marks |
|---|---|
| \(6N, 0.1m, 9N, 0.1m, 12N, 0.4m\) | B1 |
| Answer | Marks |
|---|---|
| Distance is 0.22 m | A1 ⊞ |
| A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = 30\) | M1, A1, A1 | 3 |
**(i)** | M1 | For using $M\vec{r} = \sum$ (or difference) of moments of components
Correct weights and moment distances of components:
$8N, 0.3m, 6N, 0.4m, 27N, 0.3m$
$6N, 0.1m, 9N, 0.1m, 12N, 0.4m$ | B1 |
$15\bar{x} = 9 \times 0.3 + 6 \times 0.1$ or
$15\bar{x} = 9 \times 0.1 + 6 \times 0.4$ or
$15\bar{x} = 27 \times 0.3 - 12 \times 0.4$
Distance is 0.22 m | A1 ⊞
A1 | 4 |
**(ii)** $15 \times 0.22 = 11 \times 0.6 \sin \theta$
$\theta = 30$ | M1, A1, A1 | 3 | For taking moments about B
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-3_531_791_1633_678}
A uniform lamina of weight 15 N has dimensions as shown in the diagram.\\
(i) Show that the distance of the centre of mass of the lamina from $A B$ is 0.22 m .
The lamina is freely hinged at $B$ to a fixed point. One end of a light inextensible string is attached to the lamina at $C$. The string passes over a fixed smooth pulley and a particle of mass 1.1 kg is attached to the other end of the string. The lamina is in equilibrium with $B C$ horizontal. The string is taut and makes an angle of $\theta ^ { \circ }$ with the horizontal at $C$, and the particle hangs freely below the pulley (see diagram).\\
(ii) Find the value of $\theta$.
\hfill \mbox{\textit{CAIE M2 2006 Q5 [7]}}