| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Lamina hinged at point with string support |
| Difficulty | Standard +0.8 This is a multi-part equilibrium problem requiring knowledge of the centroid of a semicircular lamina (4r/3π), taking moments about the hinge, resolving forces in two directions, and working with a three-force system. While the techniques are standard M2 content, the combination of non-trivial geometry (30° inclination), the need to recall/derive the centroid position, and careful resolution in multiple parts makes this moderately challenging—above average but not requiring novel insight. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04e Rigid body equilibrium: coplanar forces6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(d = \frac{2 \times 0.3\sin(\pi/2)}{(3\pi/2)}\) | B1 | \(d = 0.1273\) |
| \(T(0.6\cos 30°) = 0.4g(0.3\sin 30° + 0.1273\cos 30°)\) | M1 | |
| A1 | ||
| \(T = 2\) N | A1 AG [4] | \(2.003\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R = \sqrt{2^2 + (0.4g)^2}\) or \(\tan\theta = \frac{2}{0.4g}\) | M1 | Either (or \(\tan\alpha = \frac{0.4g}{2}\) with horizontal) |
| \(R = 4.47\) N | A1 | |
| \(\theta = 26.6°\) (with vertical) | A1 [3] | \(\alpha = 63.4°\) (with horizontal) |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $d = \frac{2 \times 0.3\sin(\pi/2)}{(3\pi/2)}$ | B1 | $d = 0.1273$ |
| $T(0.6\cos 30°) = 0.4g(0.3\sin 30° + 0.1273\cos 30°)$ | M1 | |
| | A1 | |
| $T = 2$ N | A1 AG **[4]** | $2.003\ldots$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R = \sqrt{2^2 + (0.4g)^2}$ or $\tan\theta = \frac{2}{0.4g}$ | M1 | Either (or $\tan\alpha = \frac{0.4g}{2}$ with horizontal) |
| $R = 4.47$ N | A1 | |
| $\theta = 26.6°$ (with vertical) | A1 **[3]** | $\alpha = 63.4°$ (with horizontal) |
---4\\
$A B$ is the diameter of a uniform semicircular lamina which has radius 0.3 m and mass 0.4 kg . The lamina is hinged to a vertical wall at $A$ with $A B$ inclined at $30 ^ { \circ }$ to the vertical. One end of a light inextensible string is attached to the lamina at $B$ and the other end of the string is attached to the wall vertically above $A$. The lamina is in equilibrium in a vertical plane perpendicular to the wall with the string horizontal (see diagram).\\
(i) Show that the tension in the string is 2.00 N correct to 3 significant figures.\\
(ii) Find the magnitude and direction of the force exerted on the lamina by the hinge.\\
\includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-3_956_540_258_804}
A small ball $B$ of mass 0.4 kg is attached to fixed points $P$ and $Q$ on a vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at $30 ^ { \circ }$ to the vertical. The ball moves in a horizontal circle (see diagram).\\
\hfill \mbox{\textit{CAIE M2 2010 Q4 [7]}}