| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| 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-step moments problem requiring: (1) finding the combined center of mass using the formula for semicircular lamina/arc centers of mass (4r/3π and 2r/π), (2) taking moments about the hinge with a tilted geometry (30° angle complicates perpendicular distances), and (3) resolving the horizontal force and weights. The geometric setup and angle work elevate this above routine moments questions, but it's still a standard M2-level problem with clear structure. |
| Spec | 3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(PE \text{ loss} = 0.8g \times (2.5 - 1.8) = 5.6J\) | B1 |
| Work done is 5.6 J | B1 | 2 marks total |
| (ii) | For using \(KE \text{ gain} = PE \text{ loss} - WD\) against resistance | M1 |
| \(\frac{1}{2} \times 0.8v^2 = 0.8g \times 2.5 - 0.6 \times 5.6\) | A1ft | |
| Speed at \(B\) is 6.45 ms\(^{-1}\) | A1 | 3 marks total |
(i) | $PE \text{ loss} = 0.8g \times (2.5 - 1.8) = 5.6J$ | B1 | |
| Work done is 5.6 J | B1 | 2 marks total |
(ii) | For using $KE \text{ gain} = PE \text{ loss} - WD$ against resistance | M1 | |
| $\frac{1}{2} \times 0.8v^2 = 0.8g \times 2.5 - 0.6 \times 5.6$ | A1ft | |
| Speed at $B$ is 6.45 ms$^{-1}$ | A1 | 3 marks total |
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\includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_426_531_258_808}
A circular object is formed from a uniform semicircular lamina of weight 12 N and a uniform semicircular arc of weight 8 N . The lamina and the arc both have centre $O$ and radius 0.6 m and are joined at the ends of their common diameter $A B$. The object is freely pivoted to a fixed point at $A$ with $A B$ inclined at $30 ^ { \circ }$ to the vertical. The object is in equilibrium acted on by a horizontal force of magnitude $F$ N applied at the lowest point of the object, and acting in the plane of the object (see diagram).\\
(i) Show that the centre of mass of the object is at $O$.\\
(ii) Calculate $F$.
\hfill \mbox{\textit{CAIE M2 2012 Q1 [6]}}