| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | March |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Hemisphere or sphere resting on plane or wall |
| Difficulty | Challenging +1.2 This is a standard mechanics equilibrium problem requiring resolution of forces, friction at limiting equilibrium (F = μR), and taking moments about a suitable point. The key insight that the center of mass is at 3r/8 from the base is standard knowledge for hemisphere problems. While it involves multiple steps (resolving horizontally/vertically, moments equation, finding μ), each step follows routine procedures without requiring novel geometric insight or complex manipulation. |
| Spec | 3.03u Static equilibrium: on rough surfaces6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(60(3 \times 0.8/8) \times 0.28 = P(0.8 - 0.8 \times 0.28)\) | M1, A1 | An attempt at taking moments |
| \(P = 8.75\) | A1 (AG) | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mu = 8.75/60\) | M1 | |
| \(\mu = 0.146\) | A1 | Total: 2 |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $60(3 \times 0.8/8) \times 0.28 = P(0.8 - 0.8 \times 0.28)$ | M1, A1 | An attempt at taking moments |
| $P = 8.75$ | A1 (AG) | **Total: 3** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mu = 8.75/60$ | M1 | |
| $\mu = 0.146$ | A1 | **Total: 2** |
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\includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_295_805_484_671}
A uniform solid hemisphere of weight 60 N and radius 0.8 m rests in limiting equilibrium with its curved surface on a rough horizontal plane. The axis of symmetry of the hemisphere is inclined at an angle of $\theta$ to the horizontal, where $\cos \theta = 0.28$. Equilibrium is maintained by a horizontal force of magnitude $P$ N applied to the lowest point of the circular rim of the hemisphere (see diagram).\\
(i) Show that $P = 8.75$.\\
(ii) Find the coefficient of friction between the hemisphere and the plane.
\hfill \mbox{\textit{CAIE M2 2016 Q2 [5]}}