| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Hemisphere or sphere resting on plane or wall |
| Difficulty | Challenging +1.8 This is a challenging mechanics problem requiring moments about a point with a tilted hemisphere, involving geometric reasoning to find moment arms, resolution of forces in a non-standard orientation, and understanding of equilibrium conditions including the constraint that normal reaction must be non-negative. It goes beyond routine textbook exercises by combining multiple concepts in a non-trivial configuration. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(F \times 0.4\sin 20° = 12 \times (0.4 / 2)\cos 20°\), \(F = 16.48\) | M1, A1, A1 | Moments about O; AG [3] |
| (ii) \(R = -16.48 + 12 + W\), \(-16.48 + 12 + W = 0\), \(W = 4.48\) | B1, M1, A1 | Equates forces vertically; Works with R = 0 [3] |
**(i)** $F \times 0.4\sin 20° = 12 \times (0.4 / 2)\cos 20°$, $F = 16.48$ | M1, A1, A1 | Moments about O; AG [3]
**(ii)** $R = -16.48 + 12 + W$, $-16.48 + 12 + W = 0$, $W = 4.48$ | B1, M1, A1 | Equates forces vertically; Works with R = 0 [3]
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\includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_786_1249_1455_447}
A smooth hemispherical shell, with centre $O$, weight 12 N and radius 0.4 m , rests on a horizontal plane. A particle of weight $W \mathrm {~N}$ lies at rest on the inner surface of the hemisphere vertically below $O$. A force of magnitude $F \mathrm {~N}$ acting vertically upwards is applied to the highest point of the hemisphere, which is in equilibrium with its axis of symmetry inclined at $20 ^ { \circ }$ to the horizontal (see diagram).\\
(i) Show, by taking moments about $O$, that $F = 16.48$ correct to 4 significant figures.\\
(ii) Find the normal contact force exerted by the plane on the hemisphere in terms of $W$. Hence find the least possible value of $W$.
\hfill \mbox{\textit{CAIE M2 2011 Q3 [6]}}