| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| 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 Further Maths mechanics problem requiring knowledge of the center of mass of a hemisphere (3a/8 from base), resolution of forces on an inclined plane, friction limits, and taking moments about a carefully chosen point. It involves multiple equilibrium conditions and algebraic manipulation to reach the given inequalities, but follows a standard approach for this type of problem once the setup is understood. |
| Spec | 6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(F\cos\alpha = R\sin\alpha\) | B2 | *EITHER:* Resolve horizontally |
| \(F = (M+m)g\sin\alpha\) | (B1) | *OR:* Resolve along plane to find friction \(F\) |
| \(R = (M+m)g\cos\alpha\) | (B1) | Resolve normal to plane for reaction \(R\) |
| \(\tan\alpha \leq \frac{1}{2}\) A.G. | M1 A1 | Use \(F/R \leq \frac{1}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(mg(a - a\sin\alpha) = Mga\sin\alpha\) | *EITHER:* Take moments about pt of contact | |
| \(mg = F = (M+m)g\sin\alpha\) | M1 A1 | *OR:* Take moments about centre |
| \(\sin\alpha \leq 1/\sqrt{5}\) | B1 | Find inequality for \(\sin\alpha\) |
| \(m \leq M/(1/\sin\alpha - 1) \leq M/(\sqrt{5}-1)\) | Combine | |
| \(m \leq M(1+\sqrt{5})/4\) A.G. | M1 A1 |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $F\cos\alpha = R\sin\alpha$ | B2 | *EITHER:* Resolve horizontally |
| $F = (M+m)g\sin\alpha$ | (B1) | *OR:* Resolve along plane to find friction $F$ |
| $R = (M+m)g\cos\alpha$ | (B1) | Resolve normal to plane for reaction $R$ |
| $\tan\alpha \leq \frac{1}{2}$ **A.G.** | M1 A1 | Use $F/R \leq \frac{1}{2}$ |
**Total part: [4]**
## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $mg(a - a\sin\alpha) = Mga\sin\alpha$ | | *EITHER:* Take moments about pt of contact |
| $mg = F = (M+m)g\sin\alpha$ | M1 A1 | *OR:* Take moments about centre |
| $\sin\alpha \leq 1/\sqrt{5}$ | B1 | Find inequality for $\sin\alpha$ |
| $m \leq M/(1/\sin\alpha - 1) \leq M/(\sqrt{5}-1)$ | | Combine |
| $m \leq M(1+\sqrt{5})/4$ **A.G.** | M1 A1 | |
**Total part: [5], Question Total: [9]**
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A uniform solid hemisphere, of radius $a$ and mass $M$, is placed with its curved surface in contact with a rough plane that is inclined at an angle $\alpha$ to the horizontal. A particle $P$ of mass $m$ is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and $P$ at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is $\frac { 1 } { 2 }$, show that\\
(i) $\tan \alpha \leqslant \frac { 1 } { 2 }$,\\
(ii) $m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }$.
\hfill \mbox{\textit{CAIE FP2 2011 Q3 [9]}}