| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Particle on inner surface of sphere/bowl |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring energy conservation, circular motion dynamics, and careful analysis of the loss of contact condition. The multi-part structure demands setting up force equations with variable normal reaction, applying conservation of energy twice in different contexts, and understanding when contact is lost (R=0). While the techniques are standard for FM students, the problem requires sustained reasoning across multiple steps and careful geometric/trigonometric setup, placing it well above average difficulty. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings6.05c Horizontal circles: conical pendulum, banked tracks6.05d Variable speed circles: energy methods |
| Answer | Marks |
|---|---|
| (iii) | 2 |
| Answer | Marks |
|---|---|
| Combine comps. to find v2: v2 = √(ga/8 + 11ga/8) = √(3ga/2) (A1) | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | [10] | |
| Page 5 | Mark Scheme: Teachers’ version | Syllabus |
| GCE A LEVEL – May/June 2012 | 9231 | 22 |
Question 3:
--- 3
(i)
(iii) ---
3
(i)
(iii) | 2
Use conservation of energy: ½mv = ½m(7ga/2)
– mga (1 – cos θ) B1
2
[v = 3ga/2 + 2ga cos θ]
2
Equate radial forces: R = mv /a + mg cos θ B1
Eliminate v to find R: R = (3mg/2) (1 + 2 cos θ) A.G. M1 A1
Find cos θ 1 when R = 0: cos θ 1 = – ½ [θ1 = 2π/3] B1
Find speed at this point: v1 2 = ½ ga, v1 = √(½ ga) M1 A1
EITHER: Use energy to find reqd speed v 2: ½mv2 2 = ½mv1 2 – mga cos θ1
or ½m(7ga/2) – mga M1
Simplify: ½v2 2 = ¼ga + ½ga, v2 = √(3ga/2) M1 A1
OR: Find horiz. comp. of v 2: v1 cos (π–θ) [= ½v1 = √(ga/8)] (M1)
Find vertical comp. of v2: √{v1 2 sin 2 (π–θ) + ga} [= √(11ga/8)] (M1)
Combine comps. to find v2: v2 = √(ga/8 + 11ga/8) = √(3ga/2) (A1) | 4
3
3 | [10]
Page 5 | Mark Scheme: Teachers’ version | Syllabus | Paper
GCE A LEVEL – May/June 2012 | 9231 | 22A particle $P$ of mass $m$ is projected horizontally with speed $\sqrt{\left(\frac{1}{2}ga\right)}$ from the lowest point of the inside of a fixed hollow smooth sphere of internal radius $a$ and centre $O$. The angle between $OP$ and the downward vertical at $O$ is denoted by $\theta$. Show that, as long as $P$ remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is $\frac{5}{2}mg(1 + 2\cos\theta)$.
[4]
Find the speed of $P$
\begin{enumerate}[label=(\roman*)]
\item when it loses contact with the sphere, [3]
\item when, in the subsequent motion, it passes through the horizontal plane containing $O$. (You may assume that this happens before $P$ comes into contact with the sphere again.) [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2012 Q3 [10]}}