| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Vertical circle – surface contact (sphere/track, leaving surface) |
| Difficulty | Challenging +1.2 This is a standard circular motion problem requiring energy conservation and Newton's second law in the radial direction. The first part (2 marks) is straightforward application of energy conservation. The second part (4 marks) requires resolving forces and using the centripetal acceleration formula, which are routine techniques for Further Maths students. While it involves multiple steps and careful algebra, it follows a well-established method without requiring novel insight. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
| Answer | Marks |
|---|---|
| 2 | 2 2 |
| Answer | Marks |
|---|---|
| Replace cos θ by 1 – 3u /8ga (A.E.F.): N = mg – mu /8a M1 A1 | 2 |
| 4 | [6] |
Question 2:
2 | 2 2
Apply conservation of energy: ½mv = ½mu – mga(1 – cos θ) M1
Put v = ½u and simplify: 8ga(1 – cos θ) = 3u 2 A.G. A1
2
Equate radial forces to find contact force N: N = mg cos θ + m(½u) /a M1 A1
2 2
Replace cos θ by 1 – 3u /8ga (A.E.F.): N = mg – mu /8a M1 A1 | 2
4 | [6]A particle $P$ of mass $m$ is projected horizontally with speed $u$ from the lowest point on the inside of a fixed hollow sphere with centre $O$. The sphere has a smooth internal surface of radius $a$. Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to $\frac{1}{2}u$ the angle $\theta$ between $OP$ and the downward vertical satisfies the equation
$$8ga(1 - \cos\theta) = 3u^2.$$ [2]
Find, in terms of $m$, $u$, $a$ and $g$, an expression for the magnitude of the contact force acting on the particle in this position. [4]
\hfill \mbox{\textit{CAIE FP2 2010 Q2 [6]}}