| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string on smooth inclined plane |
| Difficulty | Challenging +1.2 This is a multi-part elastic string problem requiring equilibrium analysis, energy conservation with elastic potential energy, and careful consideration of two extreme positions. While it involves several steps and the integration of multiple concepts (equilibrium, EPE, GPE, kinematics), the techniques are standard for M2 level and follow a predictable structure. The problem is more challenging than routine single-concept questions but doesn't require novel insight beyond applying standard energy methods. |
| Spec | 6.02g Hooke's law: T = k*x or T = lambda*x/l6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(0.8g\sin30 = 20e/0.4\) | M1 | |
| \(e = 0.08 \text{ m}\) | A1 | [2] |
| (ii) | M1 | Conservation of KE, PE, EE |
| \(0.8v^2/2 + 20 \times 0.08^2/(2 \times 0.4)\) | A1 | Correct start terms, signs accurate |
| \(= 0.8g(0.4 + 0.08)\sin30\) | A1 | Correct final term, sign accurate |
| \(v = 2.1(0) \text{ ms}^{-1}\) | A1 | [4] |
| (iii) | M1* | |
| \(0.8g d\sin30 = 20(d - 0.4)^2/(2 \times 0.4)\) | A1 | \(4d = 25(d - 0.4)^2\) |
| \(25d^2 - 24d + 4 = 0\) | D*M1 | Obtains and solves a 3 term quadratic equation |
| \(d = 0.745 \text{ m}\) | A1 | [4] |
**(i)** $0.8g\sin30 = 20e/0.4$ | M1 |
$e = 0.08 \text{ m}$ | A1 | [2]
**(ii)** | M1 | Conservation of KE, PE, EE
$0.8v^2/2 + 20 \times 0.08^2/(2 \times 0.4)$ | A1 | Correct start terms, signs accurate
$= 0.8g(0.4 + 0.08)\sin30$ | A1 | Correct final term, sign accurate
$v = 2.1(0) \text{ ms}^{-1}$ | A1 | [4]
**(iii)** | M1* |
$0.8g d\sin30 = 20(d - 0.4)^2/(2 \times 0.4)$ | A1 | $4d = 25(d - 0.4)^2$
$25d^2 - 24d + 4 = 0$ | D*M1 | Obtains and solves a 3 term quadratic equation
$d = 0.745 \text{ m}$ | A1 | [4]7 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a particle $P$ of mass 0.8 kg . The other end of the string is attached to a fixed point $O$ at the top of a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. The particle rests in equilibrium on the plane.\\
(i) Calculate the extension of the string.\\
$P$ is projected from its equilibrium position up the plane along a line of greatest slope. In the subsequent motion $P$ just reaches $O$, and later just reaches the foot of the plane. Calculate\\
(ii) the speed of projection of $P$,\\
(iii) the length of the line of greatest slope of the plane.
\hfill \mbox{\textit{CAIE M2 2011 Q7 [10]}}