| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Horizontal elastic string on rough surface |
| Difficulty | Standard +0.8 This question requires applying energy conservation with elastic potential energy, work-energy principles, and friction over multiple stages. Students must recognize when the string is slack (at natural length), calculate elastic PE correctly, set up energy equations accounting for friction work, then use dynamics to find the coefficient of friction. It combines several M2 concepts in a non-trivial way requiring careful reasoning about the physical situation. |
| Spec | 3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Initial EE \(= 6 \times 2^2 \div (2 \times 1.5)\) | B1 | |
| Final KE \(= \frac{1}{2} \times 0.4 \times 6^2\) | B1 | |
| M1 | For using WD against friction = initial EPE − final KE | |
| \(\text{WD} = 6 \times 4 \div (2 \times 1.5) - \frac{1}{2} \times 0.4 \times 6^2\) | A1 ft | Any correct form |
| WD against friction is \(0.8\) J | A1 | [5 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.8 = \mu \times 0.4g \times 2\) | M1 | For using WD \(= F \times d\) and \(F = \mu R\) |
| Coefficient is \(0.1\) | A1 ft | ft \(\mu = \text{WD} \div 8\) [2 marks] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Initial EE $= 6 \times 2^2 \div (2 \times 1.5)$ | B1 | |
| Final KE $= \frac{1}{2} \times 0.4 \times 6^2$ | B1 | |
| | M1 | For using WD against friction = initial EPE − final KE |
| $\text{WD} = 6 \times 4 \div (2 \times 1.5) - \frac{1}{2} \times 0.4 \times 6^2$ | A1 ft | Any correct form |
| WD against friction is $0.8$ J | A1 | **[5 marks]** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.8 = \mu \times 0.4g \times 2$ | M1 | For using WD $= F \times d$ and $F = \mu R$ |
| Coefficient is $0.1$ | A1 ft | ft $\mu = \text{WD} \div 8$ **[2 marks]** |
---4 A particle $P$ of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal table. $P$ is released from rest at a point on the table 3.5 m from $O$. The speed of $P$ at the instant the string becomes slack is $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find\\
(i) the work done against friction during the period from the release of $P$ until the string becomes slack,\\
(ii) the coefficient of friction between $P$ and the table.
\hfill \mbox{\textit{CAIE M2 2005 Q4 [7]}}