| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Advanced work-energy problems |
| Type | Elastic string horizontal surface projection |
| Difficulty | Challenging +1.2 This is a multi-stage mechanics problem involving elastic strings, forces in equilibrium, and work-energy principles. Part (i) requires resolving forces with the elastic string tension, part (ii) applies work-energy theorem with elastic potential energy, and part (iii) uses the work-friction relationship. While it requires careful setup and multiple techniques (elastic strings, trigonometry, energy methods), the individual steps follow standard M2 procedures without requiring novel insight. The multi-part nature and integration of several topics makes it moderately above average difficulty. |
| Spec | 3.03v Motion on rough surface: including inclined planes6.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(T = 9 \times (0.4/\sin\theta)/1.5\) | B1 | Must use the general angle \(\theta\) |
| \(R = 0.6g - T\sin\theta\) | M1 | |
| \(R = 3.6\text{N}\) | AG, A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\text{Ext(A)} = 0.4/\sin 30,\ \text{ext(B)} = 0.4\) | B1 | |
| \(\text{EE change} = 9 \times (0.8^2 - 0.4^2)/(2 \times 1.5)\) | B1 | Uses EE in two positions. EE=1.44 |
| \(WD = 1.44 - 0.6 \times (3^2 - 2.5^2)/2\) | M1 | |
| \(WD = 0.615\text{J}\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\mu = 0.615/(0.4/\tan 30)/3.6\) | M1 | |
| \(\mu = 0.247\) | A1 | [2] Accept 0.246 without wrong working |
## Question 6:
### Part (i):
| Working | Mark | Guidance |
|---------|------|----------|
| $T = 9 \times (0.4/\sin\theta)/1.5$ | B1 | Must use the general angle $\theta$ |
| $R = 0.6g - T\sin\theta$ | M1 | |
| $R = 3.6\text{N}$ | AG, A1 | [3] |
### Part (ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $\text{Ext(A)} = 0.4/\sin 30,\ \text{ext(B)} = 0.4$ | B1 | |
| $\text{EE change} = 9 \times (0.8^2 - 0.4^2)/(2 \times 1.5)$ | B1 | Uses EE in two positions. EE=1.44 |
| $WD = 1.44 - 0.6 \times (3^2 - 2.5^2)/2$ | M1 | |
| $WD = 0.615\text{J}$ | A1 | [4] |
### Part (iii):
| Working | Mark | Guidance |
|---------|------|----------|
| $\mu = 0.615/(0.4/\tan 30)/3.6$ | M1 | |
| $\mu = 0.247$ | A1 | [2] Accept 0.246 without wrong working |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-3_652_618_849_762}
A particle $P$ of mass 0.6 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 9 N . The string passes through a small smooth ring $R$ fixed at a height of 0.4 m above a rough horizontal surface. The other end of the string is attached to a fixed point $O$ which is 1.5 m vertically above $R$. The points $A$ and $B$ are on the horizontal surface, and $B$ is vertically below $R$. When $P$ is on the surface between $A$ and $B , R P$ makes an acute angle $\theta ^ { \circ }$ with the horizontal (see diagram).\\
(i) Show that the normal force exerted on $P$ by the surface has magnitude 3.6 N , for all values of $\theta$.\\
$P$ is projected with speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ towards $B$ from its initial position at $A$ where $\theta = 30$. The speed of $P$ when it passes through $B$ is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(ii) Find the work done against friction as $P$ moves from $A$ to $B$.\\
(iii) Calculate the value of the coefficient of friction between $P$ and the surface.
\hfill \mbox{\textit{CAIE M2 2014 Q6 [9]}}