| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton's laws and connected particles |
| Type | Multi-part pulley system, subsequent motion |
| Difficulty | Standard +0.3 This is a standard connected particles problem requiring Newton's second law applied to both particles, solving simultaneous equations for tension and acceleration, then using kinematics with a change in system when P hits the ground. All techniques are routine for M1 level with no novel insights required, making it slightly easier than average. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03k Connected particles: pulleys and equilibrium3.03o Advanced connected particles: and pulleys |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| M1 | For using Newton's 2nd law for P or Q; or for using \((M-m)g \times 0.8 = (M+m)a\) | |
| \(0.6g \times 0.8 - T = 0.6a\) and \(T - 0.4g \times 0.8 = 0.4a\) or \((0.6-0.4)g \times 0.8 = (0.6+0.4)a\) | A1 | |
| M1 | For solving for \(T\) or for \(a\) | |
| Tension is 3.84 N or acceleration is \(1.6\ \text{ms}^{-2}\) | A1 | |
| Acceleration is \(1.6\ \text{ms}^{-2}\) or tension is 3.84 N | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(2 = 1.6t_1 \quad (t_1 = 1.25)\) | B1ft | |
| M1 | For using \(0 + u + at\) with \(a = -0.8g\) | |
| \(0 = 2 - 0.8gt_2 \quad (t_2 = 0.25)\) | A1 | |
| Time taken is 1.5 s | A1ft [4] | ft incorrect acceleration in (i) |
## Question 6(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| | M1 | For using Newton's 2nd law for P or Q; or for using $(M-m)g \times 0.8 = (M+m)a$ |
| $0.6g \times 0.8 - T = 0.6a$ and $T - 0.4g \times 0.8 = 0.4a$ or $(0.6-0.4)g \times 0.8 = (0.6+0.4)a$ | A1 | |
| | M1 | For solving for $T$ or for $a$ |
| Tension is 3.84 N or acceleration is $1.6\ \text{ms}^{-2}$ | A1 | |
| Acceleration is $1.6\ \text{ms}^{-2}$ or tension is 3.84 N | A1 [5] | |
## Question 6(ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $2 = 1.6t_1 \quad (t_1 = 1.25)$ | B1ft | |
| | M1 | For using $0 + u + at$ with $a = -0.8g$ |
| $0 = 2 - 0.8gt_2 \quad (t_2 = 0.25)$ | A1 | |
| Time taken is 1.5 s | A1ft [4] | ft incorrect acceleration in **(i)** |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_465_849_1475_648}
Particles $P$ and $Q$, of masses 0.6 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a vertical cross-section of a triangular prism. The base of the prism is fixed on horizontal ground and each of the sloping sides is smooth. Each sloping side makes an angle $\theta$ with the ground, where $\sin \theta = 0.8$. Initially the particles are held at rest on the sloping sides, with the string taut (see diagram). The particles are released and move along lines of greatest slope.\\
(i) Find the tension in the string and the acceleration of the particles while both are moving.
The speed of $P$ when it reaches the ground is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. On reaching the ground $P$ comes to rest and remains at rest. $Q$ continues to move up the slope but does not reach the pulley.\\
(ii) Find the time taken from the instant that the particles are released until $Q$ reaches its greatest height above the ground.
\hfill \mbox{\textit{CAIE M1 2012 Q6 [9]}}