| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2015 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton's laws and connected particles |
| Type | Particle on incline, hanging counterpart |
| 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 after the string breaks. While multi-step, it follows a routine template for M1 connected particles questions with no novel insights required, making it slightly easier than average. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03k Connected particles: pulleys and equilibrium3.03l Newton's third law: extend to situations requiring force resolution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.5g \times \frac{7}{25} - T = 0.5a\) and \(T - 0.1g = 0.1a\) | M1 | For applying Newton's 2nd law to \(P\) or to \(Q\) or for applying N2 to the system; any two correct, allow \(\sin16.3\) for \(7/25\) |
| \(1.4 - 1 = 0.6a\) | A1 | |
| \(a = \frac{2}{3}\text{ ms}^{-2}\) | B1 | For eliminating \(T\) and obtaining \(a\) |
| M1 | For substituting for \(a\) to find \(T\) | |
| Tension is \(1.07\text{ N}\) | A1 | Allow \(T = 16/15\text{ N}\) |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([v^2 = 2 \times \left(\frac{2}{3}\right) \times 0.7]\) | M1 | For using \(v^2 = u^2 + 2as\) to find the speed of the particles immediately before the string breaks |
| \([2^2 = 2 \times \frac{2}{3} \times 0.7 + 2 \times 0.28g \times s]\) | M1 | For applying \(v^2 = u^2 + 2as\) for the motion of \(P\) when the string is slack and \(s\) is the distance travelled by \(P\) after the break until it reaches the floor |
| Length of string \(= 2.5 - s = 1.95\text{ m}\) | A1 | Allow length \(= 41/21\text{ m}\) |
| Total: 3 |
# Question 5:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.5g \times \frac{7}{25} - T = 0.5a$ and $T - 0.1g = 0.1a$ | M1 | For applying Newton's 2nd law to $P$ **or** to $Q$ **or** for applying N2 to the system; any two correct, allow $\sin16.3$ for $7/25$ |
| $1.4 - 1 = 0.6a$ | A1 | |
| $a = \frac{2}{3}\text{ ms}^{-2}$ | B1 | For eliminating $T$ and obtaining $a$ |
| | M1 | For substituting for $a$ to find $T$ |
| Tension is $1.07\text{ N}$ | A1 | Allow $T = 16/15\text{ N}$ |
| | **Total: 5** | |
## Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[v^2 = 2 \times \left(\frac{2}{3}\right) \times 0.7]$ | M1 | For using $v^2 = u^2 + 2as$ to find the speed of the particles immediately before the string breaks |
| $[2^2 = 2 \times \frac{2}{3} \times 0.7 + 2 \times 0.28g \times s]$ | M1 | For applying $v^2 = u^2 + 2as$ for the motion of $P$ when the string is slack and $s$ is the distance travelled by $P$ after the break until it reaches the floor |
| Length of string $= 2.5 - s = 1.95\text{ m}$ | A1 | Allow length $= 41/21\text{ m}$ |
| | **Total: 3** | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660}
A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles $P$ and $Q$, of masses 0.5 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle $Q$ is held at rest on the floor vertically below the pulley. The string is taut and $P$ is at rest on the plane (see diagram). $Q$ is released and starts to move vertically upwards towards the pulley and $P$ moves down the plane.\\
(i) Find the tension in the string and the magnitude of the acceleration of the particles before $Q$ reaches the pulley.
At the instant just before $Q$ reaches the pulley the string breaks; $P$ continues to move down the plane and reaches the floor with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(ii) Find the length of the string.
\hfill \mbox{\textit{CAIE M1 2015 Q5 [8]}}