| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle motion with kinematics only |
| Difficulty | Standard +0.3 This is a standard M1 pulley problem with routine kinematics. Part (a) requires basic differentiation and substitution; part (b) uses standard integration. The pulley system involves straightforward force resolution on an incline with friction, and the final part requires simple kinematics after the string goes slack. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.07a Derivative as gradient: of tangent to curve3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration3.03b Newton's first law: equilibrium3.03k Connected particles: pulleys and equilibrium3.03t Coefficient of friction: F <= mu*R model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a = 4 - t\) | M1 | M1 for differentiation |
| When \(a = 0,\ t = 4\) | A1 | |
| At \(t = 4,\ v = 12.5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Velocity \(= 0\) when \(4.5 + 4t - 0.5t^2 = 0\) | M1 | |
| \(t = 9\) (reject \(t = -1\)) | A1 | |
| \(\int(4.5 + 4t - 0.5t^2)\,dt\) | M1 | |
| \(4.5t + 2t^2 - \frac{1}{6}t^3\ [+c]\) | A1 | |
| Apply limits (0 and 9) | M1 | |
| Distance \(= 81\ \text{m}\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a = 4 - t$ | M1 | M1 for differentiation |
| When $a = 0,\ t = 4$ | A1 | |
| At $t = 4,\ v = 12.5$ | A1 | |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Velocity $= 0$ when $4.5 + 4t - 0.5t^2 = 0$ | M1 | |
| $t = 9$ (reject $t = -1$) | A1 | |
| $\int(4.5 + 4t - 0.5t^2)\,dt$ | M1 | |
| $4.5t + 2t^2 - \frac{1}{6}t^3\ [+c]$ | A1 | |
| Apply limits (0 and 9) | M1 | |
| Distance $= 81\ \text{m}$ | A1 | |
---6 A particle travels in a straight line $P Q$. The velocity of the particle $t \mathrm {~s}$ after leaving $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where
$$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the velocity of the particle at the instant when its acceleration is zero.\\
The particle comes to instantaneous rest at $Q$.
\item Find the distance $P Q$.\\
\includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744}
Two particles $A$ and $B$, of masses $3 m \mathrm {~kg}$ and $2 m \mathrm {~kg}$ respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle $\theta$ to the horizontal. $A$ lies on the plane and $B$ hangs vertically, 0.8 m above the floor, which is horizontal. The string between $A$ and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially $A$ and $B$ are at rest.\\
(a) Given that the plane is smooth, find the value of $\theta$ for which $A$ remains at rest.\\
It is given instead that the plane is rough, $\theta = 30 ^ { \circ }$ and the acceleration of $A$ up the plane is $0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.\\
(b) Show that the coefficient of friction between $A$ and the plane is $\frac { 1 } { 10 } \sqrt { 3 }$.
\item When $B$ reaches the floor it comes to rest.
Find the length of time after $B$ reaches the floor for which $A$ is moving up the plane. [You may assume that $A$ does not reach the pulley.]\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 2020 Q6 [9]}}