| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Multi-stage motion: particle reaches ground/pulley causing string to go slack |
| Difficulty | Standard +0.8 This is a multi-stage mechanics problem requiring resolution of forces on an inclined plane with friction, connected particle dynamics, and kinematics across two phases of motion. While the techniques are standard M1 content (Newton's second law, friction, SUVAT), the problem requires careful setup of multiple force equations, algebraic manipulation to show a specific result, and tracking motion through distinct phases (before and after Q hits the floor). The 14 total marks and three-part structure with changing conditions elevate this above routine exercises, though it remains within expected M1 problem-solving without requiring exceptional insight. |
| Spec | 3.03k Connected particles: pulleys and equilibrium3.03l Newton's third law: extend to situations requiring force resolution3.03v Motion on rough surface: including inclined planes |
| Answer | Marks |
|---|---|
| (a) Modelling assumption: string is inextensible | B1 |
| \(F = ma\); \(T = 2ma\), \(3mg \sin \theta - \frac{1}{6}(3mg \cos \theta) - T = 3ma\) | M1 A1 A1 |
| Add: \(3mg(0.8) - 0.5mg(0.6) = 5ma\) | M1 A1 A1 |
| \(5a = 2.1g\) | |
| \(a = \frac{21g}{50}\) | |
| (b) Dist \(= 1 \text{ m}\): \(v^2 = 2(\frac{21g}{50})(1)\) | M1 A1 |
| \(v = 2.87 \text{ ms}^{-1}\) | |
| (c) Time for \(Q\) to reach floor is \(t\) where \(1 = 0.21gt^2\) | M1 A1 |
| \(t = 0.697 \text{ s}\) | |
| \(0.2 \text{ m}\) at \(2.87 \text{ ms}^{-1}\) takes \(0.0697 \text{ s}\), so total time \(= 0.767 \text{ s}\) | M1 A1 |
| Total: 14 marks |
(a) Modelling assumption: string is inextensible | B1 |
| $F = ma$; $T = 2ma$, $3mg \sin \theta - \frac{1}{6}(3mg \cos \theta) - T = 3ma$ | M1 A1 A1 |
| Add: $3mg(0.8) - 0.5mg(0.6) = 5ma$ | M1 A1 A1 |
| $5a = 2.1g$ | |
| $a = \frac{21g}{50}$ | |
(b) Dist $= 1 \text{ m}$: $v^2 = 2(\frac{21g}{50})(1)$ | M1 A1 |
| $v = 2.87 \text{ ms}^{-1}$ | |
(c) Time for $Q$ to reach floor is $t$ where $1 = 0.21gt^2$ | M1 A1 |
| $t = 0.697 \text{ s}$ | |
| $0.2 \text{ m}$ at $2.87 \text{ ms}^{-1}$ takes $0.0697 \text{ s}$, so total time $= 0.767 \text{ s}$ | M1 A1 |
| | **Total: 14 marks**Two particles $P$ and $Q$, of masses $2m$ and $3m$ respectively, are connected by a light string. Initially, $P$ is at rest on a smooth horizontal table. The string passes over a small smooth pulley and $Q$ rests on a rough plane inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac{4}{3}$. The coefficient of friction between $Q$ and the inclined plane is $\frac{1}{6}$.
\includegraphics{figure_7}
The system is released from rest with $Q$ at a distance of 0.8 metres above a horizontal floor.
\begin{enumerate}[label=(\alph*)]
\item Show that the acceleration of $P$ and $Q$ is $\frac{21g}{50}$, stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
\item Find the speed with which $Q$ hits the floor. [2 marks]
\end{enumerate}
After $Q$ hits the floor and does not rebound, $P$ moves a further 0.2 m until it hits the pulley.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the total time after the system is released before $P$ hits the pulley. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q7 [14]}}