| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle on rough horizontal surface, particle hanging |
| Difficulty | Standard +0.8 This is a multi-stage pulley problem requiring Newton's second law with friction, kinematics across two phases (before and after B hits ground), and careful tracking of distances. The three-part structure with 'show that' acceleration, finding speed at impact, and post-impact motion analysis requires more problem-solving than standard M1 questions, though the techniques themselves are routine for the module. |
| Spec | 3.03d Newton's second law: 2D vectors3.03k Connected particles: pulleys and equilibrium3.03r Friction: concept and vector form |
| Answer | Marks | Guidance |
|---|---|---|
| (a) For A, resolve \(\uparrow\): \(R - 5Mg = 0 \Rightarrow R = 5Mg\) | M1 | |
| \(F = \mu R\) so \(F = \frac{3}{10}(5Mg) = \frac{3}{2}Mg\) | M1 A1 | |
| for A, resolve \(\rightarrow\): \(T - \frac{3}{2}Mg = 5Ma, \quad T - \frac{3}{4}Mg = 5Ma\) (1) | M1 A1 | |
| for B, resolve \(\downarrow\): \(3Mg - T = 3Ma\) (2) | M1 | |
| (1) + (2) gives \(\frac{9}{4}Mg = 8Ma \Rightarrow a = \frac{9}{32}g\) m s⁻² | M1 A1 | |
| (b) \(s = 1, u = 0, a = \frac{9}{32}g\), use \(v^2 = u^2 + 2as\) | M1 | |
| \(v^2 = \frac{9}{16}g \Rightarrow v = \frac{3}{4}\sqrt{g}\) (≈ 2.35 m s⁻¹) | M2 A1 | |
| (c) after string goes slack, \(F = 5Ma\) so \(a = -\frac{3Mg}{5M} = -\frac{3}{20}g\) | M2 A1 | |
| \(u^2 = \frac{9}{16}g, v = 0, a = -\frac{3}{20}g\) use \(v^2 = u^2 + 2as\) | M1 | |
| \(0 = \frac{9}{16}g - \frac{3}{10}gs \Rightarrow s = 1.875\) m + 1 m before B hits the ground | M1 A1 | |
| = 2.875 so A is 0.125 m from pulley when it comes to rest | A1 | (19 marks) |
**(a)** For A, resolve $\uparrow$: $R - 5Mg = 0 \Rightarrow R = 5Mg$ | M1 |
$F = \mu R$ so $F = \frac{3}{10}(5Mg) = \frac{3}{2}Mg$ | M1 A1 |
for A, resolve $\rightarrow$: $T - \frac{3}{2}Mg = 5Ma, \quad T - \frac{3}{4}Mg = 5Ma$ (1) | M1 A1 |
for B, resolve $\downarrow$: $3Mg - T = 3Ma$ (2) | M1 |
(1) + (2) gives $\frac{9}{4}Mg = 8Ma \Rightarrow a = \frac{9}{32}g$ m s⁻² | M1 A1 |
**(b)** $s = 1, u = 0, a = \frac{9}{32}g$, use $v^2 = u^2 + 2as$ | M1 |
$v^2 = \frac{9}{16}g \Rightarrow v = \frac{3}{4}\sqrt{g}$ (≈ 2.35 m s⁻¹) | M2 A1 |
**(c)** after string goes slack, $F = 5Ma$ so $a = -\frac{3Mg}{5M} = -\frac{3}{20}g$ | M2 A1 |
$u^2 = \frac{9}{16}g, v = 0, a = -\frac{3}{20}g$ use $v^2 = u^2 + 2as$ | M1 |
$0 = \frac{9}{16}g - \frac{3}{10}gs \Rightarrow s = 1.875$ m + 1 m before B hits the ground | M1 A1 |
= 2.875 so A is 0.125 m from pulley when it comes to rest | A1 | (19 marks)
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**Total: 75 marks**8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-4_442_924_877_443}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
Figure 3 shows two particles $A$ and $B$, of mass $5 M$ and $3 M$ respectively, attached to the ends of a light inextensible string of length 4 m . The string passes over a smooth pulley which is fixed to the edge of a rough horizontal table 2 m high. Particle $A$ lies on the table at a distance of 3 m from the pulley, whilst particle $B$ hangs freely over the edge of the table 1 m above the ground. The coefficient of friction between $A$ and the table is $\frac { 3 } { 20 }$.
The system is released from rest with the string taut.
\begin{enumerate}[label=(\alph*)]
\item Show that the initial acceleration of the system is $\frac { 9 } { 32 } \mathrm {~g} \mathrm {~ms} ^ { - 2 }$.
\item Find, in terms of $g$, the speed of $A$ immediately before $B$ hits the ground.
When $B$ hits the ground, it comes to rest and the string becomes slack.
\item Calculate how far particle $A$ is from the pulley when it comes to rest.
END
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q8 [19]}}