| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Heavier particle hits ground, lighter continues upward - vertical strings |
| Difficulty | Standard +0.3 This is a standard Atwood machine problem with a typical extension (particle hitting ground). Part (a) is routine application of F=ma to connected particles, parts (b-c) use standard equations. Part (d) requires recognizing Q continues upward after P stops, using v²=u²+2as, but this is a common M1 exercise type. Slightly above average due to the multi-stage nature and part (d) requiring careful thinking about what happens after impact, but still well within standard M1 territory. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03k Connected particles: pulleys and equilibrium3.03o Advanced connected particles: and pulleys |
| Answer | Marks |
|---|---|
| eqn. of motion for \(P\): \(3g - T = 3a\) (1) | M1 |
| eqn. of motion for \(Q\): \(T - 2g = 2a\) (2) | M1 |
| (1) + (2) gives \(g = 5a\) i.e. \(a = \frac{g}{5}\) ms\(^{-2}\) | M1 A1 |
| Answer | Marks |
|---|---|
| from (2), \(T = 2a + 2g = \frac{12}{5}g\) N (= 23.52 N) | M1 A1 |
| Answer | Marks |
|---|---|
| \(s = 1.5\), \(u = 0\), \(a = \frac{4}{5}g\) use \(v^2 = u^2 + 2as\) | M1 |
| \(v^2 = \frac{4}{5}g\) i.e. \(v = 2.42\) ms\(^{-1}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\) hits ground \(\rightarrow\) string goes slack \(\rightarrow\) \(Q\) moves freely under gravity | M1 | |
| for \(Q\): \(u' = \frac{2}{5}g\), \(v = 0\), \(a = -g\) use \(v^2 = u^2 + 2as\) | M1 | |
| \(0 = \frac{4}{5}g - 2gs \Rightarrow s = 0.3\) m | M1 A1 | |
| \(Q\) moves 1.5 m before \(P\) hits ground + 0.3 m = 1.8 m upwards ∴ closest to pulley = 0.2 m | A1 | (14) |
**Part (a):**
eqn. of motion for $P$: $3g - T = 3a$ (1) | M1 |
eqn. of motion for $Q$: $T - 2g = 2a$ (2) | M1 |
(1) + (2) gives $g = 5a$ i.e. $a = \frac{g}{5}$ ms$^{-2}$ | M1 A1 |
**Part (b):**
from (2), $T = 2a + 2g = \frac{12}{5}g$ N (= 23.52 N) | M1 A1 |
**Part (c):**
$s = 1.5$, $u = 0$, $a = \frac{4}{5}g$ use $v^2 = u^2 + 2as$ | M1 |
$v^2 = \frac{4}{5}g$ i.e. $v = 2.42$ ms$^{-1}$ | M1 A1 |
**Part (d):**
$P$ hits ground $\rightarrow$ string goes slack $\rightarrow$ $Q$ moves freely under gravity | M1 |
for $Q$: $u' = \frac{2}{5}g$, $v = 0$, $a = -g$ use $v^2 = u^2 + 2as$ | M1 |
$0 = \frac{4}{5}g - 2gs \Rightarrow s = 0.3$ m | M1 A1 |
$Q$ moves 1.5 m before $P$ hits ground + 0.3 m = 1.8 m upwards ∴ closest to pulley = 0.2 m | A1 | (14)
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**Total: (75)**\includegraphics{figure_2}
Figure 2 shows two particles $P$ and $Q$, of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with $P$ and $Q$ at the same level 1.5 metres above the ground and 2 metres below the pulley.
\begin{enumerate}[label=(\alph*)]
\item Show that the initial acceleration of the system is $\frac{g}{5}$ m s$^{-2}$. [4 marks]
\item Find the tension in the string. [2 marks]
\item Find the speed with which $P$ hits the ground. [3 marks]
\end{enumerate}
When $P$ hits the ground, it does not rebound.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item What is the closest that $Q$ gets to the pulley. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q8 [14]}}