| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Lighter particle on surface released, heavier hangs |
| Difficulty | Standard +0.3 This is a standard M1 pulley problem with connected particles requiring Newton's second law, SUVAT equations, and energy/momentum considerations. While multi-part with several steps, each part follows routine mechanics procedures: (a) standard F=ma for connected particles, (b) kinematics, (c) comparing distances/times, (d) impulse-momentum or energy methods. The problem is slightly easier than average A-level due to its predictable structure and straightforward application of standard techniques, though the multi-part nature and need to track the system through different phases adds some complexity. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03k Connected particles: pulleys and equilibrium |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(4.5g - T = 4.5a\), \(T - 4g = 4a\) | M1 A1 A1 M1 A1 | |
| Add: \(0.5g = 8.5a\) | ||
| \(a = 0.576\) ms\(^{-2}\) | ||
| (b) \(v^2 = 2as = 2(0.576)(1.9) = 2.191\) | M1 A1 A1 | |
| \(v = 1.48\) ms\(^{-1}\) | ||
| (c) \(P\) has risen 1.9 m and has speed 1.48 ms\(^{-1}\) | B1 | |
| Under gravity \(P\) rises \(s\) m where \(0 = 1.48^2 - 2(9.8)s\) | M1 A1 A1 | |
| \(s = 0.112\) m \(< 1.1\) m, so \(P\) does not hit the pulley | M1 A1 | |
| (d) Momentum conserved: \(4(1.48) + 4.5(0) = 8.5v\) | M1 A1 A1 | |
| \(v = 0.697\) ms\(^{-1}\) | Total: 17 marks |
**(a)** $4.5g - T = 4.5a$, $T - 4g = 4a$ | M1 A1 A1 M1 A1 |
Add: $0.5g = 8.5a$ | |
$a = 0.576$ ms$^{-2}$ | |
**(b)** $v^2 = 2as = 2(0.576)(1.9) = 2.191$ | M1 A1 A1 |
$v = 1.48$ ms$^{-1}$ | |
**(c)** $P$ has risen 1.9 m and has speed 1.48 ms$^{-1}$ | B1 |
Under gravity $P$ rises $s$ m where $0 = 1.48^2 - 2(9.8)s$ | M1 A1 A1 |
$s = 0.112$ m $< 1.1$ m, so $P$ does not hit the pulley | M1 A1 |
**(d)** Momentum conserved: $4(1.48) + 4.5(0) = 8.5v$ | M1 A1 A1 |
$v = 0.697$ ms$^{-1}$ | | **Total: 17 marks**7.\\
\begin{tikzpicture}[>=latex]
% Pulley
\draw[thick, fill=gray!20] (3,4.5) circle (0.4);
\fill (3,4.5) circle (1.5pt);
% Left string: from pulley down to P
\draw[thick] (2.6,4.5) -- (2.6,0);
% Right string: from pulley down to Q
\draw[thick] (3.4,4.5) -- (3.4,3.4);
% Point P (bottom of left string)
\fill (2.6,0) circle (4pt);
\node[below left] at (2.6,0) {$P$};
% Point Q (bottom of right string)
\fill (3.4,3.4) circle (4pt);
\node[below left] at (3.4,3.4) {$Q$};
% 3 m dimension arrow
\draw[<->] (1.7,4.5) -- node[left] {3\,m} (1.7,0);
% 1.1 m dimension arrow
\draw[<->] (4.4,4.5) -- node[right] {1.1\,m} (4.4,3.4);
\end{tikzpicture}
A particle $P$, of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle $Q$ of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially $Q$ is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.
\begin{enumerate}[label=(\alph*)]
\item Find the acceleration of the two particles.
\item Find the speed with which $Q$ hits the ground.
Assuming that $Q$ does not rebound from the ground while the string is slack,
\item show that $P$ does not reach the pulley before $Q$ starts to move again.
\item Find the speed with which $Q$ leaves the ground when the string again becomes taut.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q7 [17]}}