| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2021 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle on rough incline, particle hanging |
| Difficulty | Standard +0.8 This is a multi-stage pulley problem requiring force analysis in two phases (before and after B hits ground), with friction changing direction between phases. Part (b) requires setting up inequalities involving energy/kinematics and careful reasoning about when A stops, which goes beyond routine M1 exercises. The tan α = 3/4 requiring sin/cos calculation and the range-finding aspect add complexity, but the techniques are standard M1 content applied systematically. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03e Resolve forces: two dimensions3.03k 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 | Guidance |
|---|---|---|
| \(R = 2g\cos\alpha\) (Could be earned in (b) if used there) | M1A1 | Resolving perpendicular to plane; condone sign errors and sin/cos confusion |
| \(T - 2g\sin\alpha - F = 2a\) | M1A1 | Equation of motion parallel to plane |
| \(4g - T = 4a\) | M1A1 | Equation of motion vertically |
| OR \(4g - 2g\sin\alpha - F = 6a\) (whole system) | M1A1 | |
| \(F = 0.25R\) seen anywhere | B1 | |
| Solve for \(T\) | M1 | |
| \(T = 2.4g = \dfrac{12g}{5} = 24\) or \(23.5\) (N) | A1 (9) | Must have *two* equations of *motion* with \(a\) in each |
| Answer | Marks | Guidance |
|---|---|---|
| \(2.4g - 2g\sin\alpha - 0.4g = 2a\) OR \(4g - 2.4g = 4a\) | M1 | Eliminate \(T\) from equations of motion |
| \(a = 0.4g\) | A1 | May be found in (a) but must be used in (b) |
| \(v^2 = \dfrac{4gh}{5}\) | M1 | Complete method for \(v\) and \(h\) only using their \(a\) |
| \(-\dfrac{6g}{5} - \dfrac{2g}{5} = 2a'\) (\(a'\) is new acceleration of \(A\) up slope) — Allow +ve terms on LHS | B1 | Correct equation of motion for \(A\) after \(B\) hits ground |
| \(0 = \dfrac{4gh}{5} - \dfrac{8g}{5}s\) | M1 | Equation in \(s\) and \(h\) only using their \(a'\) |
| \(s = \dfrac{1}{2}h\) | A1 | |
| \(d > 1.5h\) | A1 (7) |
| Answer | Marks | Guidance |
|---|---|---|
| Weight of string; extensibility of the string; friction at pulley | B1 (1) | Simply restating what's in the question is B0 |
## Question 8:
### Part 8(a):
$R = 2g\cos\alpha$ (Could be earned in (b) if used there) | M1A1 | Resolving perpendicular to plane; condone sign errors and sin/cos confusion
$T - 2g\sin\alpha - F = 2a$ | M1A1 | Equation of motion parallel to plane
$4g - T = 4a$ | M1A1 | Equation of motion vertically
**OR** $4g - 2g\sin\alpha - F = 6a$ (whole system) | M1A1 |
$F = 0.25R$ seen anywhere | B1 |
Solve for $T$ | M1 |
$T = 2.4g = \dfrac{12g}{5} = 24$ or $23.5$ (N) | A1 (9) | Must have *two* equations of *motion* with $a$ in each
### Part 8(b):
$2.4g - 2g\sin\alpha - 0.4g = 2a$ **OR** $4g - 2.4g = 4a$ | M1 | Eliminate $T$ from equations of motion
$a = 0.4g$ | A1 | May be found in (a) but must be used in (b)
$v^2 = \dfrac{4gh}{5}$ | M1 | Complete method for $v$ and $h$ only using their $a$
$-\dfrac{6g}{5} - \dfrac{2g}{5} = 2a'$ ($a'$ is new acceleration of $A$ up slope) — Allow +ve terms on LHS | B1 | Correct equation of motion for $A$ after $B$ hits ground
$0 = \dfrac{4gh}{5} - \dfrac{8g}{5}s$ | M1 | Equation in $s$ and $h$ only using their $a'$
$s = \dfrac{1}{2}h$ | A1 |
$d > 1.5h$ | A1 (7) |
### Part 8(c):
Weight of string; extensibility of the string; friction at pulley | B1 (1) | Simply restating what's in the question is B0
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\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-20_369_1264_248_342}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Two particles, $A$ and $B$, have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$. The particle $A$ is held at rest on the plane at a distance $d$ metres from the pulley. The particle $B$ hangs freely at rest, vertically below the pulley, at a distance $h$ metres above the ground, as shown in Figure 3. The part of the string between $A$ and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between $A$ and the plane is $\frac { 1 } { 4 }$
The system is released from rest with the string taut and $B$ descends.
\begin{enumerate}[label=(\alph*)]
\item Find the tension in the string as $B$ descends.
On hitting the ground, $B$ immediately comes to rest.
Given that $A$ comes to rest before reaching the pulley,
\item find, in terms of $h$, the range of possible values of $d$.
\item State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2021 Q8 [17]}}