| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2015 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle on rough horizontal surface, particle hanging |
| Difficulty | Standard +0.3 This is a standard M1 pulley system question requiring Newton's second law for connected particles, friction calculations, and energy/kinematics. While it has multiple parts and requires careful bookkeeping, all techniques are routine for M1: finding tension using F=ma for both particles, using v²=u²+2as for the speed, and analyzing motion after Q stops. The 'show that' format provides targets to work towards, reducing problem-solving demand. Slightly easier than average due to its highly structured nature and standard M1 methods. |
| 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 | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(R=mg\) | B1 | Forces acting vertically on \(P\) |
| \(F=0.5R\) | B1 | Use of \(F=\mu R\) |
| \(4mg-T=\pm4ma\) | A1 | One equation of motion. Requires all terms but condone sign errors |
| \(T-F=\pm ma\) | A1 | A second equation of motion of \(P\). Requires all terms but condone sign errors. Signs of \(a\) must be consistent. Condone use of \(4mg-F=5ma\) in place of either of the above equations. |
| \(4mg-0.5mg=5ma\) or \(4mg-T=4T-2mg\), \(a=0.7g\) | DDM1 | Solve for \(T\). Dependent on the two preceding M marks |
| \(T=1.2mg\) | A1 | |
| (8) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(v^2=2\times0.7gh\) | M1 | Complete method to an equation in \(v\) or \(v^2\) |
| \(v=\sqrt{1.4gh}\) | A1 | Obtain given answer or exact equivalent from exact working with no errors seen. |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(-0.5mg=ma'\) | M1 | Complete method to find the deceleration of \(P\) |
| \(\Rightarrow a'=-0.5g\) | A1 | |
| M1 | Complete method to find additional distance on terms of \(h\) \((a\neq0.7g, a\neq g)\) | |
| \(0^2=1.4gh-2\times0.5g\times d\) | A1 | Correctly substituted equation. Follow their \(a\neq0.7g\), \(a\neq g\) |
| \(d=1.4h\) | A1 | |
| Hence, length of string is greater than \(1.4h+h=2.4h\) | A1 | Obtain given answer with no errors seen. Their statement needs to reflect the inequality. |
| (6) | ||
| Total: 16 |
## Question 8(a):
| Working | Mark | Guidance |
|---------|------|----------|
| $R=mg$ | B1 | Forces acting vertically on $P$ |
| $F=0.5R$ | B1 | Use of $F=\mu R$ |
| $4mg-T=\pm4ma$ | A1 | One equation of motion. Requires all terms but condone sign errors |
| $T-F=\pm ma$ | A1 | A second equation of motion of $P$. Requires all terms but condone sign errors. Signs of $a$ must be consistent. Condone use of $4mg-F=5ma$ in place of either of the above equations. |
| $4mg-0.5mg=5ma$ or $4mg-T=4T-2mg$, $a=0.7g$ | DDM1 | Solve for $T$. Dependent on the two preceding M marks |
| $T=1.2mg$ | A1 | |
| | **(8)** | |
---
## Question 8(b):
| Working | Mark | Guidance |
|---------|------|----------|
| $v^2=2\times0.7gh$ | M1 | Complete method to an equation in $v$ or $v^2$ |
| $v=\sqrt{1.4gh}$ | A1 | Obtain **given answer** or exact equivalent from exact working with no errors seen. |
| | **(2)** | |
---
## Question 8(c):
| Working | Mark | Guidance |
|---------|------|----------|
| $-0.5mg=ma'$ | M1 | Complete method to find the deceleration of $P$ |
| $\Rightarrow a'=-0.5g$ | A1 | |
| | M1 | Complete method to find additional distance on terms of $h$ $(a\neq0.7g, a\neq g)$ |
| $0^2=1.4gh-2\times0.5g\times d$ | A1 | Correctly substituted equation. Follow their $a\neq0.7g$, $a\neq g$ |
| $d=1.4h$ | A1 | |
| Hence, length of string is greater than $1.4h+h=2.4h$ | A1 | Obtain **given answer** with no errors seen. Their statement needs to reflect the inequality. |
| | **(6)** | |
| | **Total: 16** | |8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-13_648_1280_271_331}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Two particles $P$ and $Q$ have masses $m$ and $4 m$ respectively. The particles are attached to the ends of a light inextensible string. Particle $P$ is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle $Q$ hangs at rest vertically below the pulley, at a height $h$ above a horizontal plane, as shown in Figure 3. The coefficient of friction between $P$ and the table is 0.5 . Particle $P$ is released from rest with the string taut and slides along the table.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $m g$, the tension in the string while both particles are moving.
The particle $P$ does not reach the pulley before $Q$ hits the plane.
\item Show that the speed of $Q$ immediately before it hits the plane is $\sqrt { 1.4 g h }$
When $Q$ hits the plane, $Q$ does not rebound and $P$ continues to slide along the table. Given that $P$ comes to rest before it reaches the pulley,
\item show that the total length of the string must be greater than 2.4 h
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2015 Q8 [16]}}