| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Variable mass or unknown mass |
| Difficulty | Standard +0.3 This is a standard AS-level pulley problem with straightforward application of Newton's second law to two connected particles. While it requires setting up equations for both masses and solving simultaneously for tension and k, the method is routine and well-practiced. The given acceleration makes the algebra particularly simple, and part (d) is a standard modeling question. Slightly easier than average due to the helpful given information. |
| Spec | 3.03b Newton's first law: equilibrium3.03c Newton's second law: F=ma one dimension3.03d Newton's second law: 2D vectors3.03k Connected particles: pulleys and equilibrium |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Equation of motion for \(P\) | M1 | Resolving vertically for \(P\); correct no. of terms, condone sign errors; \(a\) does not need to be substituted |
| \(2mg - T = 2m \times \frac{5g}{7}\) | A1 | Correct equation; allow if they use \(7\) instead of \(\frac{5g}{7}\) |
| \(T = \frac{4mg}{7}\) | A1 | Correct answer of form \(cmg\), where \(c = \frac{4}{7}\) oe or \(0.57\) or better |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Since the string is modelled as being inextensible | B1 | B0 if any extras (wrong or irrelevant) given |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Equation of motion for \(Q\) OR for whole system | M1 | Resolving vertically for \(Q\) or whole system; correct no. of terms; neither \(T\) nor \(a\) need be substituted; N.B. inconsistent omission of \(m\) is M0 and M0 if \(k\) omitted from LHS or RHS |
| \(T - kmg = km \times \frac{5g}{7}\) OR \(2mg - kmg = (km + 2m)\frac{5g}{7}\) | A1 | Correct equation; allow if they use \(7\) instead of \(\frac{5g}{7}\) |
| \(\frac{4mg}{7} - kmg = km \times \frac{5g}{7}\) oe and solve for \(k\) | DM1 | Sub for \(T\) using answer from (a); solve to give numerical value of \(k\) (i.e. \(m\)'s must cancel) |
| \(k = \frac{1}{3}\) or \(0.333\) or better | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. The model does not take account of the mass of the string; Pulley may not be smooth; Pulley may not be light; Particles may not be moving freely e.g. air resistance; Balls may not be particles; String may not be light; String may not be inextensible | B1 | Allow converses e.g. 'pulley is smooth'; B0 if any extra incorrect answer given (but ignore incorrect consequence of a correct answer); B0: use of a more accurate value of \(g\) |
## Question 9:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation of motion for $P$ | M1 | Resolving vertically for $P$; correct no. of terms, condone sign errors; $a$ does not need to be substituted |
| $2mg - T = 2m \times \frac{5g}{7}$ | A1 | Correct equation; allow if they use $7$ instead of $\frac{5g}{7}$ |
| $T = \frac{4mg}{7}$ | A1 | Correct answer of form $cmg$, where $c = \frac{4}{7}$ oe or $0.57$ or better |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Since the string is modelled as being inextensible | B1 | B0 if any extras (wrong or irrelevant) given |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation of motion for $Q$ **OR** for whole system | M1 | Resolving vertically for $Q$ or whole system; correct no. of terms; neither $T$ nor $a$ need be substituted; N.B. inconsistent omission of $m$ is M0 and M0 if $k$ omitted from LHS or RHS |
| $T - kmg = km \times \frac{5g}{7}$ **OR** $2mg - kmg = (km + 2m)\frac{5g}{7}$ | A1 | Correct equation; allow if they use $7$ instead of $\frac{5g}{7}$ |
| $\frac{4mg}{7} - kmg = km \times \frac{5g}{7}$ oe and solve for $k$ | DM1 | Sub for $T$ using answer from (a); solve to give numerical value of $k$ (i.e. $m$'s must cancel) |
| $k = \frac{1}{3}$ or $0.333$ or better | A1 | |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. The model does not take account of the mass of the string; Pulley may not be smooth; Pulley may not be light; Particles may not be moving freely e.g. air resistance; Balls may not be particles; String may not be light; String may not be inextensible | B1 | Allow converses e.g. 'pulley is smooth'; B0 if any extra incorrect answer given (but ignore incorrect consequence of a correct answer); B0: use of a more accurate value of $g$ |9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2edcf965-9c93-4a9b-9395-2d3c023801af-26_551_276_210_890}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Two small balls, $P$ and $Q$, have masses $2 m$ and $k m$ respectively, where $k < 2$.\\
The balls are attached to the ends of a string that passes over a fixed pulley.\\
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1.
The system is released from rest and, in the subsequent motion, $P$ moves downwards with an acceleration of magnitude $\frac { 5 g } { 7 }$
The balls are modelled as particles moving freely.\\
The string is modelled as being light and inextensible.\\
The pulley is modelled as being small and smooth.\\
Using the model,
\begin{enumerate}[label=(\alph*)]
\item find, in terms of $m$ and $g$, the tension in the string,
\item explain why the acceleration of $Q$ also has magnitude $\frac { 5 g } { 7 }$
\item find the value of $k$.
\item Identify one limitation of the model that will affect the accuracy of your answer to part (c).
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 2 2018 Q9 [9]}}