| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 11 |
| Topic | Pulley systems |
| Type | Multi-stage motion: changing surface conditions or external intervention |
| Difficulty | Standard +0.3 This is a standard connected particles problem with friction requiring systematic application of Newton's second law and kinematics. While it has multiple parts and requires careful force resolution (including friction and inclined plane components), the techniques are routine A-level mechanics with no novel insight needed. The 'show that' in part (i) provides a target to verify, making it slightly easier than open-ended problems. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03k Connected particles: pulleys and equilibrium3.03r Friction: concept and vector form3.03v Motion on rough surface: including inclined planes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) For the motion of \(A\), \(T - 0.4g = 2a\) | M1A1 | For the motion of \(B\), \(4g\cos 60° - T = 4a\) |
| (ii) Applying appropriate constant acceleration equations: \(12 = 0t + \frac{1}{2} \times \frac{8}{5} \times t^2\) | M1 | \(t = 3\) |
| (iii) The net force on \(P\) is the frictional force of \(0.4g\) | M1 | The acceleration of \(P\) is \(-0.2g = -2\) |
(i) For the motion of $A$, $T - 0.4g = 2a$ | M1A1 | For the motion of $B$, $4g\cos 60° - T = 4a$ | A1 | Solve and obtain $a = \frac{8}{5}$ m s$^{-2}$ | A1 (AG) | 4 marks
(ii) Applying appropriate constant acceleration equations: $12 = 0t + \frac{1}{2} \times \frac{8}{5} \times t^2$ | M1 | $t = 3$ | A1 | $v = 0 + \frac{8}{5} \times 3 = 8$ m s$^{-1}$ | A1 | A1 | 4 marks
(iii) The net force on $P$ is the frictional force of $0.4g$ | M1 | The acceleration of $P$ is $-0.2g = -2$ | A1 | Hence $0 = 8 - 2t \Rightarrow t = 4 \Rightarrow$ total time = $7$ | A1 | 3 marksA particle $P$ of mass $2$ kg rests on a long rough horizontal table. The coefficient of friction between $P$ and the table is $0.2$. A light inextensible string has one end attached to $P$ and the other end attached to a particle $Q$ of mass $4$ kg. The particle $Q$ is placed on a smooth plane inclined at $30^{\circ}$ to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram).
\includegraphics{figure_12}
Initially the system is held in equilibrium with the string taut. The system is released from rest at time $t = 0$, where $t$ is measured in seconds. In the subsequent motion $P$ does not reach the pulley.
\begin{enumerate}[label=(\roman*)]
\item Show that the magnitude of the acceleration of the particles is $\frac{8}{3}$ m s$^{-2}$. [4]
\end{enumerate}
After the particles have moved a distance of $12$ m the string is cut.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the corresponding value of $t$ and the speed of the particles at this instant. [4]
\item Find the value of $t$ when $P$ comes to rest. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q12 [11]}}