| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2011 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle on rough incline, particle hanging |
| Difficulty | Standard +0.3 This is a standard M1 pulley system question with connected particles, requiring resolution of forces, Newton's second law, and kinematics. While it has multiple parts and involves friction on an incline, all techniques are routine for M1: finding acceleration from force equations, using v²=u²+2as, and applying equations of motion. The trigonometry is straightforward (given tan θ), and no novel problem-solving insight is required—just systematic application of standard mechanics methods. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03k Connected particles: pulleys and equilibrium3.03o Advanced connected particles: and pulleys3.03t Coefficient of friction: F <= mu*R model3.03v Motion on rough surface: including inclined planes |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | For A: \(7g - T = 7a\) | M1 A1 |
| For B: parallel to plane \(T - F - 3g\sin\theta = 3a\) | M1 A1 | |
| perpendicular to plane \(R = 3g\cos\theta\) | M1 A1 | |
| \(F = \mu R = 3g\cos\theta = 2g\cos\theta\) | M1 | |
| Eliminating T: \(7g - F - 3g\sin\theta = 10a\) | DM1 | |
| Equation in g and a: \(7g - 2g \times \frac{12}{13} - 3g \times \frac{5}{13} = 7g - \frac{39}{13}g = 4g = 10a\) | DM1 | |
| \(a = \frac{2g}{5}\) oe or \(3.9\) or \(3.92\) | A1 | (10 marks) |
| (b) | After 1 m: \(v^2 = u^2 + 2as\), \(v^2 = 0 + 2 \times \frac{2g}{5} \times 1\) → \(v = 2.8\) | M1 A1 |
| (c) | \(-(F + 3g\sin\theta) = 3a\) | M1 |
| \(\frac{2}{3} \times 3g \times \frac{12}{13} + 3g \times \frac{5}{13} = 3g = -3a\), \(a = -g\) | A1 DM1 | |
| \(v = u + at\), \(0 = 2.8 - 9.8t\) → \(t = \frac{2}{7}\) oe, \(0.29\), \(0.286\) | A1 | (4 marks) |
| [16 marks total] |
| **Part** | **Answer/Working** | **Marks** | **Guidance** |
|----------|-------------------|----------|-------------|
| (a) | For A: $7g - T = 7a$ | M1 A1 | |
| | For B: parallel to plane $T - F - 3g\sin\theta = 3a$ | M1 A1 | |
| | perpendicular to plane $R = 3g\cos\theta$ | M1 A1 | |
| | $F = \mu R = 3g\cos\theta = 2g\cos\theta$ | M1 | |
| | Eliminating T: $7g - F - 3g\sin\theta = 10a$ | DM1 | |
| | Equation in g and a: $7g - 2g \times \frac{12}{13} - 3g \times \frac{5}{13} = 7g - \frac{39}{13}g = 4g = 10a$ | DM1 | |
| | $a = \frac{2g}{5}$ oe or $3.9$ or $3.92$ | A1 | (10 marks) |
| (b) | After 1 m: $v^2 = u^2 + 2as$, $v^2 = 0 + 2 \times \frac{2g}{5} \times 1$ → $v = 2.8$ | M1 A1 | (2 marks) |
| (c) | $-(F + 3g\sin\theta) = 3a$ | M1 | |
| | $\frac{2}{3} \times 3g \times \frac{12}{13} + 3g \times \frac{5}{13} = 3g = -3a$, $a = -g$ | A1 DM1 | |
| | $v = u + at$, $0 = 2.8 - 9.8t$ → $t = \frac{2}{7}$ oe, $0.29$, $0.286$ | A1 | (4 marks) |
| | | | **[16 marks total]** |7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-12_581_1211_235_370}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Two particles $A$ and $B$, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially $B$ is held at rest on a rough fixed plane inclined at angle $\theta$ to the horizontal, where $\tan \theta = \frac { 5 } { 12 }$. The part of the string from $B$ to $P$ is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, $P$, fixed at the top of the plane. The particle $A$ hangs freely below $P$, as shown in Figure 4. The coefficient of friction between $B$ and the plane is $\frac { 2 } { 3 }$. The particles are released from rest with the string taut and $B$ moves up the plane.
\begin{enumerate}[label=(\alph*)]
\item Find the magnitude of the acceleration of $B$ immediately after release.
\item Find the speed of $B$ when it has moved 1 m up the plane.
When $B$ has moved 1 m up the plane the string breaks. Given that in the subsequent motion $B$ does not reach $P$,
\item find the time between the instants when the string breaks and when $B$ comes to instantaneous rest.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2011 Q7 [16]}}