| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2002 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle on rough incline connected to particle on horizontal surface or other incline |
| Difficulty | Standard +0.3 This is a standard M1 pulley system question requiring Newton's second law for connected particles, followed by kinematics with friction. Part (a) is routine application of F=ma to both particles with given answer to verify. Part (b) uses standard equations of motion. The question involves multiple steps but uses well-practiced techniques with no novel insight required, making it slightly easier than average for M1. |
| Spec | 3.03e Resolve forces: two dimensions3.03k Connected particles: pulleys and equilibrium3.03o Advanced connected particles: and pulleys3.03v Motion on rough surface: including inclined planes6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| \(R = 2mg \Rightarrow F = 2\mu mg\) | B1 | |
| \(A: T - 2\mu mg = 2ma\) | M1 A1 | |
| \(B: mg \times \frac{1}{2} - T = ma\) | M1 A1 | |
| Eliminating \(T\): \(3ma = \frac{1}{2}mg - 2\mu mg\) | M1 | |
| \(a = \frac{1}{6}(1 - 4\mu)g\) \((\ast)\) | A1 | (7) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mu = 0.2 \Rightarrow a = \frac{1}{30}g\) | B1 | |
| when string breaks: \(v^2 = 2 \times \frac{1}{30}g \times h = \frac{1}{15}gh\) | M1 A1 | |
| \(A\) decelerating with deceleration \(f \Rightarrow 2mf = 2\mu mg\) | ||
| \(f = \mu g = \frac{1}{5}g\) | B1 | |
| Hence distance travelled during deceleration is given by \(\frac{1}{15}gh = 2 \times \frac{1}{2}gd\) | M1 | |
| \(\Rightarrow d = \frac{1}{15}h\) | ||
| \(\therefore\) Total distance \(= \frac{2}{5}h\) | A1 cso | (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Any two from: weight of pulley; friction at pulley; friction on slope; weight of string; string extensible; 'spin' of particle | B1 B1 | (2) |
## (a)
$R = 2mg \Rightarrow F = 2\mu mg$ | B1 |
$A: T - 2\mu mg = 2ma$ | M1 A1 |
$B: mg \times \frac{1}{2} - T = ma$ | M1 A1 |
Eliminating $T$: $3ma = \frac{1}{2}mg - 2\mu mg$ | M1 |
$a = \frac{1}{6}(1 - 4\mu)g$ $(\ast)$ | A1 | (7) |
## (b)
$\mu = 0.2 \Rightarrow a = \frac{1}{30}g$ | B1 |
when string breaks: $v^2 = 2 \times \frac{1}{30}g \times h = \frac{1}{15}gh$ | M1 A1 |
$A$ decelerating with deceleration $f \Rightarrow 2mf = 2\mu mg$ | |
$f = \mu g = \frac{1}{5}g$ | B1 |
Hence distance travelled during deceleration is given by $\frac{1}{15}gh = 2 \times \frac{1}{2}gd$ | M1 |
$\Rightarrow d = \frac{1}{15}h$ | |
$\therefore$ Total distance $= \frac{2}{5}h$ | A1 cso | (6) |
## (c)
Any two from: weight of pulley; friction at pulley; friction on slope; weight of string; string extensible; 'spin' of particle | B1 B1 | (2) |
**Total: (15 marks)**
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**Note:** $(\ast)$ indicates final line is given on the paper; cso = correct solution only\includegraphics{figure_3}
Particles $A$ and $B$, of mass $2m$ and $m$ respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle $A$ is held on the table, while $B$ rests on a smooth plane inclined at $30°$ to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between $A$ and the table is $\mu$. The particle $A$ is released from rest and begins to move.
By writing down an equation of motion for each particle,
\begin{enumerate}[label=(\alph*)]
\item show that, while both particles move with the string taut. Each particle has an acceleration of magnitude $\frac{1}{5}(1 - 4\mu)g$. [7]
\end{enumerate}
When each particle has moved a distance $h$, the string breaks. The particle $A$ comes to rest before reaching the pulley. Given that $\mu = 0.2$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find, in terms of $h$, the total distance moved by $A$. [6]
\end{enumerate}
For the model described above,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2002 Q7 [15]}}