| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Friction |
| Type | Elastic string with friction |
| Difficulty | Challenging +1.2 This is a multi-stage mechanics problem requiring energy methods and friction analysis. Part (a) involves work-energy principles with elastic potential energy, gravitational potential energy, and friction workâstandard M3 content but requiring careful bookkeeping across multiple energy forms. Part (b) requires analyzing limiting equilibrium after motion ceases, comparing friction forces in extended vs natural length states. While conceptually straightforward for M3 students and following standard problem-solving patterns, the algebraic manipulation and need to coordinate multiple physical principles across two parts elevates it slightly above average difficulty. |
| Spec | 3.03v Motion on rough surface: including inclined planes6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| \(R = mg \cos \theta\) | B1 | |
| WD against friction \(= \mu xmg \cos \theta\) | B1 | |
| \(\mu xmg \cos \theta = mg\sin \theta - \frac{mgx^2}{2a}\) | M1 A2 | |
| \(x = 2a(\sin \theta - \mu \cos \theta)\) ** | A1 | Correct expression for \(x\); no errors in the working |
| (6) |
| Answer | Marks | Guidance |
|---|---|---|
| \(T = \frac{mg 2a(\sin \theta - \mu \cos \theta)}{a} = 2mg(\sin \theta - \mu \cos \theta)\) | B1 | |
| No motion if \(T \leq mg \sin \theta + \mu mg \cos \theta\) | M1 A1 | |
| \(2mg(\sin \theta - \mu \cos \theta) \leq mg \sin \theta + \mu mg \cos \theta\) | DM1 | |
| \(\frac{1}{3}\tan \theta \leq \mu\) ** | A1 | Correct inequality and no errors in the working |
| (5) | ||
| 11 |
## 3(a)
$R = mg \cos \theta$ | B1 |
WD against friction $= \mu xmg \cos \theta$ | B1 |
$\mu xmg \cos \theta = mg\sin \theta - \frac{mgx^2}{2a}$ | M1 A2 |
$x = 2a(\sin \theta - \mu \cos \theta)$ ** | A1 | Correct expression for $x$; no errors in the working
| (6) |
## 3(b)
$T = \frac{mg 2a(\sin \theta - \mu \cos \theta)}{a} = 2mg(\sin \theta - \mu \cos \theta)$ | B1 |
No motion if $T \leq mg \sin \theta + \mu mg \cos \theta$ | M1 A1 |
$2mg(\sin \theta - \mu \cos \theta) \leq mg \sin \theta + \mu mg \cos \theta$ | DM1 |
$\frac{1}{3}\tan \theta \leq \mu$ ** | A1 | Correct inequality and no errors in the working
| (5) |
| 11 |
---One end $A$ of a light elastic string $AB$, of modulus of elasticity $mg$ and natural length $a$, is fixed to a point on a rough plane inclined at an angle $\theta$ to the horizontal. The other end $B$ of the string is attached to a particle of mass $m$ which is held at rest on the plane. The string $AB$ lies along a line of greatest slope of the plane, with $B$ lower than $A$ and $AB = a$. The coefficient of friction between the particle and the plane is $\mu$, where $\mu < \tan \theta$. The particle is released from rest.
\begin{enumerate}[label=(\alph*)]
\item Show that when the particle comes to rest it has moved a distance $2a(\sin \theta - \mu \cos \theta)$ down the plane. [6]
\item Given that there is no further motion, show that $\mu \geqslant \frac{1}{3} \tan \theta$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2014 Q3 [11]}}