| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Topic | Motion on a slope |
| Type | Motion down rough slope |
| Difficulty | Moderate -0.3 This is a standard mechanics problem on inclined planes with friction. Part (i) is routine diagram drawing, part (ii) requires resolving forces and applying F=ma (standard A-level technique with 5 marks suggesting straightforward working), and part (iii) tests conceptual understanding of the friction condition. While it requires multiple steps, it follows a well-established method with no novel insight needed, making it slightly easier than average. |
| Spec | 3.03a Force: vector nature and diagrams3.03r Friction: concept and vector form3.03v Motion on rough surface: including inclined planes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Diagram showing weight, normal contact force and friction, and no others | B1 | [1] |
| (ii) Resolve perpendicular to slope: | B1 M1 | Equation of motion with 3 terms, and at least 2 correct. Condone consistent sin/cos error. |
| \(N2L\) used & resolve down slope: | ||
| \(N = mg \cos \theta\) | ||
| \(ma = mg \sin \theta - F\) | ||
| Friction law: | B1 | Limiting friction only. |
| \(F = \mu N\) | M1 | Attempt to eliminate \(N\) and \(F\), and cancel \(m\). |
| \(\therefore ma = mg \sin \theta - \mu mg \cos \theta\) | ||
| \(\therefore a = g(\sin \theta - \mu \cos \theta)\) | A1 | c.a.o. [5] |
| (iii) If \(\mu > \tan \theta\) then the particle will not move. | B1 | [1] |
**(i)** Diagram showing weight, normal contact force and friction, and no others | B1 | [1]
**(ii)** Resolve perpendicular to slope: | B1 M1 | Equation of motion with 3 terms, and at least 2 correct. Condone consistent sin/cos error.
$N2L$ used & resolve down slope: |
$N = mg \cos \theta$ |
$ma = mg \sin \theta - F$ |
Friction law: | B1 | Limiting friction only.
$F = \mu N$ | M1 | Attempt to eliminate $N$ and $F$, and cancel $m$.
$\therefore ma = mg \sin \theta - \mu mg \cos \theta$ |
$\therefore a = g(\sin \theta - \mu \cos \theta)$ | A1 | c.a.o. [5]
**(iii)** If $\mu > \tan \theta$ then the particle will not move. | B1 | [1]A particle of mass $m$ is placed on a rough inclined plane. The plane makes an angle $\theta$ with the horizontal. The coefficient of friction between the particle and the plane is $\mu$ where $\mu < \tan \theta$. The particle is released from rest and accelerates down the plane.
\begin{enumerate}[label=(\roman*)]
\item Draw a fully labelled diagram to show the forces acting on the particle. [1]
\item Find an expression in terms of $g$, $\theta$ and $\mu$ for the acceleration of the particle. [5]
\item Explain what would happen to the particle if $\mu > \tan \theta$. [1]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2014 Q9 [7]}}