| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2006 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Motion on a slope |
| Type | Range of forces for equilibrium |
| Difficulty | Standard +0.3 This is a standard M1 equilibrium problem on a slope with friction, requiring resolution of forces and understanding of limiting friction. Parts (a)(i)-(iv) are routine applications of equilibrium equations and friction inequalities, while part (b) adds straightforward dynamics. The multi-part structure and need to consider both directions of friction (preventing upward and downward motion) elevate it slightly above average, but all techniques are standard M1 material with no novel insight required. |
| Spec | 3.03b Newton's first law: equilibrium3.03e Resolve forces: two dimensions3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces3.03v Motion on rough surface: including inclined planes |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(R = 80\cos 25°\) | M1 | component attempted |
| \(R = 72.5\text{ N}\) | A1, A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(F = 0.32 \times 72.5\) | M1 | condone inequality |
| \(F = 23.2\text{ N}\) | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(T + F = 80\cos 65°\) | M2, A1 | 3 forces direction correct, component attempted |
| \(T = 10.6\text{ N}\) | A1\(\sqrt{}\) | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(T = F + 80\cos 65°\) | M1, A1 | 3 forces direction correct, component attempted |
| \(T = 57.0\text{ N}\) (57N) | A1\(\sqrt{}\) | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\text{Mass} = \frac{80}{g} = (8.16\text{ kg})\) | B1 | Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(80\cos 65° - F = \text{mass} \times \text{acceleration}\) | M1 | 3 terms, component attempted |
| \(10.6 = \frac{80}{g} \times \text{acc}\) | A1 | all correct |
| \(\text{acc} = 1.30 \text{ ms}^{-2}\) (\(1.3 \text{ ms}^{-2}\)) | A1 | Total: 3 |
## Question 8:
### Part (a)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| $R = 80\cos 25°$ | M1 | component attempted |
| $R = 72.5\text{ N}$ | A1, A1 | Total: 3 | cao |
### Part (a)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $F = 0.32 \times 72.5$ | M1 | condone inequality |
| $F = 23.2\text{ N}$ | A1 | Total: 2 | cao |
### Part (a)(iii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $T + F = 80\cos 65°$ | M2, A1 | 3 forces direction correct, component attempted |
| $T = 10.6\text{ N}$ | A1$\sqrt{}$ | Total: 4 | $\sqrt{}$ friction |
### Part (a)(iv) [tension]
| Working | Marks | Guidance |
|---------|-------|----------|
| $T = F + 80\cos 65°$ | M1, A1 | 3 forces direction correct, component attempted |
| $T = 57.0\text{ N}$ (57N) | A1$\sqrt{}$ | Total: 3 | $\sqrt{}$ friction |
### Part (a)(iv) [mass]
| Working | Marks | Guidance |
|---------|-------|----------|
| $\text{Mass} = \frac{80}{g} = (8.16\text{ kg})$ | B1 | Total: 1 | |
### Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| $80\cos 65° - F = \text{mass} \times \text{acceleration}$ | M1 | 3 terms, component attempted |
| $10.6 = \frac{80}{g} \times \text{acc}$ | A1 | all correct |
| $\text{acc} = 1.30 \text{ ms}^{-2}$ ($1.3 \text{ ms}^{-2}$) | A1 | Total: 3 | cao |8 A rough slope is inclined at an angle of $25 ^ { \circ }$ to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude $T$ newtons. The diagram shows the slope, the box and the rope.\\
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-7_307_469_500_840}
\begin{enumerate}[label=(\alph*)]
\item The box is held in equilibrium by the rope.
\begin{enumerate}[label=(\roman*)]
\item Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
\item The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
\item Find the least possible tension in the rope to prevent the box from moving down the slope.
\item Find the greatest possible tension in the rope.
\item Show that the mass of the box is approximately 8.16 kg .
\end{enumerate}\item The rope is now released and the box slides down the slope. Find the acceleration of the box.
\end{enumerate}
\hfill \mbox{\textit{AQA M1 2006 Q8 [16]}}