| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Motion on a slope |
| Type | Motion down smooth slope |
| Difficulty | Standard +0.8 This is a standard M3 banked track problem requiring resolution of forces in two directions and application of friction laws. Part (a) involves showing the banking angle matches the required angle for the given speed (routine calculation). Part (b) requires setting up equations with friction acting to prevent outward slip, solving simultaneously for μ - more algebraically demanding but follows standard M3 methodology. The 13 marks reflect substantial working, but this is a textbook application rather than requiring novel insight. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| (a) With no slip, \(R \sin 4° = \frac{mu^2}{100}\), \(R\cos 4° = mg\) | B1 B1 | |
| \(u^2 = 100g \tan 4° = 68 \cdot 53\) | \(u = 8 \cdot 28\) | M1 A1 |
| (b) \(R\cos 4° - F \sin 4° = mg\) | \(R \sin 4° + F \cos 4° = \frac{\pi(12 \cdot 9)^2}{100}\) | M1 A1 M1 A1 |
| Solve: \(R = 9 \cdot 885m\), \(F = 0 \cdot 875m\) | \(F \leq \mu R\), so \(\mu \geq F/R = 0 \cdot 089\) | M1 A1 M1 A1 |
(a) With no slip, $R \sin 4° = \frac{mu^2}{100}$, $R\cos 4° = mg$ | B1 B1
$u^2 = 100g \tan 4° = 68 \cdot 53$ | $u = 8 \cdot 28$ | M1 A1
(b) $R\cos 4° - F \sin 4° = mg$ | $R \sin 4° + F \cos 4° = \frac{\pi(12 \cdot 9)^2}{100}$ | M1 A1 M1 A1
Solve: $R = 9 \cdot 885m$, $F = 0 \cdot 875m$ | $F \leq \mu R$, so $\mu \geq F/R = 0 \cdot 089$ | M1 A1 M1 A1
**Total: 13 marks**A car moves round a circular racing track of radius 100 m, which is banked at an angle of 4° to the horizontal.
\begin{enumerate}[label=(\alph*)]
\item Show that when its speed is 8.28 ms$^{-1}$, there is no sideways force acting on the car. [4 marks]
\item When the speed of the car is 12.5 ms$^{-1}$, find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip. [9 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [13]}}