| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2010 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Banked track – with friction (find maximum/minimum speed or friction coefficient) |
| Difficulty | Standard +0.3 This is a standard M3 circular motion problem requiring application of Newton's second law with friction and resolving forces on a banked track. Part (a) is straightforward 'show that' using F=μR and centripetal force. Part (b) requires resolving forces in two directions on an inclined plane with friction, which is routine for M3 students but involves more algebraic manipulation than average. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\uparrow \quad R = mg\) | B1 | |
| Use of limiting friction \(F_r = \mu R\) | B1 | |
| \(\leftarrow \quad \mu R = \frac{m \cdot 28^2}{120}\) | M1 A1 | |
| \(\mu = \frac{28^2}{120 \times 9.8} = \frac{2}{3}\) | M1 A1 | cao (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\uparrow \quad R\cos\alpha - \mu R\sin\alpha = mg\) | M1 A1 | |
| \(\leftarrow \quad \mu R\cos\alpha + R\sin\alpha = \frac{mv^2}{r}\) | M1 A1 | |
| \(\frac{\mu\cos\alpha + \sin\alpha}{\cos\alpha - \mu\sin\alpha} = \frac{v^2}{rg}\) | M1 | Eliminating \(R\) |
| \(\frac{2\cos\alpha + 3\sin\alpha}{3\cos\alpha - 2\sin\alpha} = \frac{25}{24}\) | M1 | Substituting values |
| Leading to \(\tan\alpha = \frac{27}{122}\) | M1 A1 | awrt 0.22 (8) |
| [14] |
# Question 6:
## Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\uparrow \quad R = mg$ | B1 | |
| Use of limiting friction $F_r = \mu R$ | B1 | |
| $\leftarrow \quad \mu R = \frac{m \cdot 28^2}{120}$ | M1 A1 | |
| $\mu = \frac{28^2}{120 \times 9.8} = \frac{2}{3}$ | M1 A1 | cao **(6)** |
## Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\uparrow \quad R\cos\alpha - \mu R\sin\alpha = mg$ | M1 A1 | |
| $\leftarrow \quad \mu R\cos\alpha + R\sin\alpha = \frac{mv^2}{r}$ | M1 A1 | |
| $\frac{\mu\cos\alpha + \sin\alpha}{\cos\alpha - \mu\sin\alpha} = \frac{v^2}{rg}$ | M1 | Eliminating $R$ |
| $\frac{2\cos\alpha + 3\sin\alpha}{3\cos\alpha - 2\sin\alpha} = \frac{25}{24}$ | M1 | Substituting values |
| Leading to $\tan\alpha = \frac{27}{122}$ | M1 A1 | awrt 0.22 **(8)** |
| | **[14]** | |
---6. A bend of a race track is modelled as an arc of a horizontal circle of radius 120 m . The track is not banked at the bend. The maximum speed at which a motorcycle can be ridden round the bend without slipping sideways is $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The motorcycle and its rider are modelled as a particle and air resistance is assumed to be negligible.
\begin{enumerate}[label=(\alph*)]
\item Show that the coefficient of friction between the motorcycle and the track is $\frac { 2 } { 3 }$.
The bend is now reconstructed so that the track is banked at an angle $\alpha$ to the horizontal. The maximum speed at which the motorcycle can now be ridden round the bend without slipping sideways is $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The radius of the bend and the coefficient of friction between the motorcycle and the track are unchanged.
\item Find the value of $\tan \alpha$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2010 Q6 [14]}}