| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| 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 +0.8 This is a standard M3 banked track problem requiring resolution of forces in two directions and use of friction at limiting equilibrium. Part (a) is routine (showing μ=0.6), but parts (b) and (c) involve simultaneous equations with both banking angle and friction, requiring careful algebraic manipulation and multiple steps—moderately challenging but still a textbook exercise type. |
| Spec | 3.03t Coefficient of friction: F <= mu*R model6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{mv^2}{r} = \mu N = \mu mg\) | M1, A1 | |
| \(\mu = \frac{v^2}{rg} = \frac{21^2}{75\times 9.8} = 0.6\) | A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R(\uparrow)\ R\cos\alpha \mp 0.6R\sin\alpha = mg\) | M1, A1, A1 | |
| \(\Rightarrow R\left(\frac{4}{5} - \frac{3}{5}\cdot\frac{3}{5}\right) = mg \Rightarrow R = \frac{25mg}{11}\) | A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R(\leftarrow)\ R\sin\alpha \pm 0.6R\cos\alpha = \frac{mv^2}{r}\) | M1, A1, A1 | |
| \(v \approx 32.5\text{ m s}^{-1}\) | dM1 A1 cao | (5) |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{mv^2}{r} = \mu N = \mu mg$ | M1, A1 | |
| $\mu = \frac{v^2}{rg} = \frac{21^2}{75\times 9.8} = 0.6$ | A1 | (3) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R(\uparrow)\ R\cos\alpha \mp 0.6R\sin\alpha = mg$ | M1, A1, A1 | |
| $\Rightarrow R\left(\frac{4}{5} - \frac{3}{5}\cdot\frac{3}{5}\right) = mg \Rightarrow R = \frac{25mg}{11}$ | A1 | (4) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R(\leftarrow)\ R\sin\alpha \pm 0.6R\cos\alpha = \frac{mv^2}{r}$ | M1, A1, A1 | |
| $v \approx 32.5\text{ m s}^{-1}$ | dM1 A1 cao | (5) |
**Notes:** In part (b) M1 needs three terms of which one is $mg$. If $\cos\alpha$ and $\sin\alpha$ interchanged, awarded M1 A0 A1. In part (c) M1 needs three terms of which one is $\frac{mv^2}{r}$ or $mr\omega^2$.
**Total: 12 marks**
---5. A car of mass $m$ moves in a circular path of radius 75 m round a bend in a road. The maximum speed at which it can move without slipping sideways on the road is $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Given that this section of the road is horizontal,
\begin{enumerate}[label=(\alph*)]
\item show that the coefficient of friction between the car and the road is 0.6 .
The car comes to another bend in the road. The car's path now forms an arc of a horizontal circle of radius 44 m . The road is banked at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$. The coefficient of friction between the car and the road is again 0.6. The car moves at its maximum speed without slipping sideways.
\item Find, as a multiple of $m g$, the normal reaction between the car and road as the car moves round this bend.
\item Find the speed of the car as it goes round this bend.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2008 Q5 [12]}}