M3 January 2008 Q5
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 m s\(^{-1}\). Given that this section of the road is horizontal,
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 \(\tan\alpha = \tfrac{3}{4}\). The coefficient of friction between the car and the road is again 0.6. The car moves at its maximum speed without slipping sideways.
| Scheme | Marks |
|---|---|
| \(\dfrac{mv^2}{r} = \mu N,\ = \mu mg\) | M1, A1 |
| \(\mu = \dfrac{v^2}{rg} = \dfrac{21^2}{75 \times 9.8} = 0.6\) * | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| R(\(\uparrow\)) \(R\cos\alpha,\ \mp 0.6R\sin\alpha = mg\) | M1, A1, A1 |
| \(\Rightarrow R\left(\dfrac{4}{5} - \dfrac{3}{5}.\dfrac{3}{5}\right) = mg \Rightarrow R = \dfrac{25mg}{11}\) | A1 |
| (4) |
Notes
In part (b) M1 needs three terms of which one is mg
If cos \(\alpha\) and sin \(\alpha\) are interchanged in equation this is awarded M1 A0 A1
If they resolve along the plane and perpendicular to the plane in part (b)
| then attempt at \(R - mg\cos\alpha = \dfrac{mv^2}{r}\sin\alpha\), and \(0.6R + mg\sin\alpha = \dfrac{mv^2}{r}\cos\alpha\) and attempt to eliminate \(v\) | M1 |
| Two correct equations | A1 |
| Correct work to solve simultaneous equations | A1 |
| Answer | A1 (4) |
| Scheme | Marks |
|---|---|
| R(\(\leftarrow\)) \(R\sin\alpha,\ \pm 0.6R\cos\alpha = \dfrac{mv^2}{r}\) | M1, A1, A1 |
| \(v \approx 32.5\) m s\(^{-1}\) | dM1 A1cao |
| (5) | |
| (12 marks) |
Notes
In part (c) M1 needs three terms of which one is \(\dfrac{mv^2}{r}\) or \(mr\omega^2\)
If cos \(\alpha\) and sin \(\alpha\) are interchanged in equation this is also awarded M1 A0 A1
Alternative
| In part (c) Substitute R into one of the equations | M1 |
| Substitutes into a correct equation (earning accuracy marks in part (b)) | A1 |
| Uses \(R = \dfrac{25mg}{11}\) (or \(\dfrac{25mg}{29}\)) | A1 |
| Obtain \(v = 32.5\) | M1A1 (5) |