M3 January 2010 Q6
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 m s\(^{-1}\). The motorcycle and its rider are modelled as a particle and air resistance is assumed to be negligible.
(a) Show that the coefficient of friction between the motorcycle and the track is \(\dfrac{2}{3}\). (6)
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 m s\(^{-1}\). The radius of the bend and the coefficient of friction between the motorcycle and the track are unchanged.
(b) Find the value of \(\tan\alpha\). (8)
| Scheme | Marks |
|---|---|
| \(\uparrow \quad R = mg\) | B1 |
| Use of limiting friction, \(F_r = \mu R\) | B1 |
| \(\leftarrow \quad \mu R = \dfrac{m28^2}{120}\) | M1 A1 |
| \(\mu = \dfrac{28^2}{120 \times 9.8} = \dfrac{2}{3}\) * cao | M1 A1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(\uparrow \quad R\cos\alpha - \mu R\sin\alpha = mg\) | M1 A1 |
| \(\leftarrow \quad \mu R\cos\alpha + R\sin\alpha = \dfrac{mv^2}{r}\) | M1 A1 |
| \(\dfrac{\mu\cos\alpha + \sin\alpha}{\cos\alpha - \mu\sin\alpha} = \dfrac{v^2}{rg}\) Eliminating \(R\) | M1 |
| \(\dfrac{2\cos\alpha + 3\sin\alpha}{3\cos\alpha - 2\sin\alpha} = \dfrac{25}{24}\) Substituting values | M1 |
| leading to \(\quad \tan\alpha = \dfrac{27}{122}\) awrt 0.22 | M1 A1 |
| (8) | |
| (14 marks) |