A car of mass $m$ moves on a banked road having radius $r$ and banking angle $\theta$. To avoid slipping from the banked road, the maximum permissible speed of the car is $v_0$. The coefficient of friction $\mu$ between the wheels of the car and the banked road is
Answer: (C) $\mu = \dfrac{v_0^2 - rg\tan\theta}{rg + v_0^2\tan\theta}$
At the maximum speed, friction acts down the slope. Resolving vertically and horizontally:
$$N\cos\theta = mg + \mu N\sin\theta, \qquad N\sin\theta + \mu N\cos\theta = \frac{mv_0^2}{r}$$
Dividing:
$$\frac{v_0^2}{rg} = \frac{\sin\theta + \mu\cos\theta}{\cos\theta - \mu\sin\theta} = \frac{\tan\theta + \mu}{1 - \mu\tan\theta}$$
$$v_0^2 - \mu v_0^2\tan\theta = rg\tan\theta + \mu rg \Rightarrow \mu = \frac{v_0^2 - rg\tan\theta}{rg + v_0^2\tan\theta}$$
Solution by Sreeraj P, M.Sc Physics