A car of mass 'm' moves on a banked road having radius 'r' and banking angle $\theta$. To avoid slipping from 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 :-

Forces Acting on the Car:

When a car moves on a banked road with friction, the forces acting on it are:

1. Normal reaction ($N$): Acts perpendicular to the surface.
2. Weight ($mg$): Acts vertically downward.
3. Friction force ($f$): Acts parallel to the surface and can be either up or down the incline, depending on whether the car tends to slide outward or inward.

The frictional force can be resolved into:

• Horizontal component: $f \cos\theta$
• Vertical component: $f \sin\theta$

The normal reaction $N$ can be resolved into:

• Horizontal component: $N \sin\theta$
• Vertical component: $N \cos\theta$

Step 1: Vertical Force Balance

Since the car does not move vertically,

$$N \cos\theta + f \sin\theta = mg$$

Using $f = \mu N$,

$$N \cos\theta + \mu N \sin\theta = mg$$

 $$N (\cos\theta + \mu \sin\theta) = mg$$ $$N = \frac{mg}{\cos\theta + \mu \sin\theta}$$ 

Step 2: Centripetal Force Balance

For circular motion, the net horizontal force provides the necessary centripetal force:

$$N \sin\theta - f \cos\theta = \frac{m v_0^2}{r}$$

Substituting $f = \mu N$,

$$N \sin\theta - \mu N \cos\theta = \frac{m v_0^2}{r}$$

 $$N (\sin\theta - \mu \cos\theta) = \frac{m v_0^2}{r}$$

Using the value of $N$ from the vertical balance equation:

$$\frac{mg}{\cos\theta + \mu \sin\theta} (\sin\theta - \mu \cos\theta) = \frac{m v_0^2}{r}$$

Cancel $m$ from both sides:

$$\frac{g (\sin\theta - \mu \cos\theta)}{\cos\theta + \mu \sin\theta} = \frac{v_0^2}{r}$$

Rearranging for $\mu$:

$$\mu = \frac{\tan\theta - \frac{v_0^2}{rg}}{1 + \frac{v_0^2}{rg} \tan\theta}$$

Final Answer:

$$\mu = \frac{\tan\theta - \frac{v_0^2}{rg}}{1 + \frac{v_0^2}{rg} \tan\theta}$$

Magnitude can also be written as $\mu =\frac{v_0^2-rg\tan \theta }{rg-v_0^2\tan \theta }$

This is the required expression for the coefficient of friction $\mu$ to prevent slipping on the banked road.