Chapter 6, Problem 79A

To determine

To draw:The free body diagram of a car on a banked curve.

To find:The direction of acceleration. Also, the force exerted in the direction of the acceleration.

To identify: Whether such force will exist without friction or not.

Answer to Problem 79A

The free body diagram of a car on a banked curve is shown in Figure 1.

The direction of centripetal acceleration of the car is towards the centre of the curve.

The horizontal component of the normal force is exerted in the direction of the acceleration.

The force exerted in the direction of the acceleration is possible without friction incase of banked roads.

Explanation of Solution

Given:

On a race track, curves are banked so that it will be easy for cars to go with high speeds.

Formula used:

Centripetal acceleration is,

  $a_c=\frac{v^2}{r}$

Where, v is the velocity and r is the radius.

Calculation:

Consider the road is banked at an angle $\theta$ with the horizontal.

The free body diagram of the forces acting on the car in the banked road is shown below.

Figure 1

The gravitational force acting on the car is denoted by $F_{\text{g}}$ and the normal force acting on the car is denoted by $F_{\text{N}}$ .

Consider the centripetal force acting on the car is denoted by

$F_{\text{C}}$ . It is acting towards the centre of the curve. Hence, the centripetal acceleration is also acts towards the centre of the curve.

The horizontal component of the normal force denoted by $F_{\text{N}}\sin \theta$ , acts on the direction of the centripetal acceleration.

For the banked roads, the horizontal component of the normal force $F_{\text{N}}\sin \theta$ , acts towards the centre of the curve. This reduces the skidding of car. For a particular value of the banking angle $\theta$ , there is a particular velocity $v$ of the vehicle at which the frictional force becomes zero.

Thus, it is possible to have such force without friction.