Chapter 10, Problem 38A

A car resting on a level road has two forces acting on it: its weight (down) and the normal force (up). Recall from Chapter 4 that the normal force is the support force, which is always perpendicular to the supporting surface. If the car makes a turn on a level road, the normal force is still straight up. Friction between the wheels and road is the only centripetal force providing curved motion. Suppose the road is banked so the normal force has a component that provides centripetal force as shown in the sketch. Do you think the road could be banked so that for a given speed and a given radius of curvature, a vehicle could make the turn without friction? Explain.

To determine

Whether the road could be bankedso thatfor a given speed and radius of curvature, the car can move without friction.

Explanation of Solution

Introduction:

When car is moving on the unbanked road, the forces acting on the car is,

1. Gravitational force due to weight of the car in the downward direction.
2. Normal force into the upward direction.

The normal force balances the force due to gravity. There is no net horizontal component of force on the car.

When car moves on banked road along curved path, normal force splits into the two components - along vertical and along horizontal. Vertical component of normal force balances the weight of the car and horizontal component of force balances the centripetal force.

  $\begin{array}{l}W=mg=N\cos \left(\theta \right)\mathrm{...}\left(1\right)\\ \frac{m{v}^2}{r}=N\sin \left(\theta \right)\mathrm{...}\left(2\right)\end{array}$

Divide equation (2) by equation (1):

  $\begin{array}{l}\tan \left(\theta \right)=\frac{v^2}{rg}\\ v=\sqrt{rg\tan \left(\theta \right)}\mathrm{....}\left(3\right)\end{array}$

For given radius of curvature, when speed of car increases; the banking angle also increasesaccordingly.If the speed of car is increases beyond the ideal speed, centripetal force is not balanced by the horizontal component of normal force. In this case car skids away from the center of curve path. In order to balance the centripetal force, frictional force acts on the car.

Conclusion:

Therefore, car can turn without friction when proper selection of value of angle of banking, speed and radius of curvatureis used.