5.4

Banked Curves

When a car travels without skidding around an unbanked curve, the static frictional force between the tires and the road provides the centripetal force. The reliance on friction can be eliminated completely for a given speed, however, if the curve is banked at an angle relative to the horizontal, much in the same way that a plane is banked while making a turn.
Figure 5-11a shows a car going around a friction-free banked curve. The radius of the curve is r, where r is measured parallel to the horizontal and not to the slanted surface. Part b shows the normal force that the road applies to the car, the normal force being perpendicular to the road. Because the roadbed makes an angle with respect to the horizontal, the normal force has a component that points toward the center C of the circle and provides the centripetal force:

The vertical component of the normal force is and, since the car does not accelerate in the vertical direction, this component must balance the weight mg of the car. Therefore, . Dividing this equation into the previous one shows that

| (5.4)

Figure 5-11 (a) A car travels on a circle of radius r on a frictionless banked road. The banking angle is , and the center of the circle is at C. (b) The forces acting on the car are its weight and the normal force . A component of the normal force provides the centripetal force.

Equation 5.4 indicates that, for a given speed v, the centripetal force needed for a turn of radius r can be obtained from the normal force by banking the turn at an angle , independent of the mass of the vehicle. Greater speeds and smaller radii require more steeply banked curves—that is, larger values of . At a speed that is too small for a given , a car would slide down a frictionless banked curve; at a speed that is too large, a car would slide off the top. The next example deals with a famous banked curve.

| Example 8 | | | The Daytona 500

The Daytona 500 is the major event of the NASCAR (National Association for Stock Car Auto Racing) season. It is held at the Daytona International Speedway in Daytona, Florida. The turns in this oval track have a maximum radius (at the top) of r = 316 m and are banked steeply, with (see Figure 5-11). Suppose these maximum-radius turns were frictionless. At what speed would the cars have to travel around them?
Reasoning In the absence of friction, the horizontal component of the normal force that the track exerts on the car must provide the centripetal force. Therefore, the speed of the car is given by Equation 5.4.
Solution From Equation 5.4, it follows that

Drivers actually negotiate the turns at speeds up to 195 mph, however, which requires a greater centripetal force than that implied by Equation 5.4 for frictionless turns. Static friction provides the additional force.

The physics of the Daytona International Speedway.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.