Uniform Circular Motion

In the case of circular motion we always know that the acceleration is . Thus we always know the right hand side of Newton's second law, namely

The forces that produce circular motion get put into the left hand side.

Example In designing a curved road, engineers consider the speed v of a car and the coeffcient of friction between the car tires and the road. The radius of curvature of the road bend is chosen to be large enough so that the car will be able to derive around smoothly in a part-circle. Work out a formula for the radius of curvature in terms of the speed of the car and the coeffcient of friction.

Solution Force diagrams are shown in Fig. 1.1. The top part of the figure shows that static friction alone keeps the car in circular motion. (The forward motion of the car involves moving kinetic friction, but the sideways motion involves static friction.)

FIGURE 1.1 Car rounding a curve.

In the x direction

We get N from the y direction,

Substituting into the x equation gives

or

This formula tells an engineer how large to make the radius of curvature of the road for a given car speed v (say 5 times the speed limit) and a coeffcient of friction μ_s.