6-12. When a car brakes its rear wheels, the tires may slip on the road. The normal force on each rear tire is , and the coefficient of kinetic friction between the tires and the road is . Assume the rubber of the tires is flexible and the air pressure of  each tire follows . If the allowable shear stress of the tire follows , determine the probability of failure. Assume that P and are independent.

Solution:

The friction force between the tire and the road

Set  as the contact area between each tire and the road, then, using , we have

The average shear stress is given by

Set , then , where

Thus, the probability of failure is

                     Ans.