A circular racetrack of radius 300 m is banked at an angle of 15^@ . If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the (a) optimum speed of the race- car to avoid wear and tear on its tyres, and (b) maximum permissible speed to avoid slipping ?

A circular racetrack of radius 300 m is banked at an angle of 15^@ . I

1.	A circular racetrack of radius 300 m is banked at an angle of 15^@ . If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the (a) optimum speed of the race- car to avoid wear and tear on its tyres, and (b) maximum permissible speed to avoid slipping ?
Answer» Solution :On a BANKED road, the horizontal component of the normal force and the frictional force contribute to provide centripetal force to keep the car moving on a circular turn without slipping.At the optimum speed, the normal reaction’s component is enough to provide the needed centripetal force, and the frictional force is not needed.The optimum speed `v_0` is given by Eq. `v_0 = (Rg tan THETA)^(1//2)` Here `R = 300 m , theta = 15^@ , g =9.8 MS^(-2) ` , we have `v_0 = 28.1 ms^(-1)` The MAXIMUM permissible speed `v_(MAX) ` is given by eq. `v_(max) = (Rg (mu_s + tan theta)/(1 -mu_s tan theta))^(1//2) = 38.1 ms^(-1)`