A circular racetrack of radius 300m 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 tures, and (b) maximum. Permissible speed to avoid slipping?

Explanation

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:
`v_(0) =(Rg tan theta)^(1//2)`
Here, R = 300 m, `theta = 15^(@), g =9.8 ms^(-2)`
`v_(0)=(Rtan theta)^(1//2) = [300 xx 9.8 xx (tan 15^(@))]^(1//2)`
On simplification, `v_(0)= 28.1 ms^(-1)`
Thbe maximum permissible speed `v_("max")` is given by:
`v_("rms") = sqrt((Rg(tan theta + mu))/(1-mutan theta)) = sqrt((300 xx 9.8 xx (tan 15^(@) +0.2))/(1-0.2 tan 15^(@)))`
On simplification, `v_("max") = 38.1 ms^(-1)`