A circular racentrack 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 tures, and (b) maximum. Permissible speed to avoid slipping ?

A circular racentrack of radius 300 m is banked at an angle of 15^(@).

1.	A circular racentrack 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 tures, 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 `upsilon_(0)` is given by `upsilon_(0)=(Rg tan THETA)^(1//2)` Here `R = 300 m, theta = 15^(@), g = 9.8 ms^(-2)` `upsilon_(0)=(R g tan theta)^(1//2)=(300xx9.8xx tan 15^(@))^(1//2)` On simplification, `upsilon_(0)=28.1 ms^(-1)` The maximum PERMISSIBLE speed `upsilon_(max)` is given by `v_(max)=sqrt((Rg(tan theta + MU))/((1-mu tan theta)))` `= sqrt((300xx9.8xx(tan 15^(@)+0.2))/((1-0.2 tan 15^(@))))` On simplification `upsilon_(max)=38.1 ms^(-1)`