A circular race track of radius 400 m is banked at an angle of `10^(@)` . If the coefficient of friction between the wheels of a race car and the road is 0.2 , what is the (i) optimum speed of the race car to aviod wear and tear on its tyres .
maximum permissible speed to aviod slipping ?

Explanation

Data : r = 500 m , ` theta = 10^(@) , mu_(s) = 0.25 , g =9.8 m//s^(2) , tan 10^(@) = 0.1763` ltbr. (i) On the banked track , the maximum speed of the vechicle without slipping (skidding ) is
`v_(max) =sqrt((rg(mu+tan theta))/(1-mu_(s) tan theta))`
` = sqrt((500xx 9.8 (0.25 + 0.1763))/(1-(0.25xx 0.1763))) = sqrt((500 xx 9.8 xx 0.4263)/(0.9559)) =sqrt(2185)`
(ii) The optimum speed of the vehicle on the track is
`v_(opt) sqrt(rg tan theta)`
` sqrt(500 xx 9.8 xx 0.1763)=sqrt(839.9) = 29 . 39 m//s `