Derive an expression for maximum speed the car can have safe turn on a leveled circular road .

Explanation

In a leveled circular road , skidding mainly depends on the coefficient of static friction `mu_s` the coefficient of static friction depends on the nature of the surface which has a maximum limiting value , to avoid this is problem usually the outer edge of the road is slightly raised compared to inner edge this is called banking of roads or tracks . this introduces an inclination , and the angle is called banking angle .

Let the surface of the road make angle ` theta` with horizontal surface , then the normal force makes the same angle `theta ` with the vertical . when the car takes a turn , there are two forces acting on the car :
(a) Gravitational force Mg ( downwards )
(b ) Normal force N ( perpendicular to surface )
We can resolve the normal force into two components N` cos theta and N sin theta ` the component balances the downward gravitional force .mg. and component will provide the necessary centripetal acceleration . by using Newton second law .
The banking angle ` theta ` and radius of curvature of the road or track determines the safe speed of hte car at the turning . If the speed ofcar exceeds this safe speed then it starts to skid outward but frictional force comes into effect and provides and additional centripetal force to prevent the outward skidding .At the same time , if the speed of hte car is little lesser than safe speed , it starts to skid inward and frictional force comes into effect whcih reduces centripetal force to prevent inward skidding .However if hte speed of the vehicle is sufficiently greater than the correct speed then frictional force cannot stop the car from skidding .