Determine the angle of banking so as to minimize the wear and tear of the tyres of car negotiating a banked curve

Explanation

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.
I Gravitational force mg (downwards )
(ii) Normal force N (perpendicular to surface)
The normal force can be resolved into two components . N cos `theta` and N sin`theta ` . The gravitional force mg and compoenet N sin`theta ` will provide the necessary centripetal acceleration. By using Newton's second law .
N cos `theta = mg`
N sin `theta = (mv^(2))/(r )`
By dividing the equation we get tan `theta = (v^(2))/(r g) `
`v= sqrt( rg tan theta)`
The banking angle `theta ` and radius of curvature of the road or track determines the safe speed of the car at the turnin