Why are circular roads banked? Deduce an expression for maximum speed of a vehicle which can be achieved while taking a turn on the banked curved road neglecting friction.

Explanation

The breadths of circular roads are sometimes kept slanted with the horizontal. This is called banking. In a banked road, the normal reaction of a vehicle would have a horizontal component. This contributes to the centripetal force of circular motion, and thus helps a vehicle to have a greater safe speed.
In `theta` = banking angle, N = normal reaction, Ncos `theta` = vertical component of N, which balances the weight mg of the vehicle, Nsin`theta` = horizontal component of N, which supplies the centripetal force `(mv^(2))/(r )` for the vehicle moving with velocity v in a path of radius r.
`therefore " " " Ncos"theta = " mg" " " and " Nsin " theta = (mv^(2))/(r )`
Then, `("Nsin"theta)/("Ncos"theta) = ((mv^(2))//r)/(mg) `
or, tan`theta = (v^(2))/(rg) " "or, v = sqrt("rgtan"theta)`
this is the maximum velocity that a vehicle may achieve in the curved path.
In this treatment, we neglected the effect of friction [ However, friction plays a very important role in motions along curved paths. for example, if the road is not banked , `theta` = 0 , then our formula gives v = 0 . But in practice, vehicles can turn in curved paths even in the absence of banking . in that case, the entire centripetal force is provided by friction] .