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.

Why are circular roads banked? Deduce an expression for maximum speed

1.	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.
Answer» Solution :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] .