Question

What does ‘banking of curves’ mean? What is its need? Derive an expression
for the maximum velocity with which a vehicle can negotiate a banked curve with
angle of banking θ and coefficient of kinetic friction μ.

Answer 1

Explanation:

Banking of road means raising the outer edge of the road with respect to inner edge so that the road makes an angle with the horizontal.

When a  vehicle moves along a curve, the force of friction provides the necessary centripetal force. But this force has a limit = μmg.When the speed of the vehicle increases beyond this maximum value, the banking of roads is necessary so that the component of normal reaction of the road towards the centre provides the necessary centripetal force.

From the  figure,   for vertical  equilibrium :

Total upwards force = total downwards force

Ncosθ=mg+fsinθmg=Ncosθ−fsinθ...(1)

As the  forces Nsinθ,fcosθ  provide the necessary centripetal force.

rmv2=Nsinθ+fcosθ...(2)

Thus dividing (2) by (1),

mgrmv2=Ncosθ−fsinθNsinθ+fcosθ 

As f=μsN

Thus,

rgv2=cosθ−μssinθsinθ+μscosθ=1−μstanθtanθ+μs=1−tanλtanθtanθ+tanλ=tan(θ+λ)

v=rgtan(θ+λ)              where tanλ=μs

I hope my answer helps you.