Question

What is angle of banking derive an expression for angle of banking on a curved road with certain coefficient of friction

Answer 1

This angle is given by the following equation:   tan⁡α = µe

The associated angle is precisely the angle that would form a stable heap of such material, therefore this property is known as friction angle or coefficient of friction.

Answer 2

Answer: Refer to the diagram given below.

The angle at which the vehicle inclines on a curved road towards the inside turn is known as Angle of Banking.

Centripetal force

$\frac{mv^2}{r} = N sin \theta +\mu_s Ncos\theta\\ mg = N cos\theta-\mu_sN sin\theta$

Divide the above two equations:

$\frac{rg}{v^2}=\frac{cos\theta -\mu_s sin\theta}{sin\theta +\mu_scos\theta}$

Divide the numerator and denominator by cos θ on LHS

$\frac{rg}{v^2}=\frac{1-\mu_s tan\theta}{\mu_s+tan\theta}\\ \Rightarrow(\frac{rg}{v^2})\mu_s+(\frac{rg}{v^2})tan\theta = 1-\mu_stan\theta\\ \Rightarrow tan\theta =\frac{1-(\frac{rg}{v^2})\mu_s}{\mu_s+\frac{rg}{v^2}}\\ \Rightarrow \theta= tan^{-1}(\frac{1-(\frac{rg}{v^2})\mu_s}{\mu_s+\frac{rg}{v^2}})$