Question

A car is moving on a circular track of radius $R$. The road is banked at $\theta .\mu$ is the coefficient of friction. Find the maximum speed the car can have.

• A. ${\left[\frac{Rg(\sin \theta +\mu \cos \theta )}{\cos \theta +\mu \sin \theta }\right]}^{1/2}$
• B. ${\left[\frac{Rg(\cos \theta +\mu \sin \theta )}{\cos \theta -\mu \sin \theta }\right]}^{1/2}$
• C. ${\left[\frac{Rg(\sin \theta +\mu \cos \theta )}{\cos \theta -\mu \sin \theta }\right]}^{1/2}$
• D. $None$

Answer 1

The correct option is C ${\left[\frac{Rg(\sin \theta +\mu \cos \theta )}{\cos \theta -\mu \sin \theta }\right]}^{1/2}$

Here $f$ is frictional force $=\mu N$

Force balancing in vertical direction- $Ncos\theta =mg+\mu Nsin\theta ⟹N=\frac{mg}{cos\theta -\mu sin\theta }$ (1)

In horizontal direction- $\mu Ncos\theta +Nsin\theta =\frac{m{v}^2}{R}$ (2)

Solving both equations- $\frac{mg}{cos\theta -\mu sin\theta }(\mu cos\theta +sin\theta )=\frac{m{v}^2}{R}$

by solving- $v=\sqrt{\frac{Rg(sin\theta +\mu cos\theta )}{cos\theta -\mu sin\theta }}$