Question

If a car is moving at a velocity $v$ in a circular road with coefficient of friction $\mu$ and banking angle $\theta$. Find the radius of the road such that friction force doesn't act on the car in the radial direction.

• A. $r=\frac{v^2}{g\tan \theta }$
• B. $r=\frac{\sqrt{v}}{g\tan \theta }$
• C. $r=\sqrt{\frac{v^2}{g\tan \theta }}$
• D. $r=\frac{v^2}{\sqrt{g}}$

Answer 1

The correct option is A $r=\frac{v^2}{g\tan \theta }$

For the force of friction to be zero, the centripetal force has to be provided by the normal reaction force only. Which means $Nsin\left(\theta \right)=\frac{m{v}^2}{r}$ .....................$\left(1\right)$

now the other component of the normal reaction $Ncos\left(\theta \right)$ balances the weight of the car.

$\Rightarrow Ncos\left(\theta \right)=mg$ ....................$\left(2\right)$

Using this in $\left(1\right)$ we get

$mgtan\left(\theta \right)=\frac{m{v}^2}{r}$

$\Rightarrow r=\frac{v^2}{gtan\left(\theta \right)}$