Question

A vehicle is at rest on a banked road with angle of banking $\theta$. The normal reaction of the vehicle is $N_1$. When the vehicle takes a turn on the same road the normal reaction is $N_2$. Then $N_1/N_2$ is equal to:

• A. $1$
• B. ${\sin }^2\theta$
• C. ${\cos }^2\theta$
• D. $0.5\sin \left(2\theta \right)$

Answer 1

The correct option is C ${\cos }^2\theta$

$N_1=mg\cos \theta$ $------\left(1\right)$

When it takes a turn on the same road $\left(\frac{m{v}^2}{r}\right)$ is the centripetal force acting on it $.$

Balancing the force

$mg\sin \theta =\frac{m{v}^2}{r}\cos \theta$ $;$ $\frac{v^2}{r}=g\tan \theta$

$N_2=mg\cos \theta +\frac{m{v}^2}{r}\sin \theta =mg\cos \theta +mg\tan \theta \sin \theta$ $-----------\left(2\right)$

$\frac{N_1}{N_2}=\frac{mg\cos \theta }{mg\cos \theta +mg\sin \theta \tan \theta }=\frac{\cos \theta }{\cos \theta \frac{{\sin }^2\theta }{\cos \theta }=\frac{\cos \theta }{\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta }}}={\cos }^2\theta$

Hence$,$

Option $\left(c\right)$ is correct