Question

The upper half of an inclined plane with inclination $\theta$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of fraction for the lower half is given by

• A. $2sin\theta$
• B. $2cos\theta$
• C. $2tan\theta$
• D. $tan\theta$

Answer 1

The correct option is C $2tan\theta$

The acceleration of the block white its is sliding down the upper half of the inclined plane is $g\sin \theta$. If $\mu$ is the coefficient of kinetic friction between the block and the lower half of the plane, the retardation of the block while it is sliding down lower half $=(g\sin \theta -\mu \cos \theta )$

For the block to come to rest at bottom of the inclined plane the acceleration in the first half must be equal to the retardation in the $2nd$ half, i.e.

$g\sin \theta =-(g\sin \theta -\mu g\cos \theta )$

$\mu \cos \theta =2\sin \theta$

$\mu =2\tan \theta$.