Question

The upper half of an incline plane with inclination $\varphi$ 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 friction for the lower half is given by :

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

Answer 1

Answer: (A)

$2\tan \varphi$

The acceleration of the block while is sliding on the upper half of the inclined plane is $g\sin \varphi$

If $\mu$ is the coefficient of kinetic friction between the block and the lower half of the plane, the retardation of the block while sliding to the lower half is $-(g\sin \varphi -\mu g\cos \varphi )$

For the block to come to rest at the bottom of the initial plane, the acceleration in the second half must be equal to acceleration on the upper half :

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

$\Rightarrow \mu \cos \varphi =2\sin \varphi$

$\Rightarrow \mu =2\tan \varphi$