The u p p er half of an in clin ed p lane of inclination `theta` is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by

Options

A. `mu=tan theta`
B. `mu=(1)/(tan theta)`
C. `mu=(2)/(2 tan theta)`
D. `mu=2 tan theta`

Answer

D

Explanation

Suppose length of smooth and rough slope is equal is l .
Initially when the block moves on an inclined plane smooth produced acceleration is a = g sin ` theta`
Its velocity at distance covered l
When the block moves on the other rough surface produced acceleration a = g sin theta
When distance covered is l
`0= v^(2) + 2g (sin theta - mu_(k ) cos theta ) l`
From equation (1) and (2)
`0=2 ( g sin theta ) l + 2 g (sin theta -mu_(k ) cos theta ) l`
`mu_(k ) = 2 tan theta `