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# The upper half of an inclined plane of inclination θ is perfectly smooth while lower half is rough.A block starting from rest at the top of

The upper half of an inclined plane of inclination θ 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

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When the block is sliding down the upper half of the inclined plane;

Acceleration of the block = $gsin\theta$.

Let $\mu$ be the coefficient of kinetic friction between the block and the lower half of the plane.

When the block is sliding down the lower half;

Acceleration of the block = $-(gsin\theta -\mu cos\theta)$

For the block to come to rest at the bottom,

Acceleration in the first half must be equal to the retardation in the second half:

.$\\ gsin\theta=-(gsin\theta-\mugcos\theta) \\ \Rightarrow \mucos\theta=2sin\theta \\ \Rightarrow \mu=2tan\theta$

Therefore the coefficient of friction between the block and lower half of the plane is $\mu=2tan\theta$

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