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Question:

The upper half of an inclined plane of inclination θ is perfectly smooth while the 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 equal to:

μ=1tanθ

μ=2tanθ

μ=2tanθ

μ=tanθ

Solution:

Potential energy at top is completely used to do work against friction.
mgsinθ.L = μmgcosθ × L/2
μ = 2sinθ/cosθ = 2tanθ