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 the lower half of the plane is given by

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(1) μ=2 tan θ. Given: plane of length L, upper half smooth, lower half rough; the block starts and finishes at rest. Idea: starting and finishing at rest means the kinetic energy change is zero, so gravity's work over the whole plane is exactly cancelled by friction's work over the rough half. Work by gravity down the whole plane: mgL sin θ. Work against friction, over the lower half only: μ mg cos θ×L/2. Setting them equal: mgL sin θ=(μ mgL cos θ)/2. Cancelling mgL: sin θ=(μ cos θ)/2. μ=2 tan θ.

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 the lower half of the plane is given by

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