The upper half of an inclined plane (inclination θ) is perfectly smooth and its lower half rough. A block released from rest at the top just comes to rest at the bottom. Find coefficient of friction μ on the lower half.
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Question

The upper half of an inclined plane (inclination θ) is perfectly smooth and its lower half rough. A block released from rest at the top just comes to rest at the bottom. Find coefficient of friction μ on the lower half.

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G1 Physics · Accepted Answer

Let total inclined length be L. Gravity does work mgL sinθ over the full path. Friction acts only over the lower L/2 and dissipates μmg cosθ(L/2). Both initial and final kinetic energies are zero; equate works: mgLsinθ=μmg cosθ L/2. Thus μ=2tanθ. Answer: C.