For the smooth plane, v² = 2g(sinθ)L. For the rough plane, (v/n)² = 2g(sinθ - μcosθ)L. Dividing the second equation by the first gives 1/n² = (sinθ - μcosθ)/sinθ = 1 - μcotθ. Solving for μ gives μ = tanθ(1 - 1/n²).