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A block starts from rest on a smooth plane inclined at 45° and covers a distance d. If the same block, when placed on a rough plane of the same inclination, takes n times the time to cover the same distance, what is the coefficient of kinetic friction? [2005]

1. μₖ = √(1 - 1/n²)
2. μₖ = 1 - 1/n²
3. μₛ = √(1 - 1/n²)
4. μₛ = 1 - 1/n²

Correct answer: μₖ = 1 - 1/n²

Solution

Smooth: a1 = g sin45. Rough: a2 = g(sin45 - mu cos45). Same distance d=0.5*a*t^2 with t_rough = n*t_smooth gives a2 = a1/n^2, so (sin45 - mu cos45) = sin45/n^2. Since sin45=cos45, 1 - mu = 1/n^2, giving mu_k = 1 - 1/n^2.

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