A block slides from rest from the highest point to the lowest point of a rough inclined plane. The time taken is 50% greater than the time taken by the same block to slide the same distance from rest on an otherwise identical smooth inclined plane. Both planes are inclined at 45° to the horizontal. What is the coefficient of kinetic friction between the block and the rough plane?
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Question

A block slides from rest from the highest point to the lowest point of a rough inclined plane. The time taken is 50% greater than the time taken by the same block to slide the same distance from rest on an otherwise identical smooth inclined plane. Both planes are inclined at 45° to the horizontal. What is the coefficient of kinetic friction between the block and the rough plane?

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G1 Physics · Accepted Answer

For the rough plane, a(r) = g(sin45°-μ cos45°). For the smooth plane, a(s) = g sin45°. For the same distance s and zero initial speed, t = √(2s ÷ a). Given, t(r) = 1.5t(s). Therefore, √[a(s) ÷ a(r)] = 1.5 = 3 ÷ 2. Squaring: a(r) = (4 ÷ 9)a(s). Hence, g(sin45°-μ cos45°) = (4 ÷ 9)g sin45°. Since sin45° = cos45°, 1-μ = 4 ÷ 9. Thus, μ = 5 ÷ 9. Correct option: C.