A given object takes n times the time to slide down 45° rough inclined plane as it takes the time to slide down an identical perfectly smooth 45° inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is:
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Question

A given object takes n times the time to slide down 45° rough inclined plane as it takes the time to slide down an identical perfectly smooth 45° inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is:

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G1 Physics · Accepted Answer

For the smooth 45° incline: a₁ = g sin45° Starting from rest over the same distance ℓ: ℓ = ½a₁t₁² For the rough incline: a₂ = g(sin45° - μ cos45°) and: ℓ = ½a₂t₂² Given: t₂ = nt₁ Therefore: a₂n² = a₁ So: g(sin45° - μ cos45°)n² = g sin45° Since: sin45° = cos45° (1 - μ)n² = 1 Therefore: μ = 1 - 1/n²