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Rough versus smooth incline: friction coefficient from the ratio of sliding times

This type of question, asked 5 times: 2021 Paper 24, 29 Jan 2023, 8 Apr 2024, 4 Apr 2026 S1, 6 Apr 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 24

    24
    When a body slides down from rest along a smooth inclined plane making an angle of 30∘30^{\circ} with the horizontal, it takes time T. When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time αT\alpha T, where α\alpha is a constant greater than 1. The co-efficient of friction between the body and the rough plane is 1x(α2−1α2)\frac{1}{\sqrt{x}}\left( \frac{\alpha^{2}- 1}{\alpha^{2}} \right) where x=x =

  2. Question 2

    29 Jan 2023

    93
    The time taken by an object to slide down 45∘45^{\circ} rough inclined plane is nn times as it takes to slide down a perfectly smooth 45∘45^{\circ} incline plane. The coefficient of kinetic friction between the object and the incline plane is:

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  3. Question 3

    8 Apr 2024

    104
    A given object takes nn times the time to slide down 45∘45^{\circ} rough inclined plane as it takes the time to slide down an identical perfectly smooth 45∘45^{\circ} inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is :

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  4. Question 4

    4 Apr 2026 S1

    3
    The time taken by a block of mass mm to slide down from the highest point to the lowest point on a rough inclined plane is 50%50\% more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at 45∘45^{\circ} with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is ____\_\_\_\_ .

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  5. Question 5

    6 Apr 2026 S2

    21
    A block takes tt time to slide down a plane inclined at 45∘45^{\circ} to the horizontal. If the surface is made smooth (frictionless), the block takes times t2\frac{t}{2} to slide down the plane. The coefficient of friction between the block and the inclined plane is (α100)\left( \frac{\alpha}{100} \right) . The value of α\alpha is ____\_\_\_\_.