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: (a) 1 - 1/n² (b) √(1 - 1/n²) (c) 1 - n² (d) √(1 - n²)

Answer: (a). At 45°: smooth a₀ = g/√(2), rough a = g/√(2)(1-μ). Same length: (tr)/(ts) = √(a₀/a) = n ⇒ (1)/(1-μ) = n² So the answer is (a) μ = 1 - 1/n².

A given object takes n times the time to slide down 45° rough… | JEE Main 2024 (8 Apr, Shift 2)

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Friction on an Incline

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

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Answer: (a)

At $45^\circ$ : smooth $a_0 = \dfrac{g}{\sqrt{2}}$ , rough $a = \dfrac{g}{\sqrt{2}}(1-\mu)$ .
Same length: $\dfrac{t_r}{t_s} = \sqrt{\dfrac{a_0}{a}} = n \Rightarrow \dfrac{1}{1-\mu} = n^2$
So the answer is (a) $\mu = 1 - \dfrac{1}{n^2}$ .

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