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 the inclined plane

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Solution

Q.
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 the inclined plane is ______ [2024].

1 $\sqrt{1 - n^{2}}$

2 $\sqrt{1 - \frac{1}{n^{2}}}$

3 $1 - n^{2}$

4 $1 - \frac{1}{n^{2}}$

Ans.
(4)

Case-1: No friction

$a = g \sin θ$

$ℓ = \frac{1}{2} (g \sin θ) t_{1}^{2} \Rightarrow t_{1} = \sqrt{\frac{2 ℓ}{g \sin θ}}$

Case-2: With friction

$a = g \sin θ - μ g \cos θ$

$ℓ = \frac{1}{2} (g \sin θ - μ g \cos θ) t_{2}^{2} \Rightarrow t_{2} = \sqrt{\frac{2 ℓ}{(g \sin θ - μ g \cos θ)}}$

$t_{2} = n t_{1} \Rightarrow \sqrt{\frac{2 ℓ}{(g \sin θ - μ g \cos θ)}} = n \sqrt{\frac{2 ℓ}{g \sin θ}}$

$\Rightarrow (\sin θ - μ \cos θ) = \frac{1}{n^{2}} \sin θ$

$\Rightarrow (1 - μ) = \frac{1}{n^{2}} \Rightarrow μ = 1 - \frac{1}{n^{2}}$

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