The time taken by an object to slide down a rough inclined plane is times as it takes to slide down a perfectly smooth inclined plane. The coefficient of kinetic friction between the object and the inclined plane is [2023]

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Solution

Q.
The time taken by an object to slide down a $45 °$ rough inclined plane is $n$ times as it takes to slide down a perfectly smooth $45 °$ inclined plane. The coefficient of kinetic friction between the object and the inclined plane is [2023]

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

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

3 $1 + \frac{1}{n^{2}}$

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

Ans.
(4)

$a_{1} = g \sin θ = \frac{g}{\sqrt{2}}$

$a_{2} = g \sin θ - K g \cos θ = \frac{g}{\sqrt{2}} - \frac{K g}{\sqrt{2}}$

$t_{2} = n t_{1} and a_{1} t_{1}^{2} = a_{2} t_{2}^{2}$

$\frac{g}{\sqrt{2}} t_{1}^{2} = (\frac{g}{\sqrt{2}} - \frac{K g}{\sqrt{2}}) n^{2} t_{1}^{2}$

$\Rightarrow K = 1 - \frac{1}{n^{2}}$

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