Question 135 of 210

A given object takes $\mathrm{n}$ 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 :

A$\sqrt{1-\frac{1}{n^2}}$B$1-n^2$C$1-\frac{1}{n^2}$D$\sqrt{1-n^2}$

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Answer: (C) $1-\frac{1}{n^2}$

Case-1 : No friction
$\begin{aligned}
& \mathrm{a}=\mathrm{g} \sin \theta \\
& \ell=\frac{1}{2}(\mathrm{~g} \sin \theta) \mathrm{t}_1^2 \\
& \mathrm{t}_1=\sqrt{\frac{2 \ell}{\mathrm{g} \sin \theta}}
\end{aligned}$
Case-2 : With friction
$\begin{aligned}
& \mathrm{a}=\mathrm{g} \sin \theta-\mu \mathrm{g} \cos \theta \\
& \ell=\frac{1}{2}(\mathrm{~g} \sin \theta-\mu \mathrm{g} \cos \theta) \mathrm{t}_2^2 \\
& \sqrt{\frac{2 \ell}{\mathrm{g} \sin \theta-\mu g \cos \theta}}=\mathrm{n} \sqrt{\frac{2 \ell}{\mathrm{g} \sin \theta}} \\
& \mu=1-\frac{1}{\mathrm{n}^2}
\end{aligned}$