Question
There are two inclined surfaces of equal length $(L)$ and same angle of inclination $45^{\circ}$ with the horizontal. One of them is rough and the other is perfectly smooth. A given body takes 2 times as much time to slide down on rough surface than on the smooth surface. The coefficient of kinetic friction $\left( \mu_{k} \right)$ between the object and the rough surface is close to
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Solution
$t_{\text{rough}\text{~}} = 2t_{\text{smooth}\text{~}}$
$$a_{\text{smooth}\text{~}} = gsin\theta$$
$$\begin{matrix} & t \propto \frac{1}{\sqrt{a}} \Rightarrow t_{\text{smooth}\text{~}} \propto \frac{1}{\sqrt{gsin\theta}} \\ & a_{\text{rough}\text{~}} = gsin\theta - \mu_{k}gcos\theta \\ & \frac{t_{\text{rough}\text{~}}}{t_{\text{smooth}\text{~}}} = \frac{\sqrt{sin\theta}}{\sqrt{sin\theta - \mu_{k}cos\theta}} = 2 \end{matrix}$$
Squaring both sides
$$\begin{matrix} & \frac{sin\theta}{sin\theta - \mu_{k}cos\theta} = 4 \Rightarrow \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}} - \mu_{k} \times \frac{1}{\sqrt{2}}} = 4 \\ & \ \Rightarrow 1 - \mu_{k} = \frac{1}{4} \\ & \mu_{k} = \frac{3}{4} \\ & \ = 0.75 \end{matrix}$$
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