There are two inclined surface 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 :

• A

  $0.25$
• B

  $0.40$
• C

  $0.5$
• Correct answer
  D

  $0.75$

Solution

$\begin{aligned} & L=\frac{1}{2} a_S t_S^2=\frac{1}{2} a_R t_R^2 \\ \Rightarrow & \left(\frac{t_{\mathrm{s}}}{t_R}\right)^2=\frac{a_R}{a_{\mathrm{S}}} \\ \Rightarrow & \left(\frac{1}{2}\right)^2=\frac{g \sin \theta-\mu_k g \cos \theta}{g \sin \theta} \\ \Rightarrow & \frac{1}{4}=\frac{\sin 45^{\circ}-\mu_k \cos 45^{\circ}}{\sin 45^{\circ}} \\ \Rightarrow & \frac{1}{4}=\frac{\left(\frac{1}{\sqrt{2}}\right)}{\left(\frac{1}{\sqrt{2}}\right)}\left(\frac{1-\mu_k}{1}\right) \\ \Rightarrow & \frac{1}{4}=1-\mu_k \Rightarrow \mu_k=1-\frac{1}{4}=\frac{3}{4}=0.75\end{aligned}$

Hence, the correct answer is option: 'D'.