Question

A body takes time t to reach the bottom of a smooth inclined plane making an angle $\theta$ with the horizontal. If the plane is made rough, time taken now is $2t$. Find the coefficient of friction of the rough surface.

• A. $\frac{3}{4}\tan \theta$
• B. $\frac{2}{3}\tan \theta$
• C. $\frac{1}{4}\tan \theta$
• D. $\frac{1}{2}\tan \theta$

Answer 1

Answer: (A)

$\frac{3}{4}\tan \theta$

Let the distance traveled along the inclined plane be s.
Smooth Surface:
$a=gsin\theta u=0s=ut+\left(1/2\right)a{t}^{2}2s=\left(gsin\theta \right)t^2$
Rough Surface: Let coefficient of friction be $\mu$
Friction force= $f=\mu N=\mu mgcos\theta$
$\therefore a=(mgsin\theta -\mu mgcos\theta )/m=gsin\theta -\mu gcos\theta u=0s=ut+\left(1/2\right)a{\left(2t\right)}^{2}2s=4(gsin\theta -\mu gcos\theta )t^2$
Comparing both equations:
$\left(gsin\theta \right)t^2=4(gsin\theta -\mu gcos\theta )t^{2}gsin\theta =4gsin\theta -4\mu gcos\theta 3gsin\theta =4\mu gcos\theta \mu =\frac{3}{4}tan\theta$