Question

The time taken by a body to slide down a rough ${45}^{\circ }$ inclined plane is thrice of that is required to slide down a smooth ${45}^{\circ }$ inclined plane. The coefficient of kinetic friction between the object and rough plane is given by:

• A. $\frac{8}{9}$
• B. $\frac{3}{4}$
• C. $\sqrt{\frac{3}{4}}$
• D. $\sqrt{\frac{2}{4}}$

Answer 1

The correct option is A $\frac{8}{9}$

Acceleration of block moving down rough plane $=a_1=g\sin \theta -\mu g\cos \theta$
Acceleration of block moving down smooth surface $=a_2=g\sin \theta$

Time taken to slide down the plane a distance of $s$ is found by
$s=\frac{1}{2}a{t}^2$
$\Rightarrow t=\sqrt{\frac{2s}{a}}$

Thus $t_1=\sqrt{\frac{2s}{g\sin {45}^{\circ }-\mu g\cos {45}^{\circ }}}...\left(i\right)$
$t_2=\sqrt{\frac{2s}{g\sin {45}^{\circ }}}...\left(ii\right)$

As it is given that the time taken by a body to slide down a rough ${45}^{\circ }$ inclined plane is thrice that required to slide down a smooth ${45}^{\circ }$ inclined plane. $t_1=3t_2$

By squaring and dividing the equation $\left(i\right)/\left(ii\right)$ we get,

$\Rightarrow 3^2=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\frac{1}{\sqrt{2}}-\mu \frac{1}{\sqrt{2}}}}$

$\Rightarrow 1-\mu =\frac{1}{9}$
$\Rightarrow \mu =\frac{8}{9}$

Hence option A is the correct answer.