Question

The time taken by a body to slide down a rough ${45}^{\circ }$ inclined plane is twice that 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. $\frac{1}{3}$
• B. $\frac{3}{4}$
• C. $\sqrt{\frac{3}{4}}$
• D. $\sqrt{\frac{2}{3}}$

Answer 1

Answer: (B)

$\frac{3}{4}$

Acceleration of block moving down rought plane=$a_1=gsin\theta -\mu gcos\theta$

Acceleration of block moving down smooth surface=$a_2=gsin\theta$

Time taken to slide down the plane a distance of $s$ is found by

$s=\frac{1}{2}a{t}^2$

$⟹t=\sqrt{\frac{2s}{a}}$

Thus $t_1=\sqrt{\frac{2s}{gsin{45}^{\circ }-\mu gcos{45}^{\circ }}}$

$t_2=\sqrt{\frac{2s}{gsin{45}^{\circ }}}$

$⟹2^2=\frac{1/\sqrt{2}}{\frac{1}{\sqrt{2}}-\mu \frac{1}{\sqrt{2}}}$

$⟹1-\mu =\frac{1}{4}$

$⟹\mu =\frac{3}{4}$