Question

A block at rest is released from a smooth inclined plane of inclination ${45}^{\circ }$ and is allowed to slide a distance 'd'. Now same block is released on a similar rough inclined plane and allowed to slide the same distance. The time taken by block to slide on rough plane is 'n' times the time taken by it on a smooth plane. Then what would be the coefficient of friction for the rough inclined plane?

• A. ${\mu }_k=\sqrt{1-\frac{1}{n^2}}$
• B. ${\mu }_k=1-\frac{1}{n^2}$
• C. ${\mu }_s=\sqrt{1-\frac{1}{n^2}}$
• D. ${\mu }_s=1-\frac{1}{n^2}$

Answer 1

The correct option is B ${\mu }_k=1-\frac{1}{n^2}$


$d=\frac{1}{2}\left(gsin\theta \right)t_1^2,d=\frac{1}{2}(gsin\theta -\mu gcos\theta )t_2^2$
$t_1=\sqrt{\frac{2d}{gsin\theta }},t_2=\sqrt{\frac{2d}{gsin\theta -\mu gcos\theta }}$
According to question, $t_2=n{t}_1$
$n\sqrt{\frac{2d}{gsin\theta }}=\sqrt{\frac{2d}{gsin\theta -\mu gcos\theta }}$
$n=\frac{1}{\sqrt{1-{\mu }_k}}$ $(\because cos{45}^{\circ }={45}^{\circ }=\frac{1}{\sqrt{2}})$
$n^2=\frac{1}{1-{\mu }_k}or1-{\mu }_k=\frac{1}{n^2}or{\mu }_k=1-\frac{1}{n^2}$