Question

A smooth block is released at rest on a ${45}^{\circ }$ incline and then slides a distance d. The time taken by the block to slide equal distances on rough incline is n times as much to slide on smooth incline of same angle of inclination. Then the coefficient of friction is

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

Answer 1

The correct option is A

${\mu }_k=1-\frac{1}{n^2}$

$S=\frac{1}{2}gsin45t^2.......\left(1\right)$
For rough inclined plane
$S=\frac{1}{2}[gsin{45}^{\circ }-\mu gcos{45}^{\circ }]t^2{n}^2......\left(2\right)$
$[a=gsin\theta -\mu gcos\theta$ where $\theta ={45}^{\circ }]$
By (1) and (2)
$\frac{g}{2}\times \frac{1}{\sqrt{2}}{t}^2=\frac{g}{2}[\frac{1}{\sqrt{2}}-\frac{\mu }{\sqrt{2}}]n^2{t}^2\frac{1}{\sqrt{2}}=\frac{(1-\mu )}{\sqrt{2}}{n}^2\frac{1}{n^2}=1-\mu \mu =1-\frac{1}{n^2}$