Question

A smooth block is released from rest on a ${45}^0$ inclined plane and it slides a distance $d$. The time taken to slide is $n$ times as much to slide on a rough incline than on a smooth incline. The coefficient of friction is :

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

Answer 1

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

When surface is smooth
$d=\frac{1}{2}g\sin \theta \times t^2$

And when surface is rough
$d=\frac{1}{2}g(\sin \theta -{\mu }_k\cos \theta )t_1^2$

Given: $t_1=nt$
$\therefore \frac{1}{2}g\sin \theta t^2=\frac{1}{2}g(\sin \theta -{\mu }_k\cos \theta )(nt{)}^2$
$\theta ={45}^0{\mu }_k=1-\frac{1}{n^2}$