Question

 A smooth block is released at rest on a 45∘∘ incline and then slides a distance dd. The time taken to slide is nn times as much to slide on rough incline than on a smooth incline. The coefficient of friction is  

(a) µs = 1 - 1/n²

(b) µs = √(1 - 1/n²)

(c) µk = 1 - 1/n²

(d) µk = √(1 - 1/n²)

Answer 1

The correct answer is  a)  µk = 1 - 1/n² 

^Solution::

On a smooth plane

d = ½ (g sinθ) t₁

_{$t_1=\sqrt{\frac{2d}{g\sin \theta }}$} 

On a rough surface

d = ½ (g sinθ – μg cosθ) t₂

_{$t_2=\sqrt{\frac{2d}{g\sin \theta -\mu g\cos \theta }}$} 

from the question , we know 

t₂=nt₁

$t_2=\sqrt{\frac{2d}{g\sin \theta -\mu g\cos \theta }}=n\sqrt{\frac{2d}{g\sin \theta }}$ 

$n=\frac{1}{\sqrt{1-\mu k}}$ 

$n^2=\frac{1}{1-\mu k}$ 

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