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

_{\(t_1=\sqrt{ 2d\over g\sin\theta}\)} 

On a rough surface

d = ½ (g sinθ – μg cosθ) t₂

_{\(t_2=\sqrt{ 2d\over g\sin\theta - \mu g\cos\theta}\)} 

from the question , we know 

t₂=nt₁

\(t_2=\sqrt{ 2d\over g\sin\theta - \mu g\cos\theta}= n\sqrt { 2d\over g\sin\theta}\) 

\(n={1\over \sqrt{ 1-\mu k}}\) 

\(n^2={ 1\over 1-\mu k}\) 

\(\mu k = 1- {1\over n^2}\)