Question

A body taken n times as much time to slide down a ${45}^{\circ }$ rough incline as it takes to slide down a smooth ${45}^{\circ }$ incline. The coefficient of friction between the body and the incline will be:

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

Answer 1

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

Without friction acceleration down the incline is $g\sin \theta$

For a displacement of $s$, the time taken to reach the bottom with zero initial velocity is

$T_1=\sqrt{\frac{2s}{g\sin \theta }}$

For the same displacement with friction acting on the body and coefficient of friction $=\mu$

$T_2=\sqrt{\frac{2s}{g\sin \theta -\mu g\cos \theta }}$

$n{T}_1=T_2$

$\therefore \frac{n^2}{g\sin \theta }=\frac{1}{g\sin \theta -\mu g\cos \theta }$

For $\theta ={45}^o$

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

The coefficient of friction
$\mu =(1-\frac{1}{n^2})$
Here, $\theta ={45}^{\circ }$
$\therefore \mu =\tan {45}^{\circ }(1-\frac{1}{n^2})=1-\frac{1}{n^2}$