Question

A given object takes n times more time to slide down ${45}^0$ rough inclined plane as it takes to slide down a perfectly smooth ${45}^0$ inclined. The coefficient of kinetic friction between the object and the incline is:

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

Answer 1

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

Given,

Time taken by block to slide on frictionless incline plane is $T$

Time taken by block to slide on rough incline plane is $nT$

Net acceleration on smooth incline plane, $a=g\sin \theta$

Apply kinematic equation

$s=ut+\frac{1}{2}a{t}^2$

$s=0+\frac{1}{2}g\sin \theta \times T^2....\left(1\right)$

Net acceleration on rough inclined plane $a=g\sin \theta -\mu \cos \theta$

Apply kinematic Equation.

$s=ut+\frac{1}{2}a{t}^2$

$s=0+\frac{1}{2}(g\sin \theta -\mu g\cos \theta )\times {\left(nT\right)}^2$

Put value of $s$ from equation $\left(1\right)$

$\frac{1}{2}g\sin \theta \times T^2=\frac{1}{2}(g\sin \theta -\mu g\cos \theta ){\left(nT\right)}^2$

$1=(1-\frac{\mu }{\tan {45}^o})n^2$

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

Coefficient of friction $\mu =(1-\frac{1}{n^2})$