Question

The time taken by an object to slide down $45^{\circ}$ rough inclined plane is $n$ times as it takes to slide down a perfectly smooth $45^{\circ}$ incline plane. The coefficient of kinetic friction between the object and the incline plane is

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

Answer 1

Answer: (D)

$ a _1= g \sin \theta= g / \sqrt{2} $
$ a _2= g \sin \theta- Kg \cos \theta=\frac{ g }{\sqrt{2}}-\frac{ Kg }{\sqrt{2}} $
$ t _2= nt _1 \,\, \& \,\,\, a _1 t _1^2= a _2 t _2^2 $
$ \frac{ g }{\sqrt{2}} t _1^2=\left(\frac{ g }{\sqrt{2}}-\frac{ kg }{\sqrt{2}}\right) n ^2 t _1^2$
$ K =1-\frac{1}{ n ^2} $