Question

A given object takes $n$ times as much time to slide down at ${45}^{\circ }$ rough incline as it takes to slide down a perfectly smooth ${45}^{\circ }$ incline. The coefficient of kinetic friction between the object and the incline is given by :

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

Answer 1

The correct option is A $(1-\frac{1}{n^2})$

For rough surface,

$mg\sin \theta -\mu mg\cos \theta =ma$

$\to \frac{g}{\sqrt{2}}-\frac{\mu g}{\sqrt{2}}=a......\left(1\right)$

or ${}^{\prime }a=\frac{g}{\sqrt{2}}[1-\mu ],T=nt$ [Acc. to question]

For smooth surface

$mg\sin \theta =m{a}^{\prime }$

$\frac{g}{\sqrt{2}}=a^{\prime }T=t$

since both starts From $O$ i.e. from rest and the distance $\left(S\right)$ is same then

$S_1=S_2$

$\frac{1}{2}a{t}^2=\frac{1}{2}a(nt{)}^2$

$\to \frac{g}{\sqrt{2}}{t}^2=\frac{g}{\sqrt{2}}(1-\mu )n^2{t}^2$

$\to n^2[1-\mu ]=1$

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

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

(A) Ans.