Question

A smooth block is released at rest on a $45^°$ incline and then slides a distance $'d'$. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is :

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

Answer 1

Answer: (A)

When friction is absent $a_{1}=g\,sin\,\theta$
$\therefore s_{1}=\frac{1}{2} a_{1}t^{2}_{1}\,...\left(i\right)$
When friction is present
$a_{2}=g\,sin\,\theta-\mu_{k}\,g\,cos\,\theta $
$\therefore s_{1}=\frac{1}{2} a_{1}t^{2}_{1}\,...\left(ii\right)$
From Eqs. $\left(i\right)$ and $\left(ii\right)$
$\frac{1}{2}a_{1}\,t^{2}_{1}=\frac{1}{2}a_{2}\,t^{2}_{2}$
or $a_{1}\,t^{2}_{1}=a_{2}\left(n\,t_{1}\right)^{2}\left(\because t_{2}=nt_{1}\right)$
or $a_{1}=n^{2}a_{2}$
or $\frac{a_{2}}{a_{1}}=\frac{g\,sin\,\theta-\mu_{k}\,g\,cos\,\theta}{g\,sin\,\theta } =\frac{1}{n^{2}}$
or $\frac{g\,sin\,45^{\circ}-\mu_{k}\,g\,cos\,45^{\circ}}{g\,sin\,45^{\circ} } =\frac{1}{n^{2}}$
or $1-\mu_{k}=\frac{1}{n^{2}}$
$\mu_{k}=1-\frac{1}{n^{2}}$