Question

A smooth block is released at rest on a $45^{°}$ incline and then slides a distance ${}^{\prime }{d}^{\prime }$. 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 $a_1=gsin\theta$
$\therefore s_1=\frac{1}{2}{a}_1{t}_1^2...\left(i\right)$
When friction is present
$a_2=gsin\theta -{\mu }_{k}gcos\theta$
$\therefore s_1=\frac{1}{2}{a}_1{t}_1^2...\left(ii\right)$
From Eqs. $\left(i\right)$ and $\left(ii\right)$
$\frac{1}{2}{a}_1{t}_1^2=\frac{1}{2}{a}_2{t}_2^2$
or $a_1{t}_1^2=a_2{\left(n{t}_1\right)}^2\left(\because t_2=n{t}_1\right)$
or $a_1=n^2{a}_2$
or $\frac{a_2}{a_1}=\frac{gsin\theta -{\mu }_{k}gcos\theta }{gsin\theta }=\frac{1}{n^2}$
or $\frac{gsin45^{\circ }-{\mu }_{k}gcos45^{\circ }}{gsin45^{\circ }}=\frac{1}{n^2}$
or $1-{\mu }_k=\frac{1}{n^2}$
${\mu }_k=1-\frac{1}{n^2}$