Question

A smooth block is released at rest on a $45^{\circ}$ 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)

$\mu_{k}=1-\frac{1}{n^{2}}$

Answer 2

When friction is absent $a_{1}=g \sin \theta$ $\therefore s_{1}=\frac{1}{2} a_{1} t_{1}^{2}$ ... (i) When friction is present $a_{2} =g \sin \theta-\mu_{k} g \cos \theta$ $\therefore s_{2} =\frac{1}{2} a_{2} t_{2}^{2}$...(ii) From Eqs. (i) and (ii) $\frac{1}{2} a_{1} t_{1}^{2}=\frac{1}{2} a_{2} t_{2}^{2}$ or $a_1 t_1^2 = a_2( nt_1)^2$ $\left(\because t_{2}=n t_{1}\right)$ 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 $1-\mu_{k}=\frac{1}{n^{2}}$ or $\mu_{k}=1-\frac{1}{n^{2}}$