Question

A cylinder is released from rest from the top of an incline plane of inclination $60^{\circ}$ where friction coefficient varies with distance $x$ as $\mu=\frac{2-3 x}{\sqrt{3}}$. Find the distance travelled by the cylinder on incline before it starts slipping :

• A. $1 / 3 m$
• B. $1 / \sqrt{3} m$
• C. $3 m$
• D. $\sqrt{3} m$

Answer 1

Answer: (A)

Let friction is acting in backward direction

$ m g \sin 60^{\circ}-f =M a \,\,\,...(2)$
$ f R =I \alpha $
$\Rightarrow f R =\frac{M R^{2}}{2}\left(\frac{a}{R}\right) $
$\Rightarrow f =\frac{M a}{2} \,\,\,...(3)$
$M g \sin 60^{\circ} =\frac{3}{2} M a$
$\Rightarrow a=\frac{2}{3} g \sin 60^{\circ}=\frac{g}{\sqrt{3}}$
$\Rightarrow f=\frac{M g}{2 \sqrt{3}}$
For sliding, $f=\mu M g \cos 60^{\circ}$
$\frac{M g}{2 \sqrt{3}}=\frac{2-3 x}{\sqrt{3}} \times \frac{1}{2}$
$ \Rightarrow 2-3 x =1 $
$ \Rightarrow 3 x =1 $
$\Rightarrow x =\frac{1}{3} m $