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A cylinder of mass 1 \mathrm{~kg} is resting on an rough incline plane of inclination angle 60^{\circ}?
The cylinder is connected with another block of mass 'm' ?. The value of 'm' is kept minimum to keep the cylinder in equilibrium.
What should be the minimum coefficient of friction so that the cylinder can stay in the equilibrium?

Option: 1

2 \sqrt{3} \mathrm{~kg}, \mu \geqslant \sqrt{3}


Option: 2

\sqrt{3} \mathrm{~kg}, \mu \leqslant \sqrt{3}


Option: 3

\sqrt{3} \mathrm{~kg}, \mu \geqslant \frac{1}{\sqrt{3}}


Option: 4

2 \sqrt{3} \mathrm{~kg}, \mu \leqslant \frac{1}{\sqrt{3}}


Answers (1)

For translator y motion of the cylinder.

\begin{aligned} N&=M g \cos \theta+T \sin \theta \\ f+T \cos \theta&=M g \sin \theta \end{aligned}

\begin{aligned} As, \quad T&=m g\\ m&= \frac{M \sin \theta}{1+\cos \theta}=\frac{1 \times \sqrt{3} / 2}{1+3 / 2}=\sqrt{3} \\ \because \quad f &=m g \leqslant \mu N \\ \frac{\sin \theta}{1+\cos \theta} &\leqslant \mu \\ \frac{\sin 60^{\circ}}{1+\cos 60^{\circ}} &\leqslant \mu \\ \mu &\geqslant \frac{\sqrt{3} / 2}{3 / 2} \\ \mu& \geqslant \frac{1}{\sqrt{3}} \end{aligned}

Posted by

Sumit Saini

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