Mr. $A, B$ and $C$ are trying to put ha heavy piston into a cylinder at a mechanical workshop in a railway yard. If they apply forces $F_{1}$ , $F_{2}$ and $F_{3}$ respectively on ropes then for which set of forces at that instant, they will be able to perform the said job?

\sqrt{3} F_{1} = F_{2} + 2 F_{3}

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2 F_{1} = F_{2} + F_{3}

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2 F_{2} = \sqrt{3} F_{1} - \frac{F_{3}}{2}

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F_{3} = 2 F_{1} - \sqrt{3} F_{2}

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Solution

The correct option is A $\sqrt{3} F_{1} = F_{2} + 2 F_{3}$

Given, Three forces $F_{1}, F_{2}, F_{3}$ are applied respectively

So, $F_{1} sin 60^{\circ} = F_{3} + F_{2} cos 60^{0} \Rightarrow F_{1} \frac{\sqrt{3}}{2} = F_{3} + \frac{F_{2}}{2}$ Since $sin 60^{\circ} = \frac{\sqrt{3}}{2} cos 60^{\circ} = \frac{1}{2}$

$\frac{\sqrt{3}}{2} F_{1} = \frac{2 F_{3} + F_{2}}{2} \Rightarrow \sqrt{3} F_{1} = 2 F_{3} + F_{2}$

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