Mr. $A, B and C$ are trying to put a 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 the ropes, then, what is the relation between the forces if piston decends vertically with uniform speed?

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

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\sqrt{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 $\sqrt{3} F_{1} = F_{2} + 2 F_{3}$
FBD of the piston can be shown as

Piston is vertically above the cylinder, so to drop it inside the cylinder, net horizontal force must be zero on the piston
Thus, $F_{1} sin 60^{\circ} = F_{2} cos 60^{\circ} + F_{3}$
$\Rightarrow F_{1} \frac{\sqrt{3}}{2} = \frac{F_{2}}{2} + F_{3}$
$\Rightarrow \sqrt{3} F_{1} = F_{2} + 2 F_{3}$

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