The resultant of two forces $F_{1}$ and $F_{2}$ is F. If $F_{2}$ is reversed then the resultant is 2F. If $F_{2}$ is doubled then also the resultant is also doubled. Find the ratio $F_{1} : F_{2} : F$

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$\sqrt{3} : \sqrt{2} : 1$

$\sqrt{2} : \sqrt{3} : \sqrt{2}$

$1 : \sqrt{2} : \sqrt{3}$

$\sqrt{3} : \sqrt{2} : \sqrt{2}$

answer is D.

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Detailed Solution

$F^{2} = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} \cos θ$

4 $F^{2} = F_{1}^{2} + F_{2}^{2} - 2 F_{1} F_{2} \cos θ$

If $F_{2}$ is doubled then resultant is 2F

$4 F^{2} = F_{1}^{2} + 4 F_{2}^{2} + 4 F_{1} F_{2} \cos θ$

on simplification, eliminate $' θ'$ then $F_{1} : F_{2} : F = \sqrt{2} : \sqrt{3} : \sqrt{2}$

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