Two forces F1 and F2 acting at a point have a resultant F. If F2 is doubled, F is also doubled. If F2 is reversed in direction, then also F is doubled. Then F1:F2:F =

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Solution

$Resultant of F_{1} and F_{2} at an angle α is F . Resultant of F_{1} and 2 F_{2} at an angle α is 2 F . Resultant of F_{1} and F_{2} at an angle (π - α) is 2 F . So, F^{2} = {F_{1}}^{2} + {F_{2}}^{2} + 2 F_{1} \times F_{2} cosα - - - - (i) 4 F^{2} = {F_{1}}^{2} + 4 {F_{2}}^{2} + 2 F_{1} \times 2 F_{2} cosα - - - (ii) 4 F^{2} = {F_{1}}^{2} + {F_{2}}^{2} + 2 F_{1} \times F_{2} \cos (π - α) or 4 F^{2} = {F_{1}}^{2} + {F_{2}}^{2} - 2 F_{1} \times F_{2} cosα - - - - (iii) Adding (i) and (iii) and adding twice of (iii) in (ii) 2 {F_{1}}^{2} + 2 {F_{2}}^{2} - 5 F^{2} = 0 and {F_{1}}^{2} + 2 {F_{2}}^{2} - 4 F^{2} = 0 \frac{{F_{1}}^{2}}{2} = \frac{{F_{2}}^{2}}{3} = \frac{F^{2}}{2} Or F_{1} : F_{2} : F = \sqrt{2} : \sqrt{3} : \sqrt{2}$

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