Two forces whose magnitudes are in the ratio $3 : 5$ give a resultant of $28 N$ . If the angle of their inclination is $60^{o}$ . Find the magnitude of each force.

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Solution

Let the forces be $F_{1}$ and $F_{2}$
We know that the resultant force $(F) = \sqrt{F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} cos θ}$
Also, $3 F_{2} = 5 F_{1}$
$\Rightarrow F_{2} = \frac{5}{3} F_{1}$
$\Rightarrow F$ $= \sqrt{F_{1}^{2} + (\frac{5}{3} F_{1})^{2} + 2 F_{1} (\frac{5 F_{1}}{3}) cos 60^{0}} = 28$
$\Rightarrow (\frac{7}{3}) F_{1} = 28 N$
$\Rightarrow F_{1} = 12 N, F_{2} = 20 N$

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