Question

If two forces act in opposite direction then their resultant is $10N$ and if they act mutually perpendicular then their resultant is $50N$. Find the magnitudes of both the forces.

Answer 1

If the forces act as in the opposite direction then,

$F_1-F_2=10N...........1$

And if they both act as a perpendicular then $\sqrt{F_1^2+F_2^2}=50N..............2$

From equation 1 we get $F_1=10+F_2$ put this value in equation 2 then we get,

$(F_2+10{)}^2+F_2^2={50}^2$

$\Rightarrow F_2^2+F_2^2+100+2F_2=2500$

$\Rightarrow 2F_2^2+2F_2-2400=0$

$\Rightarrow F_2^2+40F_2-30F_2-1200=0$

$\Rightarrow F_2(F_2+40)-30(F_2+40)$

$\Rightarrow (F_2-30)(F_2+40)$

$\Rightarrow F_2=30N$

Put this value in equation 1 then we get,

$F_1=10+30=40N$