First slide

Motion in a plane

Question

The resultant of two forces F₁ and F₂ is F. If F₂ is reversed then the magnitude of resultant is 2F. If F₂ is doubled then the resultant is also doubled. Find the ratio F₁ : F₂: F

Moderate

A

1: √2 :√3
B

√3 :2 :1
C

√3 :√2 :2
D

√2 :√3 :√2

Solution

If the angle between F₁ and F₂ is $θ$
$\large (2F)^2=F_1^2+F_2^2+2F_1F_2cos(\pi-\theta)$
$\large 4F^2=F_1^2+F_2^2-2F_1F_2cos\theta..................(2)$
$\large Also , (2F)^2=F_1^2+(2F_2)^2+2F_1(2F_2).cos\theta$
$\large \Rightarrow 4F^2=F_1^2+4F_2^2+4F_1F_2cos\theta.......(3)$
$\large Solving , F_1=F,F_2=\frac {\sqrt3.F}{\sqrt2}$
Hence, the ratio is $\sqrt{2} : \sqrt{3} : \sqrt{2}$

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