Question

Two forces, each of magnitude $F$ have a resultant of the same magnitude $F$. The angle between the two forces is

• A. ${45}^{\circ }$
• B. ${120}^{\circ }$
• C. ${150}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is B ${120}^{\circ }$

$\text{Step 1: Resultant of two vectors}$

Resultant, $R=\sqrt{A^2+B^2+2AB\cos \theta }$

$Step2:Substitutingthevalues$
By substituting, $A=F,B=F$ and $R=F$ we get

$F=\sqrt{F^2+F^2+2FF\cos \theta }$

Squaring both sides

$\Rightarrow$ $-F^2=2F^{2}cos\theta$
$\Rightarrow$ $\cos \theta =\frac{-1}{2}$

$\therefore \theta ={120}^o$

Hence option $B$ is correct.

$\text{Alternate solution: [Refer figure]}$

Since two vectors are $\overrightarrow{F}$ and $\overrightarrow{F}$ and their resultant is also $\overrightarrow{F}$.

They will form an equilateral triangle.

$\therefore$ $\theta ={180}^o-{60}^o={120}^o$

Hence option $B$ is correct.