Question

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

Answer 1

We have to find the angle between the two forces.

Step 1: Given data.

$R$= F, $A=B=$F.

The resultant formula is given by,$R=\sqrt{A^2+B^2+2\times A\times B\times cos\theta }$.

Where,

$R$= Resultant force

$AandB$= two forces

$\theta$=Angle between the two forces

Step 2:

Replacing these values in the formula we get,

$$\begin{gathered} R=\sqrt{A^2+B^2+2\times A\times B\times cos\theta } \\ F=\sqrt{F^2+F^2+2\times F\times F\times \cos \theta } \\ F=\sqrt{2F^2+2F^2\cos \theta } \\ F=\sqrt{2F^2\left(1+\cos \theta \right)} \\ {F}^2=2F^2\left(1+\cos \theta \right)(Squaringonbothsides) \\ \frac{F^2}{2F^2}=\left(1+\cos \theta \right) \\ \frac{1}{2}=\left(1+\cos \theta \right) \\ \frac{1}{2}-1=\cos \theta \\ -\frac{1}{2}=\cos \theta \\ \theta ={\cos }^{-1}\left(\frac{-1}{2}\right) \\ \theta =120° \end{gathered}$$

Therefore the angle between the two forces is $120°$.