Question

Two forces with equal magnitudes $F$ act on a body and the magnitude of the resultant force is $\frac{F}{3}$. The angle between the two forces is:

• A. ${\cos }^{-1}(\frac{1}{2\sqrt{3}}-1)$
• B. $co{s}^{-}1(-\frac{1}{3})$
• C. ${\cos }^{-1}(\frac{1}{2\sqrt{3}}+1)$
• D. $co{s}^{-}1(-\frac{8}{9})$

Answer 1

The correct option is A ${\cos }^{-1}(\frac{1}{2\sqrt{3}}-1)$

Given,

Both forces are of equal magnitude $F$

Resultant force is $\frac{F}{3}$

$F_R=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos \theta }$

$\frac{F}{3}=\sqrt{F^2+F^2+2FF\cos \theta }$

$\frac{F^2}{\sqrt{3}}=2F^2(1+\cos \theta )$

$\cos \theta =(\frac{1}{2\sqrt{3}}-1)$

$\theta ={\cos }^{-1}(\frac{1}{2\sqrt{3}}-1)$

Hence, angle between vectors is ${\cos }^{-1}(\frac{1}{2\sqrt{3}}-1)$