Question

Let the two forces have equal magnitude $A$. If the magnitude of the resultant is $\dfrac{2A{3}$, then the angle between those two forces is}

• A. $\cos^{-1}\left(\dfrac{7}{9}\right)$
• B. $\cos^{-1}\left(-\dfrac{7}{9}\right)$
• C. $\cos^{-1}\left(-\dfrac{5}{9}\right)$
• D. $\cos^{-1}\left(\dfrac{5}{9}\right)$

Hint:

If resultant is less than individual forces, the angle between forces is obtuse.

Answer 1

The Correct Option is B

Step 1: Resultant of two equal forces.
\[ R^2 = A^2 + A^2 + 2A^2\cos\theta \] \[ R^2 = 2A^2(1+\cos\theta) \]
Step 2: Substituting given resultant.
\[ \left(\frac{2A}{3}\right)^2 = 2A^2(1+\cos\theta) \] \[ \frac{4A^2}{9} = 2A^2(1+\cos\theta) \]
Step 3: Solving for $\cos\theta$.
\[ 1+\cos\theta = \frac{2}{9} \] \[ \cos\theta = -\frac{7}{9} \]
Step 4: Conclusion.
\[ \theta = \cos^{-1}\left(-\frac{7}{9}\right) \]