Question

The angle between two forces of equal magnitude \( R \), if the magnitude of their resultant is \( \dfrac{R}{2} \), is

• A. \( \cos^{-1}\left(-\dfrac{7}{8}\right) \)
• B. \( \cos^{-1}\left(-\dfrac{5}{7}\right) \)
• C. \( \cos^{-1}\left(-\dfrac{3}{7}\right) \)
• D. \( \cos^{-1}\left(-\dfrac{3}{4}\right) \)

Hint:

For equal forces, resultant depends only on the angle between them.

Answer 1

The Correct Option is A

Step 1: Write formula for resultant of two forces.
For two equal forces \( R \) making an angle \( \theta \) between them, resultant is \[ R_{\text{res}} = \sqrt{R^2 + R^2 + 2R^2\cos\theta} \]
Step 2: Substitute given resultant.
\[ \frac{R}{2} = \sqrt{2R^2(1+\cos\theta)} \]
Step 3: Square both sides.
\[ \frac{R^2}{4} = 2R^2(1+\cos\theta) \]
Step 4: Simplify.
\[ \frac{1}{4} = 2(1+\cos\theta) \Rightarrow 1+\cos\theta = \frac{1}{8} \] \[ \cos\theta = -\frac{7}{8} \]
Step 5: Conclusion.
\[ \theta = \cos^{-1}\left(-\frac{7}{8}\right) \]