Question

The magnitude of the resultant of two equal forces each of magnitude \( F \) is \( \sqrt{2}F \). Then the angle between their line of action is:

• A. \( 120^\circ \)
• B. \( 60^\circ \)
• C. \( 45^\circ \)
• D. \( 90^\circ \)

Hint:

Use the formula for resultant of two vectors: \[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta \] When \( R = \sqrt{2F \) and \( A = B = F \), then \( \theta = 90^\circ \).

Answer 1

The Correct Option is D

Step 1: Use the vector addition formula.
When two forces of equal magnitude \( F \) act at an angle \( \theta \), the magnitude \( R \) of the resultant is given by: \[ R = \sqrt{F^2 + F^2 + 2F^2 \cos \theta} = \sqrt{2F^2(1 + \cos \theta)} \]
Step 2: Substitute the given resultant magnitude. We are given: \[ R = \sqrt{2}F \] So, \[ \sqrt{2F^2(1 + \cos \theta)} = \sqrt{2}F \]
Step 3: Square both sides to simplify. \[ 2F^2(1 + \cos \theta) = 2F^2 \Rightarrow 1 + \cos \theta = 1 \Rightarrow \cos \theta = 0 \Rightarrow \theta = 90^\circ \] Conclusion: The angle between the two equal forces is: \[ \boxed{90^\circ} \]