Question

If the resultant of two forces of equal magnitude has the same magnitude as either of the two forces, then the angle between the two forces is

• A. \( 30^\circ \)
• B. \( 60^\circ \)
• C. \( 90^\circ \)
• D. \( 120^\circ \)

Hint:

When the resultant of two equal forces equals the magnitude of one force, the angle between them is always obtuse and equals \( 120^\circ \).

Answer 1

The Correct Option is D

Step 1: Writing the formula for resultant of two forces.
If two forces of equal magnitude \( F \) act at an angle \( \theta \), the magnitude of their resultant \( R \) is given by: \[ R = \sqrt{F^2 + F^2 + 2F^2 \cos\theta} \]
Step 2: Applying the given condition.
It is given that the resultant has the same magnitude as either of the forces. Therefore: \[ R = F \] Substituting into the formula: \[ F = \sqrt{2F^2(1 + \cos\theta)} \]
Step 3: Solving for the angle.
Squaring both sides and simplifying: \[ F^2 = 2F^2(1 + \cos\theta) \] \[ 1 = 2(1 + \cos\theta) \] \[ \cos\theta = -\frac{1}{2} \] This gives: \[ \theta = 120^\circ \]
Step 4: Conclusion.
The angle between the two equal forces is \( 120^\circ \).

Answer 2

Two forces of equal magnitude \( F \) act at some angle \( \theta \) between them, and it is given that their resultant also has magnitude \( F \). Rather than deriving the angle algebraically, test each candidate angle directly using the resultant magnitude formula \( R = \sqrt{2F^2(1+\cos\theta)} \).

1. Option \( 30^\circ \): Substituting, \( \cos(30^\circ) \approx 0.866 \), giving \( R = \sqrt{2F^2(1.866)} \approx 1.93F \), which is much larger than \( F \); this angle does not satisfy the condition.
2. Option \( 60^\circ \): Substituting, \( \cos(60^\circ) = 0.5 \), giving \( R = \sqrt{2F^2(1.5)} = \sqrt{3}\,F \approx 1.73F \), still larger than \( F \); this does not satisfy the given equality either.
3. Option \( 90^\circ \): Substituting, \( \cos(90^\circ) = 0 \), giving \( R = \sqrt{2F^2(1)} = \sqrt{2}\,F \approx 1.41F \), which is again larger than \( F \), so this angle also fails to match.
4. Option \( 120^\circ \): Substituting, \( \cos(120^\circ) = -0.5 \), giving \( R = \sqrt{2F^2(1-0.5)} = \sqrt{2F^2(0.5)} = \sqrt{F^2} = F \); this exactly reproduces the given condition that the resultant equals either individual force in magnitude.

Only one of the four candidate angles, when substituted into the resultant formula, produces a resultant magnitude exactly equal to the magnitude of either original force.

So the correct answer is \( 120^\circ \).