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: For two equal forces \( F \) at angle \( \theta \), the resultant magnitude is \( R = \sqrt{2F^2(1+\cos\theta)} \).
Step 2: Setting \( R = F \) as given, square both sides: \( F^2 = 2F^2(1+\cos\theta) \), which simplifies to \( 1 = 2(1+\cos\theta) \).
Step 3: Solving, \( \cos\theta = -\dfrac{1}{2} \), which corresponds to an angle in the second quadrant. \[ \boxed{\theta = 120^\circ} \]

Answer 2

An alternative, purely geometric route is to notice that three vectors of the same magnitude \( F \) (the two original forces and their resultant) form an equilateral triangle when added tip-to-tail, and to check each angle option against that geometric picture.

1. Option \( 30^\circ \): If the angle between the two original forces were only \( 30^\circ \), the triangle formed by the two forces and the resultant (using the parallelogram law) would be a narrow, elongated shape with the resultant side much longer than the other two, not matching a resultant equal in length to the original forces.
2. Option \( 60^\circ \): Interestingly, if the angle between the two forces were \( 60^\circ \), the parallelogram's diagonal (the resultant) would form an equilateral-triangle relationship with each individual force, giving a resultant of magnitude \( F\sqrt{3} \), not \( F \); this angle actually corresponds to the case where each force equals the OTHER diagonal, not the case asked here.
3. Option \( 90^\circ \): At a right angle between the two forces, the resultant forms the hypotenuse of a right triangle with two equal legs \( F \), giving \( R = F\sqrt{2} \), a right isosceles triangle relationship, not the equal-sided case needed.
4. Option \( 120^\circ \): When the angle between the two forces is \( 120^\circ \), the triangle formed by the two force vectors and the resultant (via the parallelogram/triangle rule) becomes equilateral, meaning all three sides, including the resultant, share the same length \( F \); this geometric configuration is exactly the one described in the question.

Visualising the vector triangle formed by the two forces and their resultant shows that only one specific angle produces a resultant of equal length to the two original forces, forming an equilateral triangle.

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