Question

The square of resultant of two equal forces is three times their product. The angle between the forces is?

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

Hint:

When dealing with two forces of equal magnitude, use the law of cosines to calculate the resultant force and solve for the angle between the forces.

Answer 1

The Correct Option is B

Let the two equal forces be \( F \) and \( F \), and the angle between them be \( \theta \). The resultant force \( R \) of two forces acting at an angle is given by the law of cosines: \[ R^2 = F^2 + F^2 + 2F \cdot F \cdot \cos(\theta) \] Simplifying: \[ R^2 = 2F^2 (1 + \cos(\theta)) \] We are given that the square of the resultant force is three times the product of the two forces, so: \[ R^2 = 3F^2 \] Equating the two expressions for \( R^2 \): \[ 2F^2 (1 + \cos(\theta)) = 3F^2 \] Cancelling \( F^2 \) from both sides: \[ 2(1 + \cos(\theta)) = 3 \] Simplifying: \[ 1 + \cos(\theta) = \frac{3}{2} \] \[ \cos(\theta) = \frac{1}{2} \] Therefore, the angle \( \theta \) is: \[ \theta = 60^\circ \] Thus, the angle between the forces is \( 60^\circ \). So the correct answer is (B).