The magnitiude of resultant of two equals force is equal to either to either of the force . the angle b/w two force will be .?

The scenario you're describing involves two equal forces acting at an angle to each other, and you're interested in finding out what that angle is when the magnitude of the resultant force equals the magnitude of either of the individual forces. This is a classic problem in physics, particularly in the study of vector addition.

Understanding the Forces

Let's denote the magnitude of each of the two equal forces as \( F \). When two forces are equal in magnitude and act at an angle \( \theta \) to each other, the resultant force \( R \) can be calculated using the formula derived from the law of cosines:

Resultant Force Formula

The formula for the magnitude of the resultant \( R \) is given by:

R = √(F² + F² + 2F²cos(θ))

Since both forces are equal, we can simplify this to:

R = √(2F²(1 + cos(θ)))

Now, we want to find the angle \( \theta \) such that the resultant \( R \) is equal to \( F \). Therefore, we set:

F = √(2F²(1 + cos(θ)))

Solving for the Angle

To solve for \( \theta \), we first square both sides of the equation:

F² = 2F²(1 + cos(θ))

Next, we can divide both sides by \( F² \) (assuming \( F \neq 0 \)):

1 = 2(1 + cos(θ))

Now, simplifying gives us:

1 = 2 + 2cos(θ)

Rearranging this leads to:

2cos(θ) = 1 - 2

Which simplifies to:

2cos(θ) = -1

Thus, we find:

cos(θ) = -1/2

Finding the Angle

The cosine of \( \theta \) being equal to \(-1/2\) corresponds to specific angles in the unit circle. The angles that satisfy this condition are:

• θ = 120°
• θ = 240°

However, in the context of forces, we typically consider the angle between the two forces to be less than 180°, so the relevant angle here is:

θ = 120°

Visualizing the Forces

To visualize this, imagine two arrows (representing the forces) starting from the same point and forming an angle of 120° between them. The resultant arrow, which represents the combined effect of these two forces, will have the same length as either of the individual arrows, confirming our calculation.

In summary, when two equal forces act at an angle of 120° to each other, the magnitude of their resultant is equal to the magnitude of either force. This principle is fundamental in understanding vector addition in physics.