Two concurrent coplanar forces of the same nature are acting at a point at an angle θ between them. What should be the value of angle θ, so that their resultant force has the minimum value?

Concept:

Let F₁ and F₂ two concurrent coplanar forces acting at an angle θ to each other shown in the figure below.

“If two forces, acting simultaneously on a particle, be represented in magnitude and direction by the two adjacent sides of a parallelogram; their resultant may be represented in magnitude and direction by the diagonal of the parallelogram, which passes through their point of intersection.” Mathematically, resultant force,

Squaring both side of equation (1)

For the minimum and maximum value of resultant;

Differentiate the above equation (2) w.r.t θ

0 = -2F₁F₂sinθ

2F1F2sinθ = 0 ................ (3)

2F₁F₂ ≠ 0

sinθ = 0

θ = 0^∘ or 180^∘ 

If we consider θ = 0^∘ then two forces become a single force, So for the minimum value of the resultant, θ is equal to 180^∘. Hence option (4) is correct.