Question

Three concurrent forces of the same magnitude are in equilibrium. What is the angle between the force? Also name the triangle formed by the force as sides
(1) $ {60^ \circ } $ equilateral triangle
(2) $ {120^ \circ } $ equilateral triangle
(3) $ {120^ \circ },{30^ \circ },{30^ \circ } $ isosceles triangle
(4) $ {120^ \circ } $ an obtuse angled triangle

Answer 1

Hint: This question will be solved by using Lami’s theorem which states that when 3 forces related to the vector magnitude acting at the point of equilibrium, each force of the system is always proportional to the sine of the angle that lies between the other two forces.

Complete answer:
Lami's Theorem is related to the magnitudes of concurrent, coplanar, and non-collinear forces that maintain an object in static equilibrium. The Theorem is so useful to analyze most of the mechanical and structural systems as well. The proportionality constant is similar for all the given three forces.
Also, if the size and direction of the forces acting on an object are exactly balanced, then there is no net force acting on the object and the object is said to be in equilibrium.
The concurrent forces always pass through a common point.
According to Lami's theorem ,
 $ \dfrac{C}{{\operatorname{Sin} a}} = \dfrac{B}{{\operatorname{Sin} b}} = \dfrac{A}{{\operatorname{Sin} c}} $
Since, it is given that the magnitude of the forces are same, so,
 $ A = B = C $
And from this, we can say that $ a = b = c $
Also, $ a + b + c = {360^ \circ } $
So, $ a = b = c = \dfrac{{360}}{3} = {120^ \circ } $
As all forces have the same magnitude and same angle between them so it will make an equilateral triangle.
Thus, the final answer is (2) $ {120^ \circ } $ equilateral triangle.

Note:
It must be noted that there should be only three forces acting on the object. If there are more than three forces or less than three forces, we may not apply the Lami's theorem. Further, Lami's theorem is applicable to only coplanar, concurrent and non-collinear forces.