Question

State the principle of equilibrium of concurrent forces. Derive it.

Answer 1

"If a number of forces act at the same point, they are called concurrent forces".

Consider that a body is under the action of a number of forces. Suppose that the body is in equilibrium under the action of these forces, i.e., the body remains in its state of rest or of uniform motion along a straight line, when acted upon these forces.

"The condition that the body may be in equilibrium or the number of forces acting on the body may be in equilibrium is that these forces should produce zero resultant force."

Consider three concurrent forces vector(F₁, F₂ and F₃) acting at the point 'O' (see fig). In order top find the resultant of vector(F₁, F₂ and F₃), we first find the resultant of vector(F₁ and F₂) by completing the parallelogram OAC'B. The diagonal OC' of the parallelogram OAC'B gives vector(F₁ + F₂) i.e., the resultant of vector(F₁ and F₂). If vectorF₃ (represented by OC) is equal and opposite to vector(F₁ + F₂) (represented by vectorOC) then resultant of force vector(F₁, F₂ and F₃) acting at point O

will be zero. Therefore, for vector(F₁, F₂ and F₃) to be in equilibrium, vector(OC = OC)

or, vector(F₃ = -(F₁ + F₂))

or, vector(F₁ + F₂ + F₃) = 0    ...(i)

The condition for equilibrium of three concurrent forces triangle law of addition of vectors. Suppose that the forces vector(F₁, F₂ and F₃) are such that they can be represented by the three sides vector(AB, BC and CA) of the triangle ABC taken in same order, i.e., vector(AB = F₁, BC = F₂ and CA = F₃)

According to the triangle law of addition of vectors

vector(AB + BC = AC or, AB + BC - AC) = 0    ...(ii)

Now the vectors vector(AC and CA) are equal in magnitude but opposite in direction.

or, vector(AC = -CA)

Hence, the equation (ii) becomes,

vector(AB + BC = AC = 0 or, F₁ + F₂ + F₃) = 0

Therefore, the resultant of three concurrent forces will be zero and hence they will be in equilibrium, if they can be represented completely by the three sides of a triangle taken in same order.

In general, the concurrent forces, vector(F₁, F₂, F₃ .... F_n) will be in equilibrium, if their resultant is zero, i.e.,

vector(F₁ + F₂ + F₃ + ....+ F_n) = 0    ...(iii)

In case, a number of forces act at a point, then they will be in equilibrium, if they can be represented completely by the sides of a closed polygon taken in order.