Analytical Method for Finding Resultant Force - Parallelogram Law of Forces

The resultant force of a given system of forces may be found out analytically by Parallelogram law of forces.

Parallelogram Law of Forces

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

Let forces ‘P’ and ‘Q’ acting at a point O be represented in magnitude and direction by OA and OB respectively as shown in Fig.1. Then, according to the theorem of parallelogram of forces, the diagonal OC drawn through O represents the resultant of P and Q in magnitude and direction.

Fig.1

Determination of the Resultant of Two Concurrent Forces with the Help of Law of Parallelogram of Forces

Consider two forces P and Q acting at and away from point A as shown in Fig. 2. Let the forces P and Q are represented by the two adjacent sides of a parallelogram AD and AB respectively as shown in Fig. 2. Let ‘θ’ be the angle between the force P and Q and ‘α’ be the angle between R and P. Extend line AB and drop perpendicular from point C on the extended line AB to meet at point E.

Fig. 2

Consider Right angle triangle ACE,

           AC² = AE² + CE2

                   = (AB + BE)² + CE2

                   = AB² + BE² + 2.AB.BE + CE2

                   = AB² + BE² + CE² + 2.AB.BE …………………….. (1)

Consider right angle triangle BCE,

            BC² = BE² + CE²    and     BE = BC. Cos θ

Putting BC² = BE² + CE² in equation (1), we get

            AC² = AB² + BC² + 2.AB.BE      ……………………….. (2)

Putting BE = BC. Cos θ in equation (2)

            AC² = AB² + BC² + 2.AB. BC. Cos θ

But,   AB = P, BC = Q and AC = R

               R = √ (P2 + Q2 + 2PQ Cos θ)

In triangle ACE

      

Now let us consider two forces F1 and F2 are represented by the two adjacent sides of a parallelogram

i.e. F1 and F2 = Forces whose resultant is required to be found out,

θ = Angle between the forces F1 and F2

α = Angle which the resultant force makes with one of the forces (say F1).

Then resultant force

            R= √ (F1² + F2² + 2F1 F2 Cos θ)

                  

If ‘α’ is the angle which the resultant force makes with the other force F2, then

                 

Cases

1) If θ = 0⁰ i.e., when the forces act along the same line, then

                                R_max = F1 + F2

2) If θ = 90⁰ i.e., when the forces act at right angle, then

                              R= √ (F1² + F2²)

3) If θ = 180⁰ i.e., when the forces act along the same straight line but in opposite directions, then

                               R_min= F1 – F2

In this case, the resultant force will act in the direction of the greater force.

4) If the two forces are equal i.e., when F1 = F2 = F then