The Law of Parallelogram of Forces, defined in physics, states that the resultant of any two non-collinear concurrent forces may be found by this law, which states that if two forces acting simultaneously on a body at a point are represented in magnitude and direction by two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through the point of intersection of the two sides representing the forces.

Parallelogram Law of Forces Definition

The specified relationship for the angle α made by R with P is given by the equation Q sinθ / P + Q cosθ.

The magnitude of the resultant of the two concurrent forces R is given by the equation R² = P² + Q² + 2PQ cosθ.

Parallelogram Law of Forces Triangle Law Solution

The Triangle Law of Force, which is a corollary of the Parallelogram Law, states that if two forces are represented by their force vector placed tip to tail, their resultant is the vector directed from the tail of the first vector to the tip of the second vector. This can be solved using the cosine rule, which is expressed as 180-θ.

Parallelogram Law of Forces Examples

Sample Problem No. 01

Find the resultant of the given forces and the angle it makes with the horizontal using parallelogram law and triangle law.

a. Solution using Parallelogram Law
R = √(8² + 6² + 2(8)(6) cos 90)
R = 10 N
tan α = 6 / 8
α = 36.87⁰

b. Using Triangle Law
By Cosine Rule
R = √(8² + 6² - 2(8)(6) cos 90)
R = 10 N
sin α = 6 / 10
α = 36.87⁰

Sample Problem No. 02

Find the resultant of the given forces and the angle it makes with the horizontal using parallelogram law and triangle law.
a. Solution using Parallelogram Law
R = √(80² + 100² + 2(80)(100) cos 60)
R = 156.20 N
tan α = 100 sin 60 / 80 + 100 cos 60
α = 33.67⁰

b. Using Triangle Law
By Cosine Rule
R = √(80² + 100² - 2(80)(100) cos 90)
R = 156.20 N
sin α = 100 sin 120 / 156.20
α = 33.67⁰