Law of Parallelogram of Forces
Law of Parallelogram of Forces states that if two forces acting simultaneously on a body at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, 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.
Formula for Magnitude of Resultant Force R
Where R is the resultant force and θ is the angle between P and Q.
Formula for Direction of Resultant Force R
Cases for Parallelogram Law of Vector Addition
1. When two vectors are acting in the same direction, then θ = 0° and cosθ = 1
When two vectors are parallel and have the same direction, the law simplifies to simple addition. The resultant vector will have the same direction as the original vectors and a magnitude equal to the sum of their magnitudes.
R2 = P2 + Q2 + 2PQ [As cosθ = 1]
or, R2 = (P + Q)2
or, R = P + Q
Thus, when two vectors are in the same direction, the resultant vector is their algebraic sum.
2. When two vectors are acting in opposite directions, then θ = 180° and cosθ = –1
When two vectors are in opposite directions, the magnitude of the resultant vector equals the difference between their individual magnitudes.
R2 = P2 + Q2 – 2PQ [As cosθ = –1]
or, R2 = (P – Q)2
or, R = P – Q
3. When two vectors act at right angles to each other, θ = 90° and cosθ = 0
When two vectors are perpendicular to each other, by using the Pythagorean theorem, the magnitude of the resultant vector is equal to the square root of the sum of the squares of the magnitudes of the two vectors.
R2 = P2 + Q2 [As cosθ = 0]
We have seen in the above cases that the magnitude of the resultant of two vectors is maximum when the vectors act in the same direction, and is minimum when they act in opposite directions.
Limitations of Parallelogram Law of Forces
The parallelogram law of forces can only be used when the two forces can form adjacent sides of a distinct parallelogram. If the two forces are parallel to each other or opposite to each other (i.e., the angle between them is 0° or 180°), a visual geometric parallelogram cannot form, and the forces must simply be compiled algebraically.
Parallelogram law of forces states that if two forces acting simultaneously at a point be represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant may be represented in magnitude and direction by:
A. longer side of the other two sides
B. shorter side of the other two sides
C. diagonal of the parallelogram which does not pass through their point of intersection
D. diagonal for the parallelogram which passes through their point of intersection
Show Answer
Option D is the correct answer.
The diagonal for the parallelogram which passes through their point of intersection. See the statement of the Law of Parallelogram of Forces.