📌 Quick Answer
The parallelogram law of forces states that if two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is given by the diagonal passing through that point.
The resultant magnitude is R = √(P² + Q² + 2PQ·cosθ), where θ is the angle between the forces.
🔹 Key Takeaways
- Two forces at a point = a single resultant along the diagonal of the parallelogram.
- Resultant: R = √(P² + Q² + 2PQ cosθ).
- Direction: tanα = (Q sinθ) / (P + Q cosθ).
- It is a fundamental law for adding vectors (forces, velocities).
Statement of the Parallelogram Law of Forces
The parallelogram law of forces states that if two forces acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.
Formula for the Resultant
R = √(P² + Q² + 2PQ·cosθ)
tanα = (Q·sinθ) / (P + Q·cosθ)
Here P and Q are the two forces, θ is the angle between them, R is the resultant and α is the angle the resultant makes with force P.
Derivation
Let forces P and Q act at point O with angle θ between them, forming parallelogram OACB. Drop a perpendicular from C to the extension of OA at point D. Using right triangle geometry, CD = Q·sinθ and AD = Q·cosθ. Then in right triangle OCD: R² = OD² + CD² = (P + Q·cosθ)² + (Q·sinθ)². Expanding and using sin²θ + cos²θ = 1 gives R² = P² + Q² + 2PQ·cosθ.
Special Cases
| Angle θ | Resultant R |
|---|---|
| 0° (same direction) | P + Q (maximum) |
| 90° (perpendicular) | √(P² + Q²) |
| 180° (opposite) | P − Q (minimum) |
Frequently Asked Questions
What is the parallelogram law of forces?
It states that if two forces acting at a point are represented by the adjacent sides of a parallelogram, their resultant is represented by the diagonal through that point.
What is the formula for the resultant of two forces?
R = √(P² + Q² + 2PQ·cosθ), where P and Q are the forces and θ is the angle between them.
When is the resultant maximum and minimum?
The resultant is maximum (P + Q) when the forces act in the same direction (θ = 0°) and minimum (P − Q) when they are opposite (θ = 180°).
What is the parallelogram law used for?
It is used to add two vectors — such as forces or velocities — acting at a point to find a single resultant.