📐 Parallelogram Law of Vectors

Verify Vector Addition using Force Table and Parallelogram Method

✅ FREE Experiment • 🎨 Interactive Diagram • 🎓 NEB Class 11 Practical

Interactive Simulation

Adjust forces to see parallelogram and resultant vector

Experiment Controls

Range: 1 - 10 Newtons

Range: 1 - 10 Newtons

Angle between F₁ and F₂

Vector Components
Force F₁: 5.0 N
Force F₂: 5.0 N
Angle θ: 60°
Resultant R: 0.0 N
Direction α: 0°

Calculated Results

Force F₁
5.0
N
Force F₂
5.0
N
Angle θ
60
degrees
Resultant R
0.0
N
Direction α
0.0
degrees

📚 Theory & Concepts

What is Parallelogram Law of Vectors?

The Parallelogram Law states that if two vectors acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

Resultant Magnitude Formula

R = √(F₁² + F₂² + 2F₁F₂cosθ)

Where:
• R = Magnitude of resultant vector
• F₁ = Magnitude of first force
• F₂ = Magnitude of second force
• θ = Angle between F₁ and F₂

Direction Formula

tan(α) = (F₂ sin θ) / (F₁ + F₂ cos θ)

Where α is the angle the resultant makes with F₁.

Special Cases

Properties of Vector Addition

1. Commutative: F₁ + F₂ = F₂ + F₁
2. Associative: (F₁ + F₂) + F₃ = F₁ + (F₂ + F₃)
3. Zero Vector: F + 0 = F
4. Negative Vector: F + (-F) = 0

🔬 Procedure

  1. Set up the force table on a horizontal surface
  2. Attach two spring balances at fixed angles
  3. Apply known forces F₁ and F₂ at angle θ
  4. Measure the angle θ accurately using protractor
  5. Note down F₁, F₂, and θ values
  6. Calculate resultant using R = √(F₁² + F₂² + 2F₁F₂cosθ)
  7. Calculate direction using tan(α) = (F₂sinθ)/(F₁+F₂cosθ)
  8. Verify by finding equilibrant force experimentally
  9. Equilibrant should be equal and opposite to resultant
  10. Draw parallelogram diagram to scale
  11. Measure diagonal length to verify calculated resultant
  12. Repeat with different force magnitudes and angles

💬 Viva Questions & Answers

Q1: What is a vector quantity?
A vector quantity is a physical quantity that has both magnitude and direction. Examples include force, velocity, acceleration, displacement, and momentum.
Q2: State the Parallelogram Law of Vectors.
If two vectors acting simultaneously at a point are represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through that point.
Q3: What is the formula for resultant?
R = √(F₁² + F₂² + 2F₁F₂cosθ), where F₁ and F₂ are the magnitudes of two forces and θ is the angle between them.
Q4: What is meant by equilibrant?
The equilibrant is a single force that is equal in magnitude but opposite in direction to the resultant. It brings the system into equilibrium.
Q5: When is resultant maximum and minimum?
Resultant is maximum (F₁ + F₂) when θ = 0° (vectors in same direction) and minimum (|F₁ - F₂|) when θ = 180° (vectors in opposite directions).
Q6: What if two equal forces act at 90°?
If F₁ = F₂ = F and θ = 90°, then R = √(F² + F²) = F√2. The resultant makes 45° angle with each force.
Q7: Difference between scalar and vector?
Scalars have only magnitude (mass, time, speed, energy). Vectors have both magnitude and direction (force, velocity, acceleration, displacement).
Q8: What is Triangle Law of Vectors?
If two vectors are represented by two sides of a triangle taken in order, their resultant is represented by the third side taken in opposite order. It's equivalent to parallelogram law.
Q9: Can vectors be added like scalars?
No, vectors cannot be added arithmetically. They must be added using vector addition rules (parallelogram, triangle, or polygon law) considering their directions.
Q10: What are applications of vector addition?
Applications include: finding net force on objects, determining velocity of boat in river, analyzing forces in structures, projectile motion, navigation, and electromagnetic field analysis.