Lesson 6 – Centre of Mass & System of Particles
Stage 2 : Mathematical Definition of Centre of Mass
(Intermediate Physics – Page 5)

In this page, we extend the mathematical concept of centre of mass to systems of three particles and to motion in the x–y plane. These ideas are frequently tested in Intermediate examinations.

Centre of Mass of a Three-Particle System (1D)

Consider three particles of masses m₁, m₂ and m₃ placed at positions x₁, x₂ and x₃ along the x-axis.

The position of the centre of mass is given by:

x_cm = ( m₁x₁ + m₂x₂ + m₃x₃ ) / ( m₁ + m₂ + m₃ )

This expression can be extended to any number of particles by adding more terms in the numerator and denominator.

Simple Numerical Example (Three Particles)

Example:
Three particles of masses 1 kg, 2 kg and 3 kg are placed at x = 0 m, x = 2 m and x = 4 m respectively. Find the position of the centre of mass.

Solution:

x_cm = (1×0 + 2×2 + 3×4) / (1 + 2 + 3)

x_cm = (0 + 4 + 12) / 6 = 16 / 6 ≈ 2.67 m

Thus, the centre of mass lies at 2.67 m from the origin.

Centre of Mass in Two Dimensions (x–y Plane)

When particles are located in a plane, the centre of mass has both x and y coordinates.

If a particle of mass m_i is located at (x_i, y_i), then the coordinates of the centre of mass are:

x_cm = Σ(m_ix_i) / Σm_i
y_cm = Σ(m_iy_i) / Σm_i

The x and y coordinates are calculated independently.

Numerical Example (2D Centre of Mass)

Example:
Two particles of masses 2 kg and 4 kg are located at points (0, 0) and (4, 2) respectively. Find the coordinates of the centre of mass.

Solution:

x_cm = (2×0 + 4×4) / (2 + 4) = 16 / 6 ≈ 2.67

y_cm = (2×0 + 4×2) / (2 + 4) = 8 / 6 ≈ 1.33

Therefore, the centre of mass is at (2.67, 1.33).

Important Observations (Exam-Oriented)

• COM coordinates are mass-weighted averages
• x and y directions are treated separately
• Choice of origin affects calculation but not physical position
• Heavier mass has greater influence on COM position

Common Mistakes to Avoid

• Forgetting to divide by total mass
• Mixing x and y coordinates
• Using wrong sign convention
• Skipping units in final answer

Stage 2 – Page 5 Recap

✔ COM formula extends naturally to many particles
✔ Three-particle and 2D problems are board favorites
✔ Separate x and y calculations simplify problems
✔ Clear steps ensure full marks

What Comes Next

In the next page, we will study the centre of mass of continuous mass distributions such as rods and laminae, which completes the Intermediate-level mathematical treatment.

📘 Centre of Mass & System of Particles – Complete Physics Library

This library is a complete learning package for Intermediate + IIT JEE Physics covering Centre of Mass and System of Particles.

It includes theory, solved examples, objective questions, IIT-level problems, revision strategies, tricks, and exam-oriented guidance.

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1. Part 1
2. Part 2
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10. Part 10

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