Lesson 6 – Centre of Mass & System of Particles
Stage 2 : Mathematical Definition of Centre of Mass
(Intermediate Physics – Page 4)

In Stage 1, we understood the physical meaning of centre of mass. In this page, we develop the mathematical definition of centre of mass, which is required to solve numerical problems in Intermediate examinations.

Centre of Mass of a Two-Particle System

Consider two particles of masses m₁ and m₂ placed at positions x₁ and x₂ along the x-axis.

Let x_cm be the position of the centre of mass.

Mathematical Expression

The position of the centre of mass is given by:

x_cm = ( m₁x₁ + m₂x₂ ) / ( m₁ + m₂ )

This formula shows that the centre of mass is a mass-weighted average of positions.

Important Observations

• If m₁ = m₂, then x_cm is the midpoint
• If one mass is larger, x_cm shifts towards that mass
• Centre of mass always lies between the two particles

Simple Numerical Example

Example:
Two particles of masses 2 kg and 3 kg are placed at x = 0 m and x = 5 m respectively. Find the position of the centre of mass.

Solution:

x_cm = (2 × 0 + 3 × 5) / (2 + 3)

x_cm = 15 / 5 = 3 m

Therefore, the centre of mass lies 3 m from the origin, closer to the heavier mass.

Centre of Mass Along a Straight Line

When all particles lie along a straight line, the centre of mass can be calculated using one-dimensional coordinates.

This type of problem is very common in Intermediate examinations.

Intermediate Exam Tips

• Always choose a clear origin
• Use sign convention properly
• Write formula before substitution
• Units must be written in final answer

Key Points to Remember

✔ Centre of mass is a mass-weighted average position
✔ Mathematical formula is essential for numericals
✔ Heavier mass pulls COM closer to itself
✔ Correct sign convention avoids mistakes