Centre of Mass of Rigid Body

Two Particle System

Suppose that two particles are at heights z₁ and z₂ from a horizontal surface. Suppose further that the gravitational force is uniform in the small region in which the two particles move about. The force on each particle will be mg. The total force acting on the system is 2mg.

There is a point C somewhere in the system so that if a force 2mg acts at that point located at a height z from the horizontal surface, the motion of the system would be the same as with two forces. The potential energies of particles 1 and 2 are mgz₁ and mgz₂, respectively. The potential energy of the particle at C is 2mgz. Since this must be equal to the combined potential energy of the two particles,

2mgz = mgz₁ + mgz₂

The point C lies midway between the two particles. If the two masses were unequal, this point would not have been in the middle. If the mass of particle 1 is m₁ and that of particle 2 is m₂, then

(m₁ + m₂)gz = m₁gz₁ + m₂gz₂

The point C is called the centre of mass (CM) of the system. It is just a mathematical tool and there is no physical point as CM.