how to derive the formula for centre of mass?

(m1r1+m2r2)/(m1+m2)

 

If we try to balance the system on a pivot at a point x_p in between x₁ and x₂ , both particles exert torques that tend to tip the beam, rotating it around the pivot. In order to balance the system on the pivot, we want the torques caused by each of the particles to cancel each other. Since the force exerted by each particle is given by its mass times the acceleration of gravity, x_p must be chosen so that

m₁ g (x₁ - x_p ) = m₂ g (x₂ - x_p )

Rearranging this equation gives

x_p (m₁ + m₂ ) = (x₁ m₁ + x₂ m₂ )

This line of reasoning easily generalizes. If a rigid body is composed of n particles connected in a straight line, then the location of the center of mass for the system is xcm where

(m₁ + m₂ + . . . + m_n)x_cm = m₁ x₁ + . . . + m_n x_n

 

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