We can find the centre of mass of a system of particles by taking moments about a point for each particle, treating the position of each particle as a position vector, and summing the results, then equating this sum to the position vector of the whole system of particles. For example, consider the system of particles below, with each mass at the corner of a rectangle.
Defining the origin,
to be at the position of the mass
the position vectors of the remaing masses are:
Taking moments about O gives
The mass of the system of particles is
If the entire system has centre of mass at position vector
then taking moments for the entire system and equating to the sum of moments for the individual particles gives
from which we have