
The 1st mass moment of a particle is defined as the product of its mass and position vector (Figure 1):
where q = the first mass moment, m = the mass, and r = the position vector. The total mass moment of the system of particles shown in Figure 2 is then
where i = point number. Center of mass is an imaginary point which suffices the following relationship:
where m_{total} = the mass of the system, r_{CM} = the position vector of the CM. In other words, the CM is a single point which effectively represents the whole system. From [3]:
Based on Figure 3, [2] can be rewritten as
The second term of [5] is the total 1st mass moment of the system with respect to its CM. From [3] and [5]:
In other words, the sum of the 1st mass moments of the particles with respect to their CM always becomes zero. 
© YoungHoo Kwon, 1998 