The sum of the moments of masses of a system of particles about the center of mass is indeed zero. This property is known as the principle of conservation of angular momentum.
The center of mass of a system of particles is the point that behaves as if all the system's mass were concentrated there. When considering the moments of masses about the center of mass, we are essentially calculating the individual masses' contributions to the system's overall angular momentum.
The principle of conservation of angular momentum states that the total angular momentum of an isolated system remains constant if no external torques act on it. In the case of a system of particles, the total angular momentum is the sum of the moments of masses of the particles about the center of mass.
To understand why the sum of the moments of masses about the center of mass is zero, we can consider the definition of the center of mass. By definition, the center of mass is the point where the sum of the moments of masses is zero.
If we choose the center of mass as the reference point, each mass in the system will contribute a moment that is equal in magnitude but opposite in direction to another mass in the system. The moments will cancel each other out, resulting in a net sum of zero.
This property holds true regardless of the distribution of masses within the system. As long as no external torques act on the system, the sum of the moments of masses about the center of mass will remain zero, preserving the conservation of angular momentum.