The Intuition:

• a point where body is balanced ( if we apply force at that point ( given it is on the body since Center of Mass can even lie outside the body ), it performs only translation and no rotation )

• effectively the point where gravity can be assumed to act upon ( strictly speaking center of mass and center of gravity are different concepts, but if dimensions of body are small enough to assume constant gravity, we can assume both points as equivalent )

•  some more complex meaning in n-particle systems and gravitational fields

The concept of Mass Moment:

• Mass moment is a vector quantity of a point mass about a reference origin ( any coordinate point ) with magnitude as m*r where m is the mass of point object and r is the distance of the point object from the reference origin. The direction of the mass moment is the direction of vector r from refence origin to that point mass.

• The interesting part is that cumulative mass moment of a system of particles ( point masses, again also a composite body of non-negligible dimensions can be considered as a system of infinite dimensionless particles ) about its centre of mass is 0

• This property of mass moment about centre of mass helps us derive the mathematical formulation to calculate centre of mass of a system

• To put it in calculating centre of mass of a body about its centre of mass itself as reference origin gives 0 i.e., the reference origin itself is the centre of mass

• This also plays a part when we derive parallel axis theorem for moment of inertia in rotational dynamics

Derivation: