For a continuous system of matter Equation (1) becomes
In Cartesian components :
The computations for C.M. are usually tedious, but when the distribution of matter is symmetric about
any axis, then the C.M. must lie on that axis; If more than one intersecting axes of symmetries exist than
the point of intersection must be the centre of mass. This is obvious consequence of C.M. lying on the axis
Thus the C.M. of ring, disc, rod, cylinder, sphere etc. are their respective geometric centers.
Equivalence of system to C.M. in case of translational motion
For Kth particle in the system,