o
If we increase the number of elements n, the element size Dmi decreases, and coordinates of COM is given by:
Where x, y, z = coordinates of COM of small element dm
o
The vector expression equivalent to these three scalar expressions is Where o o
R = position of COM of body r = position of COM of small element of mass dm
Consider a thin rod of length l, taking the origin to be at the geometric centre of the rod and x-axis to be along the length of the rod, we can say that on account of reflection symmetry, for every element dm of the rod at x, there is an element of the same mass dm located at – x.
The net contribution of every such pair to the integral and hence the integral x dm itself is zero. Thus the COM coincides with the geometric centre.