Center of mass for uniform distribution of mass
If the mass is uniformly distributed in a bulk object, then a small mass (∆m) of the body can be treated as a point mass and the summations can be done to obtain the expressions for the coordinates of center of mass.
On the other hand, if the small mass taken is infinitesimally* small (dm) then, the summations can be replaced by integrations as given below.
Locate the center of mass of a uniform rod of mass M and length l.
Consider a uniform rod of mass M and length whose one end coincides with the origin as shown in Figure. The rod is kept along the x axis. To find the center of mass
of this rod, we choose an infinitesimally small mass dm of elemental length dx at a distance x from the origin.
Now, we can write the center of mass equation for this mass distribution as,
As the position l/2 is the geometric center of the rod, it is concluded that the center of mass of the uniform rod is located at its geometric center itself.