Continuous Rigid Body

It was quite easy to calculate the location of center of mass in multi-particle systems. But that can be quite useless because in real life and the physical world we live in, almost nothing is of particle systems. With powerful tools of integral calculus, center of mass of continuous rigid bodies with sufficient symmetry can be found. Or, for highly symmetric objects such as pool balls, the center of mass can be found utilizing simple geometric intuition.

When using the general method of calculating the center of mass of an object, it is assumed that a continuous object is a composition of many particles held by massless entities. Thus a solid object is a sum of many miniscule particles and integration is utilized for this. Let's first learn the general method of calculating the center of mass using integration on highly symmetric object then we can check the answer through geometry.

Center of Mass of a rod:

Before we start any calculation, we must consider the density of the object. Through the density, we can determine a relationship between the amount of substance and its location.

l = M/L
M = lL
Now we take a small particle of the total mass M,
dm = ldx
x dm = x ldx

Mxcm = lL2/2
Mxcm = ML/2
xcm = L/2

This makes sense because a rod is highly symmetric and the center would be in the half.
And for pool balls, which are also highly symmetric, the center of mass would exist in its center.

Sample Problem
Find the center of mass of a semi-circular rod of radius R and mass M