Why Area Can "Stand in" for Mass

We know that the center of mass of a system of n discrete objects is:

(1)

but suppose that we have divided some "homogeneous thin laminar plate" into sections and determined the center of mass of each piece. Now, mass density, , is given by:

So:

Substituting this into equation (1), and realizing the the density, , and thickness, t, of each piece are the same gives:

so, if the continuous object is homogeneous and has a constant thickness, areas can "stand in" for masses when calculating centers of mass.

last update February 1, 2010 by JL Stanbrough