Find the centroid of a polygon with weighted vertices

Question

I know how to find the centroid (center of mass) of a regular polygon. This assumes that every part of the polygon weighs the same. But how do I calculate the centroid of a weightless polygon (made from aerogel perhaps :), where each vertex has a weight?

Simplified illustration of what I mean using straight line:

5kg-----------------5kg
           ^center of gravity

10kg---------------5kg
        ^center of gravity offset du to weight of vertices

Of course, I know how to calculate the center of gravity on a straight line with weighted vertices, but how do I do it on a polygon with weighted vertices?

Thanks for your time!

Answer 1

You want take a weighted average over all the vertices. So say your vertices are v1, v2, v3 .... vn with masses m1, m2 ...mn and have x and y coordinates v1x, v1y, v2x, v2y etc then to get the center of mass (cx, cy) you want:

cx = (v1x*m1 + v2x*m2 + ... vnx*mn) / (m1 + m2 .... mn) 
cy = (v1y*m1 + v2y*m2 + ... vny*mn) / (m1 + m2 .... mn)

It's essentially the same principle as when you do it for a line.

Answer 2

1) Generate a vector for each vertex

2) Multiply each vector for the weight of the vertex

3) Sum the vectors

4) Divide for total mass

5) There's your center of mass!