Centroids and moments
The purpose of this note is to provide some physical motivation for the standard
formulas which give the center of mass of an object in the plane. Suppose A is a
bounded planar object whose center of mass (or centroid) has coordinates
(x*, y*),
and choose positive numbers
a
and
b
such that all points on A belong to the solid
rectangular region B defined by the following inequalities:
x* – a
≤
x
≤
x* + a
y* – b
≤
y
≤
y* + b
Think of B as a flat, firm rectangular sheet with uniform density (made of glass, metal,
wood, plastic,
etc
.) such that A rests on top of B.
Next, suppose that we have a triangular rod C with equilateral ends, positioned so that
one of the lateral faces is horizontal and the opposite edge E lies above this face.
As suggested by the figure on the next page, suppose that we now we rest B and A on
the edge E of C along the vertical line defined by the equation
x = x*
.
