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The two figures below illustrate the geometry,
elements of volume, area, and length as
well as the position vectors involved in
calculating the center of gravity (cg), also called centroid,
for volumes, areas, and curved
lines.
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Here cg denotes the center of gravity (centroid), rbar the position vector from the origin to
the centroid, and
r the position vector to dA,
dV, dL ˗ the
elements of volume, area, and
arc length respectively.
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The principle of first moments applies to
three-dimensional cases such as volumes (as illustrated below) as well as those
involving two-dimensional applications (areas), and one-dimensional applications
(straight lines) such as the yard stick example previously mentioned.
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V rbar
= ʃ r
dV
i.e. moment of sum =
sum of moments
For the x-component only: V xCG = ʃ
x dV
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Click here for a discussion of centroids
involving composite bodies.
Click here for examples.
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