|
|
Key
Concepts: The
centroid (or center of mass) of a body is the
idealized location where
all its mass can be thought to be
concentrated. To locate the centroid use the principle
of first moments: (summarized as
follows)
|
The moment of the sum equals the sum of the
moments of individual parts. |
|
|
In
a Nutshell: Two
common methods can be used to located the centroid
– the method of integration and the method of summation for composite
bodies where masses volumes, areas
or lines
are joined together. The concept of centroid applies to masses, volumes, areas, curves, and
lines. If there is a void (i.e. a
hole in the volume or area), then the void has a negative contribution.
To illustrate: Pick up a yard stick. To “balance” it at its centroid, position your finger 18 inches from each
end. Equal amounts of mass on each
side result in the balance.
|
|
|
The
Method of Integration: The
integral form of the principle of moments can be applied
to locate the centroids
of masses,volumes, areas, curves, and lines.
|
For
Masses: M
rbar =
∫ r dm
where M is the total mass, rbar is the vector location of the centroid
and r
is the vector to the element of mass, dm
|
|
For
Volumes: V rbar = ∫ r dV
where V is the total volume, rbar is the vector location of the centroid
and r
is the vector to the element of volume, dV
|
|
For
Areas: A rbar = ∫ r dA
where A is the total area, rbar is the vector location of the centroid
and r
is the vector to the element of area, dA
|
|
For
Curves: S rbar = ∫ r ds
where S is the total length of the curved
line, rbar is the vector location
of the centroid
and r
is the vector to the element of arc length, ds
|
|
For
Lines: L rbar = ∫ r dL
where L is the total length of the line
(curve), rbar is the vector location
of the centroid
and r
is the vector to the element of length, dL
|
Click
here to continue discussion of centroids.
|