Chapter 5 DISTRIBUTED FORCES:
CENTROIDS AND CENTERS OF GRAVITY
The center of gravity of a rigid body is the point G where a
single force W, called the weight of the body, can be applied to
represent the effect of the earth’s attraction on the body.
Consider two-dimensional bodies, such as flat plates and
wires contained in the xy plane. By adding forces in the vertical
z direction and moments about the horizontal y and x axes, the
following relations were derived
W = dW
xW = x dW
yW =  y dW
which define the weight of the body and the coordinates x and y
of its center of gravity.
For a homogeneous flat plate of uniform thickness, the
center of gravity G of the plate coincides with the centroid C
of the area A of the plate, the coordinates of which are defined
by the relations
xA =  x dA
yA =  y dA
These integrals are referred to as the first moments of area
A with respect to the y and x axes, and are denoted by Qy
and Qx , respectively
Qy = xA
Qx = yA
Similarly, the determination of the center of gravity of a
homogeneous wire of uniform cross section contained in a
plane reduces to the determination of the centroid C of the
line L representing the wire; we have
xL =  x dL
yL = y dL
y
x
C
L
y
O
x
z
W3 y
z
y
W1 W2
SW
G3
X
O
G
Y
O
x
G2
G1
x
The areas and the centroids of various common shapes are
tabulated. When a flat plate can be divided into several of these
shapes, the coordinates X and Y of its center of gravity G can be
determined from the coordinates x1, x2 . . . and y1, y2 . . . of the
centers of gravity of the various parts using
X SW = S xW
Y SW = S yW
z
y
G
O
x
If the plate is homogeneous and of uniform thickness, its
center of gravity coincides with the centroid C of the area of
the plate, and the first moments of the composite area are
Qy = X SA = S xA
Qx = Y SA = S yA
When the area is bounded by analytical curves, the
coordinates of its centroid can be determined by
integration. This can be done by evaluating either double
integrals or a single integral which uses a thin rectangular
or pie-shaped element of area. Denoting by xel and yel the
coordinates of the centroid of the element dA, we have
Qy = xA = xel dA
Qx = yA = yel dA
L
C
y
x
2py
The theorems of Pappus-Guldinus
relate the determination of the area
of a surface of revolution or the
volume of a body of revolution to the
determination of the centroid of the
generating curve or area. The area A
of the surface generated by rotating
a curve of length L about a fixed axis
is
A = 2pyL
where y represents the distance from the centroid C of the
curve to the fixed axis.
C
A
y
x
2py
The volume V of the body generated by rotating an area A
about a fixed axis is
V = 2pyA
where y represents the distance from the centroid C of
the area to the fixed axis.
dW
w
w
W
x
w
O
dx
C
B
x
O
P
W=A
B
x
x
L
L
The concept of centroid of an area can also be used to solve
problems other than those dealing with the weight of flat plates.
For example, to determine the reactions at the supports of a
beam, we replace a distributed load w by a concentrated load
W equal in magnitude to the area A under the load curve and
passing through the centroid C of that area. The same approach
can be used to determine the resultant of the hydrostatic forces
exerted on a rectangular plate submerged in a liquid.
The coordinates of the center of gravity G of a threedimensional body are determined from
xW = x dW
yW =  y dW
zW =  z dW
For a homogeneous body, the center of gravity G coincides with
the centroid C of the volume V of the body; the coordinates of C
are defined by the relations
xV = x dV
yV =  y dV
zV = z dV
If the volume possesses a plane of symmetry, its centroid C
will lie in that plane; if it possesses two planes of symmetry,
C will be located on the line of intersection of the two planes; if it
possesses three planes of symmetry which intersect at only one
point, C will coincide with that point.
The volumes and centroids of various common threedimensional shapes are tabulated. When a body can be
divided into several of these shapes, the coordinates X, Y, Z of
its center of gravity G can be determined from the
corresponding coordinates of the centers of gravity of the
various parts, by using
X SW = S xW
Y SW = S yW
Z SW = S zW
If the body is made of a homogeneous material, its center
of gravity coincides with the centroid C of its volume, and
we write
X SV = S xV
Y SV = S yV
Z SV = S zV
z
xel = x
xV = xel dV
yV =  yel dV
When a volume is
P(x,y,z)
bounded by analytical
yel = y
surfaces, the coordinates
1
zel = 2 z
of its centroid can be
z
dV = z dx dy determined by integration.
zel
To avoid the computation
y
xel
of triple integrals, we can
use elements of volume
dx
x yel
in the shape of thin
dy
filaments (as shown).
Denoting by xel , yel , and zel the coordinates of the centroid of
the element dV, we write
zV = zel dV
If the volume possesses two planes of symmetry, its centroid C
is located on their line of intersection.
y
xel
xel = x
dV = pr 2 dx
x
z
dx
If the volume possesses two planes of symmetry, its centroid C
is located on their line of intersection. Choosing the x axis to lie
along that line and dividing the volume into thin slabs
parallel to the xz plane, the centroid C can be determined from
xV = xel dV
For a body of revolution, these slabs are circular.