7. CENTROIDS AND CENTERS OF MASS
CENTROID - an average position of any set of quantities with which we can
associate positions
(*)
x
 xi ci
i
 ci
y
 y i ci
i
 ci
i
i
xi, yi - coordinates
ci - any quantity associated with position
(e.g. area, volume, mass, weight, etc.)
7.1 CENTROIDS OF AREAS
In order to determine the centroid (average position of the area), consider an arbitrary
area A in the x-y plane, divide the plane area into parts A1, A2, …, AN and denote the
positions of the parts by (x1, y1), (x2, y2), …, (xN, yN). The centroid can be obtained using
Eqs. (*), but there is a question: What are the exact positions of the areas A1, A2, …, AN ?
Some obvious cases: circle, rectangle, areas that are symmetric about one or two axes.
To determine the exact location of the centroid of an arbitrary area, we must take the limit
as the sizes of the parts approach zero:
x
 x dA
A
 dA
A
y
 y dA
A
 dA
A
x, y - coordinates of the differential
element of area dA
7.2 CENTROIDS OF COMPOSITE AREAS
Integration process becomes difficult for complicated areas.
COMPOSITE AREA
x
- an area that consists of a combination of simple areas
- the centroid can be determined without integration, if the centroids
of its parts are known (some of them tabulated in Appendix B).
 x dA A x dA  A x dA    A x dA

A
 dA
1
A
y 
2
n
 dA   dA     dA
A1
A2
An
xi Ai
x1 A1  x2 A2    xn An 

 i
A1  A2    An
 Ai
i
 yi Ai
i
 Ai
(**)
i
If the composite area contains some cutouts, the centroid of that area can be determined
using the same Eqs. by treating the cutouts as negative areas.
Procedure:
1. Choose the parts - divide the area into parts whose centroids are known or can be
easily determined
2. Determine the values (the centroid and the area) for the parts
3. Calculate the centroid using the Eqs. (**)
7.3 DISTRIBUTED LOADS - loads continuously distributed along lines
- it is described by a function w such that the force exerted
on an infinitesimal element dx of the beam is w dx
- the dimensions of w are (force)/(length), i.e. [N/m] or [lb/ft]
The total force exerted by the distributed load (assuming that the function w is known):
F   w dx
L
(integrate the loading curve (graph of w) with respect to x)
 the total force exerted by the distributed load is equal to the “area”
between the loading curve and the x axis
The total moment about a point (origin) due to the distributed load:
M   xw dx
L
Distributed load can be represented by a single equivalent force F acting at the position
x
 xw dx
L
 w dx
L
(moments of F and distributed load about the origin, have to be equal)
7.4 CENTROIDS OF VOLUMES AND LINES
CENTROIDS OF VOLUMES - considering a volume V and its differential element dV with
coordinates x, y and z, by analogy with Eqs. for a centroid of
an area, the coordinates of the centroid of V are:
 x dV
xV
 dV
 y dV
yV
V
 dV
 z dV
z V
V
 dV
V
If a volume has the form of a plate with uniform thickness and cross-sectional area A, its
centroid coincides with the centroid of A and lies at the midpoint between the two faces.
CENTROIDS OF LINES
x
- considering a differential element dL with coordinates x, y and
z, the coordinates of the centroid of a line L are:
 x dL
L
 dL
L
y
 y dL
L
 dL
L
z
 z dL
L
 dL
L
The centroids of some simple volumes and lines are tabulated in Appendixes B and C.
CENTROIDS OF COMPOSITE VOLUMES AND LINES - the same approach and procedure
as for the composite areas
x
 xiVi
i
Vi
y
 yiVi
i
Vi
x
i
 Li
i
i
Vi
i
i
 xi Li
z
 ziVi
y
 yi Li
i
 Li
- for a composite volume
i
z
 zi Li
i
 Li
i
- for a composite line
i
7.5 THE PAPPUS-GULDINUS THEOREMS
- two simple and useful theorems relating surfaces and volumes of revolution
to the centroids of the lines and areas that generate them
FIRST THEOREM
- consider a line L in the x-y plane rotating about the x-axis (the line does
not intersect the x-axis and the coordinates of its centroid are x, y )
- the area of the surface of revolution is A  2 y L
SECOND THEOREM - consider an area A in the x-y plane rotating about the x-axis (the area
does not intersect the x-axis and the coordinates of its centroid are x, y )
- the volume of the volume of revolution is V  2 y A
7.6 DEFINITION OF CENTER OF MASS of an object is the centroid of its mass
 x dm
 y dm
xm
ym
 dm
m
 dm
m
 z dm
zm
 dm
m
x, y, z - coordinates of the differential
element of mass dm
The weight of an object can be represented by a single equivalent force
acting at its center of mass.
7.7 CENTERS OF MASS OF OBJECTS
The variable of integration is changed from mass to volume by introducing
the mass density ; dimensions of  are (mass)/(volume), i.e. [kg/m3] or [slug/ft3].
dm   dV
the mass of a differential element of a volume
m   dm    dV
total mass of an object
m
V
m    dV  V
total mass of a homogeneous object (its mass density is uniform)
V
The weight density is   g and can be expressed in [N/m3] or [lb/ft3].
The coordinates of the center of mass (in terms of volume integrals) are:
  x dV
xV
  dV
  y dV
yV
V
  z dV
z V
  dV
  dV
V
V
In some particular cases, centers of mass coincide with centroids of volumes, areas and lines



the center of mass of a homogeneous object coincides with the centroid of its volume
the center of mass of a homogeneous plate of uniform thickness coincides with the
centroid of its cross-sectional area
the center of mass of a homogeneous slender bar of uniform cross-sectional area
coincides approximately with the centroid of the axis of the bar
7.8 CENTERS OF MASS OF COMPOSITE OBJECTS
x
 xi mi
i
 mi
y
 yi mi
i
 mi
i
x
 xiWi
i
Wi
i
z
 zi mi
i
 mi
i
y
 yiWi
i
Wi
i
- center of mass of a composite object
i
z
 ziWi
i
Wi
i
- center of weights of a composite object