Distributed Forces: Center of Mass, Center
of Gravity, Centroid of Areas and Lines by
Integration
Introduction
o
In previous discussions, all forces were
treated as concentrated along their lines of
action.
o
This treatment provided a reasonable model
for those forces.
o
“CONCENTRATED” forces do not exist in the
exact sense, since every external force
applied mechanically to a body is distributed
over a finite contact area, however small.
Introduction
o
when forces are applied over a region whose dimensions are not
negligible compared with other pertinent dimensions, the actual
manner in which the force is distributed must be accounted for
o
This is done by summing the effects of the distributed force over
the entire region using mathematical integration
o
This requires that we know the intensity of the force at any
location
o
There are three categories of such problems
Introduction
o
Line distribution – when a force is distributed
along a line, the intensity w of the loading is
expressed as force per unit length of line (N/m).
o
Area distribution – when a force is distributed over
an area, the intensity is expressed as force per unit
area (N/m2).
o
Volume distribution - when a force is distributed
over the volume of a body (body force), the
intensity is expressed as force per unit area
(N/m3).
Centers of mass and centroids
o
Consider a 3D body of any size and shape,
having a mass m.
o
If this body is suspended as shown from any
point such as A, the body will be in
equilibrium under the action of tension in the
cord and the resultant W of the gravitational
forces acting on all particles of the body
o
The lines of action will be concurrent at a
single point G, which is called the center of
gravity of the body.
Determining the center of gravity
•
Center of gravity of a plate
 M y x W   xW
  x dW
 M y yW   yW
  y dW
•
Center of gravity of a wire
 M y x W   xW
  x dW
 M y yW   yW
  y dW
Centroids and First Moments of Areas and Lines
o
Centroid of an area
o
Centroid of a line
x W   x dW
x At    x t dA
x A   x dA  Q y
first moment with respect to y
yA   y dA  Qx
first moment with respect to x
x W   x dW
x  La    x  a dL
x L   x dL
yL   y dL
First Moments of Areas and Lines
o
An area is symmetric with respect to an axis
BB’ if for every point P there exists a point P’
such that PP’ is perpendicular to BB’ and is
divided into two equal parts by BB’.
o
The first moment of an area with respect to a
line of symmetry is zero.
o
If an area possesses a line of symmetry, its
centroid lies on that axis
o
If an area possesses two lines of symmetry, its
centroid lies at their intersection.
o
An area is symmetric with respect to a center
O if for every element dA at (x,y) there exists
an area dA’ of equal area at (-x,-y).
o
The centroid of the area coincides with the
center of symmetry.
Practice Problem
Centroid of a circular arc: Locate the centroid of the circular arc
as shown in the figure. 
Solution
Practice Problem
Centroid of a triangular area: Determine the distance h from
the base of a triangle of altitude h to the centroid of its area. 
Practice Problem
Centroid of the area of a circular sector: Locate the centroid
of the area of a circular section with respect to its vertex. 
Solution
Practice Problem
Locate the centroid of the area under the curve x = ky3 from x = 0
to x = a.
Solution
Centroids of Common Shapes of Areas
Centroids of Common Shapes of Areas
Centroids of Common Shapes of Areas
Composite Plates and Areas
o
Composite plates
X W   x W
Y W   y W
o
Composite area
X  A   xA
Y  A   yA
Sample Problem
For the plane area shown,
determine the first moments
with respect to the x and y axes
and the location of the centroid.
SOLUTION:
• Divide the area into a triangle,
rectangle, and semicircle with a
circular cutout.
• Calculate the first moments of each
area with respect to the axes.
• Find the total area and first moments
of the triangle, rectangle, and
semicircle.
• Subtract the area and first moment
of the circular cutout.
• Compute the coordinates of the area
centroid by dividing the first
moments by the total area.
Sample Problem
•
Find the total area and first moments of the
triangle, rectangle, and semicircle. Subtract the
area and first moment of the circular cutout.
Qx  506.2  103 mm 3
Q y  757.7  103 mm 3
Sample Problem
•
Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
x A  757.7 103 mm 3

X 

 A 13.828 103 mm 2
X  54.8 mm
y A  506.2 103 mm 3

Y 

 A 13.828 103 mm 2
Y  36.6 mm
Determination of Centroids by Integration
x A   xdA   x dxdy   xel dA
yA   ydA   y dxdy   yel dA
Double integration to find the first moment may be
avoided by defining dA as a thin rectangle or strip.
x A   xel dA
x A   xel dA
x A   xel dA
yA   yel dA
ax
 a  x dy 
2
yA   yel dA
  x  ydx 
y
   ydx 
2

  y a  x dy 
2r
1 2 
  cos  r d 
3
2

yA   yel dA

2r
1

sin   r 2 d 
3
2

Sample Problem
Determine by direct integration
the location of the centroid of a
parabolic spandrel.
SOLUTION:
• Determine the constant k.
• Evaluate the total area.
• Using
either vertical or
horizontal strips, perform a
single integration to find the
first moments.
• Evaluate
the
centroid
coordinates.
Sample Problem
SOLUTION:
• Determine the constant k.
y  k x2
b
b  k a2  k  2
a
b
a
y  2 x 2 or x  1 2 y1 2
a
b
• Evaluate the total area.
A   dA
3 a

b
b x
  y dx   2 x 2 dx   2 
 a 3  0
0a
ab

3
a
Sample Problem
o
Using vertical strips, perform a single integration
to find the first moments.
a
 b

Q y   xel dA   xydx   x 2 x 2 dx

0 a
a
 b x4 
a 2b
 2
 
4
 a 4  0
2
a
y
1 b

Q x   yel dA   ydx    2 x 2  dx
2

02a
a
 b2 x5 
ab 2
 4  
 2a 5  0 10
Sample Problem
o
Or, using horizontal strips, perform a single
integration to find the first moments.
b 2
ax
a  x2
a  x dy  
Q y   xel dA  
dy
2
2
0
1 b  2 a 2
  a 
2 0 
b
2

a
b

y dy 

4

a


Q x   yel dA   y a  x dy   y a  1 2 y1 2 dy


b
a 3 2
ab 2

   ay  1 2 y dy 
10

b
0
b
Sample Problem
o
Evaluate the centroid coordinates.
xA  Q y
ab a 2b
x

3
4
3
x a
4
yA  Q x
ab ab 2
y

3
10
y
3
b
10
Practice Problem
The figure shown is made from a piece of thin,
homogeneous wire. Determine the location of its
center of gravity. x = 10 in, y = 3 in.
C
10 in.
A
B
24 in.
Practice Problem
Determine the x- and ycoordinates of the centroid of the
𝑎
shaded area. 𝑥 = 6 𝜋7𝑎− 1
𝑦=
𝜋−1
Practice Problem
Locate the centroid of the shaded area.
𝑥 = 7.50 𝑖𝑛. 𝑦 = 5.08 𝑖𝑛.
Solution
𝑦
𝑥𝐴
𝑦𝐴
in
in3
in3
6
5
720
600
30
14
10/3
420
100
3
-14.14
6
1.273
-84.8
-18
4
-8
12
4
-96
-32
TOTALS
127.9
969
650
A
𝑥
Part
in2
in
1
120
2
Practice Problem
Determine the coordinates of the mass center of the
bracket, which is constructed from sheet metal of
uniform thickness.
𝑥 = 2.48 𝑖𝑛.
𝑦 = 2.71 𝑖𝑛.
𝑧 = −0.882 𝑖𝑛.
Solution
A
𝑥
𝑧
𝑥𝐴
Part
𝑦
in2
in
in
in
in3
in3
in3
1
16
2
2
0
32
32
0
2
6
5
4/3
0
30
8
0
3
12
2
4
-1.5
24
48
-18
4
2p
2
4
-3.85
12.57
25.1
-24.2
5
-p
2
4
-3
-6.28
-12.56
9.42
92.3
100.6
-32.8
TOTALS
𝑦𝐴
𝑧𝐴
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