Chapter 9 – Center of Gravity and Centroid
Center of gravity- the point at which the weight of the body can be concentrated.
CG of 2-D body
z
y
(x1 y1) weight = dW1
(x2 y2) weight = dW2
x
y
W
x
Fz
W  dW1  dW2  ......  dWn
M y : x W  x1 dW1  x 2 dW2  ....  x n dWn
M x : yW  y1 dW1  y 2 dW2  ....  y n dWn
If we take the limit as dW  0
W   dW
xW   x dW
z W   z dW
yW   y dW
Centroids of areas and lines
- homogeneous, uniform thickness
dw   t dA
  density
t  thickness
dA  differential area
Substituting into M y , M x
M Y : xptA  x1 ptdA1  x2 ptdA2  ......  xn ptdAn
M x : yptA  y1 ptdA1  y 2 ptdA2  .....  y n ptdAn
xA   x dA
yA   y dA
zA   z dA
Similarly for wires
xL   x dL
dw  pa dL
a  cross-sectional area
dL  differential length
yL   y dL
z L   z dL
First moments of areas and lines
The integral  x dA is known as the first moment of the area A with respect to the
y axis.
Denoted by Qy
Qy   x dA  xA
Q x   y dA  yA
First moments of areas extremely useful in mechanics of solids.
Centroids of common shapes can be found on the inside back cover of the book.
These will be useful.
Centroids by integration
The centroids of areas bounded by analytical curves is usually determined by computing
the integrals:
XA   x dA
Y A   y dA
If the element of area dA is chosen to be a small square of dx by dy, then the
determination of the integrals requires a double integration.
By choosing a thin rectangular strip, this can be avoided
Qy  XA   xel dA
Q x  Y A   y el dA
y
y
P(x, y)
P(x, y)
dy
xel
yel
dx
xel  x
y
2
dA  y dx
y el 
yel
x
x
xel
a
ax
2
yel  y
xel 
dA  (a  x) dy
Example
1). Given:
y
y = mx
b
y = kx3
x
a
Find: Determine by integration, the centroid.
y
a
A
b
y2
y1
yel
x
x
y1  kx3
at pt. A
y2  mx
at pt. A
b
b
 y1  3 x 3
3
a
a
b
b
b  ma  m 
 y2  x
a
a
b  ka3  k 
b
b
dA  ( y2  y1 )dx  ( x  3 x 3 )dx
a
a
xel  x
y el 
1
1 b
b
( y 2  y1 )  ( x  3 x 3 )
2
2 a
a
b a
x3 
b  x2
x4  a 1



  ab
x

dx


a 0 
a  2 4a 2  0 4
a 2 
A   dA 
a b
b 3
b a  2 x4 
b  x3
x5  a 2 2


x
dA

x
x

x
dx

x

dx





  a b
 el
0  a a 3 
a 0 
a  3 5a 2  0 15
a 2 
 yel dA  
a
0

b2
2a 2
1b
b 3  b
b 3
b2
 x  3 x  x  3 x  dx  2
2a
a
a
2a
 a


 2 x6 
b2  x3
x7  a
2


x

dx


 ab 2
2 
4 
0  a 4 
2a  3 7a  0 21
a
1  2
XA  xel dx  X  ab   a 2 b
 4  15
1  2
Y A   y el dx  Y  ab   ab 2
 4  21
8
a
15
8
Y  b
21
X 
a
0

x4
x 2 1  2
 a

 dx
