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) w eight = dW1
x
(x2 y2) w eight = dW2
y
W
x
Fz
W  dW1  dW2  ......  dWn
M y : x W  x1 dW1  x2 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   tdA
 = 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
Lecture18.doc
yA   y dA
zA   z dA
1
Similarly for wires
xL   x dL
dw  padL
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 Pg. 509. These will be useful.
Centroids by integration
The centroids of area 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
Lecture18.doc
Q  YA   yel dA
2
y
P(x, y)
dy
x
a
y
P(x, y)
ax
2
yel  y
xel  x
xel 
y
2
dA  y dx
y el 
dA  (a  x) dy
x
dx
Lecture18.doc
3
Example
1). Given:
y
y = mx
b
y = kx3
x
a
Find: Determine by integration, the centroid.
y
a
A
b
y2
yel
y1
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
Lecture18.doc
4
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


a
0

x4
x 2 1  2
 a

 dx

 2 x6 
b2  x3
x7  a
2


x

dx


 ab 2
2 
4 
0  a 4 
2a  3 7 a  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
Lecture18.doc
8
a
15
8
Y  b
21
X 
5