# Centroids; Centers of Gravity

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```					     Centroids; Centers of
Gravity

How do you know where to locate
the resultant weight of a body?
Center of Gravity of a
Planar Body
   Assume body is composed of small
elements such that all element weight
vectors are parallel.
y

x
z
… more
 Resultant weight: W=W1+W2+…Wn
 Find location by moment balance.

y
xW  x1ΔW1  ...  xn ΔWn
yW  y 1ΔW1  ...  y n ΔWn
y   x
x
z           W
… more
   The quantities x , y locate the center
of gravity of the planar body B.
W   dW
B

xW   x dW
B

y W   y dW
B
Centroids of Areas
   Areas.   W = gtA , W = gtA
Agt   gtdA       A   dA
B                     B
xAgt   xgt dA        xA   x dA
B                  B
y Agt   y gtdA       yA   y dA
B                     B
x , y locates centroid of plane area
Centroids of Lines

   Lines.   W = gAL , W = gAL

z                L   dL
B
W       y
xL   x dL
B
y L   y dL
x               B
First Moments; Symmetry
   First moments: Qx = ydA Qy = xdA
Qx  yA Qy  xA
   An area is symmetric with respect to an
axis if there is a line joining every 2 points
which is perpendicular to the axis and
bisected by it.
… more
   For areas with 2 axes of symmetry the
centroid is at the intersection of the axes.

   An area has a center of symmetry if
element at (x,y) has mirror image (-x,-y).
Composite Areas
   Compute the centroids of areas
composed of simple shapes.
n
A   Ai
y                   i 1
n
xA   xi A i
i 1
x                n
yA   y i A i
i 1

```
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