Centroids; Centers of
How do you know where to locate
the resultant weight of a body?
Center of Gravity of a
Assume body is composed of small
elements such that all element weight
vectors are parallel.
Resultant weight: W=W1+W2+…Wn
Find location by moment balance.
xW x1ΔW1 ... xn ΔWn
yW y 1ΔW1 ... y n ΔWn
The quantities x , y locate the center
of gravity of the planar body B.
xW x dW
y W y dW
Centroids of Areas
Areas. W = gtA , W = gtA
Agt gtdA A dA
xAgt xgt dA xA x dA
y Agt y gtdA yA y dA
x , y locates centroid of plane area
Centroids of Lines
Lines. W = gAL , W = gAL
z L dL
xL x dL
y L y dL
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.
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).
Compute the centroids of areas
composed of simple shapes.
y i 1
xA xi A i
yA y i A i