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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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posted:12/11/2011
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