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STATICS - THEORY

    Equilibrium

   

In the previous section, Centroid: Line, Area and Volume , it was shown that the centroid of an object involves evaluating integrals of the form

     

where Q is a line, area, or volume, depending on the centroid that is required. The same equation can also be used for the other two directions by just substituting y or z for x. If there are several objects, then the centroid of the entire system is given by

     

where n is the number of objects in the system. This equation can be simplify as

     

where is the centroid location of the ith object, and Qi is the length, area, or volume of the ith object, depending on the type of centroid.

     
    Centroid of a System of Lines

   

For a system of lines, the coordinates of the centroid are

          

Here and are the centroid coordinates of the ith line, and Li is the length of the ith line.

     
    Area Centroid of a System of Objects

   

For a system of objects, the coordinates of the area centroid are

     

Here and are the area centroid coordinates of the ith object, and Ai is the area of the ith object.

     
    Volume Centroid of a System of Objects


Centroid of Multiple Objects
 

For a system of objects, the coordinates of the volume centroid are

     

Here and are the volume centroid coordinates of the ith volume, and Vi is the volume of the ith object.

     
    Subtraction of Material (Holes)

   

If an object or system of objects has a cutout or hole, then the centroid of the system can be found by considering the cutout or hole as a negative area, volume, or line length.

For example, if a system consists of a cube with a centroid at x1, y1, z1 and a volume of V1; and a hole within the cube that has a centroid at x2, y2, z2 and a volume of V2; then the coordinates of the centroid for the system are

     

     
   
 
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