
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 i^{th} object, and Q_{i} 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 L_{i} 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 A_{i}
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 V_{i} 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 x_{1}, y_{1}, z_{1} and a volume of V_{1}; and a hole within the cube that has a centroid at x_{2}, y_{2}, z_{2} and a volume of V_{2}; then the coordinates of the centroid for the system are




