STATICS  THEORY



Recall,
the moment of a force about a point is given by the magnitude of the force times the perpendicular distance from the point to the force.
Using the same definition, the moment of an area about a point is the magnitude of the area times the perpendicular distance to the point. When the moment of an area about a point is zero, that point is called the centroid of the area.
The same method can be used to determine the centroid of a line or the centroid of a volume by taking the moment of the line or the moment of the volume.
Finding the location of a centroid is very similar to finding the location of the
force resultant of a distributed force as covered in the moment chapter. 



Centroid of a Line 

Centroid of a Line


The coordinates for the centroid of a line can be determined by using
three scalar equations,



Centroid of an Area

Centroid of an Area 

The centroid of an area can be determined by using three similar equations:



Centroid of a Volume

Centroid of a Volume 

Similarly, centroid of a volume can be determined by



Use of Symmetry

Centroid and Planes of Symmetry 

Finding the centroid of a body is greatly simplified when the body has planes of symmetry. If a body has a single plane of symmetry, then the centroid is located somewhere on that plane. If a body has more than one plane of symmetry, then the centroid is located at the intersection of the planes.



Center of Mass

Three Planes of Symmetry 

The centroid of a volume defines the point at which the total moment of volume is zero. Similarly, the center of mass of a body is the point at which the total moment of the body's mass about that point is zero. The location of a body's center of mass can be determined by using the following equations,
Here ρ is the density of the body. If ρ is constant throughout the body, then the center of mass is exactly the same as the centroid. 





Center of Gravity



The center of gravity of a body is the point at which the total moment
of the force of gravity is zero. The coordinates for
the center of gravity of an object can be determined with
Here g is the acceleration of gravity (9.81m/s^{2} or 32.2 ft/s^{2}).
If g is constant throughout the body, then the center of gravity is exactly the same as the center of mass.



