Center of Gravity

 Center of Gravity

The terms "center of mass" and "center of gravity" are used synonymously in a uniform gravity field to represent the unique point in an object or system which can be used to describe the system's response to external forces and torques. The concept of the center of mass is that of an average of the masses factored by their distances from a reference point. In one plane, that is like the balancing of a see saw about a pivot point with respect to the torques produced.

center of mass

for two masses

                                           

Center of Mass: Continuous

For a continuous distribution of mass, the expression for the center of mass of a collection of particles :

                     

                     

                      

WHERE
M  : TOTAL MASS
Xc : x coordinate distance of  mass center
mi : mass of  particles 
xi   : x coordinate distance of particles 

Center of Gravity.—The center of gravity of a body, volume, area, or line is that point at which if the body, volume, area, or line were suspended it would be perfectly balanced in

all positions. For symmetrical bodies of uniform material it is at the geometric center. The center of gravity of a uniform round rod, for example, is at the center of its diameter halfway along its length; the center of gravity of a sphere is at the center of the sphere. For solids, areas, and arcs that are not symmetrical, the determination of the center of gravity may be made experimentally or may be calculated by the use of formulas.

The tables that follow give such formulas for some of the more important shapes. 

Center of Gravity of Figures of any Outline.—If the figure is symmetrical about a center line, as in Fig. 1, the center of gravity will be located on that line. To find the exact location on that line, the simplest method is by taking moments with reference to any convenient axis at right angles to this center line. Divide the area into geometrical figures, the centers of gravity of which can be easily found. In the example shown, divide the figure into three rectangles KLMN, EFGH and OPRS. Call the areas of these rectangles A, B and C, respectively, and find the center of gravity of each. Then select any convenient axis, as X–X, at right angles to the center line Y–Y, and determine distances a, b and c.

If the figure, the center of gravity of which is to be found, is not symmetrical about any axis, as in Fig. 2, then moments must be taken with relation to two axes X–X and Y–Y, centers of gravity of which can be easily found, the same as before. The center of gravity is determined by the equations:


                                                                   

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