FIRST AND SECOND MOMENTS 



19 



If m^ m z , ., m n denote the masses of the n particles, and M their 

 sum, then, since 



Vi > = *,# 



where g denotes the acceleration due to gravity, the above relations 

 for determining the center of gravity become 



_^mgx 



Mg = 



or, since g is constant, 

 (11) M = 



mg, 



m 



The point determined from these relations by taking the system of 

 particles in two or more positions is called the center of inertia or 

 center of mass. Since these relations are identical with those given 

 above, it is evident that the center of mass is identical with the 

 center of gravity. 



13. Centroid. It is often necessary to determine the point called 

 the center of gravity or center of mass without reference to either 

 the mass or weight of the body, but simply with respect to its 

 geometric form. 



For a solid body let Av denote an element of volume, Aw its 

 mass, and D the density of the body. Then, since mass is jointly 

 proportional to volume and density, 



Am = DAv. 

 Therefore the formulas given above may be written 



x 



or, since the density D is constant, these become 



(12) 



Since the point previously called the center of gravity or center of 

 mass is now determined simply from the geometric form of the 

 body, it is designated by the special name centroid. 



