10 INTRODUCTION TO DYNAMICS. [21. 



origin, and let G' be the point obtained as centroid from this 

 origin, so that 



2[mr], M-O'G' = 2|W]. 

 As we have the geometric equation [r'] = [d] + [>], we find 



Hence subtracting the first equation and dividing by M, 



[O f G']-[OG] = [d], or [O'G'] = [d] + [OG] = [O'G] 

 .so that G and G f coincide. 



It follows from this consideration that a given system has 

 only one centroid. 



21. Regarding again the mass of the centroid as equal to 

 that of the whole system, we may now define the centroid of a 

 system as a point such that its moment with respect to any point 

 or plane is equal to the sum of the moments of all the points 

 constituting the system; the sum being understood to be a 

 geometric sum for moments with respect to a point, and an 

 algebraic sum for moments with respect to a plane. 



Taking the centroid itself as origin, we have the proposition 

 that the geometric sum of the moments of a system with respect 

 to the centroid is equal to zero. It has been proved before 

 (Art. 14) that the algebraic sum of the moments of a system 

 vanishes for any plane passing through the centroid. 



22. In determining the centroid of a given system it will 

 often be found convenient to break the system up into a number 

 of partial systems whose centroids are either known or can 

 be found more readily. The moment of the whole system is 

 obviously equal to the sum of the moments of the partial systems. 



Thus let the given mass M be divided into k partial masses 

 M lt M 9 ...M so that M= M^ + M^ + ^+M ; let G, G v G 2 , ... 



