220 



UNIVERSITY OF COLORADO STUDIES 



1. Theorem. 



If a part of a hody {^or sysiem) he moved ^ so that the Center 

 of Mass of the part is transferred to a new positon, the Center of 

 Mass of the hody or system, still considered: as a whole, is trans- 

 ferred in a parallel direction^ and to a distance which is to the 

 distance traveled hy the Center of Mass of the part, as the mass of 

 the part is to that of the whole. 



Let the body represented in Fig. 1 be divided into two parte Mj 

 aud M2 by the dotted line, and let Aj and K^ be their respective Cen- 

 ters of Mass ; then A the Center of Mass of the whole divides AjAj 



, A,A M, 

 BO that = 



A,A, M, + M, 



If now Ml be transferred till its Center of Mass occupies the 

 position A,' we have if A' be the new position of A 



A, A 



M, 



Ao -A Aqii- 



Aj Ai AjAj 



AA' Mi + M, 

 and .'. the triangles AjAA' and A2A1A/ are similar. 



.•.AA' is parallel to AjA/ and 

 proposition. 



AA' 



M, 



AjA/ M,+ M, 



which proves the 



2. To find the center of 7nass of a uniform circular arc. 



Pig. 2. 



Let ADB (Fig. 2) be the arc, O the center of the circle and G 

 the Center of Mass. OG is plainly perpendicular to AB. Let the 



