﻿Calculation of Centroids. 251 



the position of the centroid of each by putting- a = 7r. Thus 



0G=^. 



O 7T 



For the portion of a sphere shown by the firm lines of 

 figure 7, we have, if OG is the distance of the centroid from 

 the centre of the complete sphere, 



where A is the area abed, and K is its radius of gyration 

 about its diameter. Thus 



OP — 71 " r2 S ^ llS 2 a 4 r2 Sni2 2 U 



3 r 



For the portion of a sphere enclosed by the firm lines of 



figure 8 we have 



,2 \r 2 sin | 

 x = 



•nr 4 *?' sin ka. 



o 2lT — OL 

 7TT 6 



3 2tt 



3 irr sin \a. 



3 27T — OC 



where x is the distance of the centroid from 00'. For a 

 hemisphere a = 7r, and 



x = i r. 



Finally, for the portion of an anchor ring shown in figure 9 

 we have, if r is the radius of the ring and a the radius of 

 the section, 



OP — 7ra,2 (i a 2 + 7 ' 2 i ^ sin £* 

 7ra J X 27rr X 



Z7T 



_ 2{r* + ja*) sin Ja 

 y(27T~a) ' 



in 



which reduces to 2rjir when a = 7r and a is very small 

 comparison with r. 



If a body is floating partly immersed in a liquid, the 



