SECTOE OF A SPHERE 



135 



The sector of a sphere OPQ has now been replaced by the 

 uniform spherical shell pq, and the center of gravity of this shell 

 is known to be G, the middle point of re in fig. 79. This point G 

 is, accordingly, the center of gravity required. 



If the semivertical angle of the cone by which the sector is 

 bounded is a, and if a is the radius of the sphere, we have 



Oe = J a, Or = J a cos a, 

 so that OG = f a (1 + cos a). 



In particular, if a = > the sector becomes a hemisphere, and 



' 



Thus the center of gravity of a hemisphere is three eighths of 

 the way along the radius which is perpendicular to its base. 



V 



CENTEK OF GRAVITY OF AREAS AND VOLUMES OBTAINED BY 

 DIRECT INTEGRATION 



105. Center of gravity of a lamina. To find the center of 

 gravity of a lamina of any shape by integration, we take any con- 

 venient set of axes Ox, Oy in the plane of the lamina, and imagine 

 the lamina divided into small elements 

 by two series of lines, one parallel to the 

 axis Ox, and the other parallel to the 

 axis Oy. 



Consider the small rectangular element 

 for which the values of x for the two 

 edges parallel to Oy are x and x -f- dx, 

 and the values of y for the two other 

 edges are y and y + dy. The area of 

 this element is dxdy, so that if p is the mass of the lamina per unit 

 area at this point, the mass of the element will be p dxdy. More- 

 over, when dx, dy are made vanishingly small in the limit, the 

 mass may be treated as a particle. Thus the whole mass of the 

 lamina may be regarded as the masses of a number of particles. 



FIG. 80 



