134 



CENTER OF GRAVITY 



Thus the center of gravity of the pyramid is in the line AG, 

 three quarters of the way down from A. 



104. Center of gravity of the sector of a sphere. We can now 

 find the center of gravity of the sector of a sphere, the volume 

 cut out of a solid sphere by a right circular cone having its ver- 

 tex at the center of the sphere. To do this we divide the base 

 PQ of the sector into a number of small elements of area, and 

 then divide the volume of the sector into a number of pyra- 

 mids of small cross section by taking these elements of area as 

 bases and joining them to the common vertex 0. These pyra- 

 mids are all of the same 

 height, so that their masses 

 are proportional to their 

 bases. The center of grav- 

 ity of each pyramid is three 

 quarters of the distance 

 down from to its base, 

 and is, therefore, at a dis- 

 tance from equal to three 

 quarters of the radius of 

 the sphere. Thus, if we con- 

 struct a second sphere hav- 

 ing as its center and of 

 radius equal to three quar- 

 ters of the radius of the original sphere, the center of gravity 

 of each small pyramid will lie on this new sphere. Each pyra- 

 mid may be replaced by a particle at its center of gravity, so 

 that the whole spherical sector may be replaced by a series of 

 particles lying on this sphere and forming the spherical cap peq 

 (fig- 79). ' 



The mass of each pyramid is proportional to the base, and this 

 again is proportional to the part of the spherical shell peq which 

 is intercepted by the pyramid. Thus the spherical shell peq which 

 is to replace the original volume must be supposed to be of uni- 

 form density. 



FIG. 79 



