132 



CENTER OF GRAVITY 



CENTEK OF GRAVITY OF A SOLID 



103. Center of gravity of a pyramid on a plane base. Let a 



pyramid be formed having any plane figure OPQR as base and 



any point A as vertex. We can find the center of gravity of a 



homogeneous pyramid by dividing it into thin layers parallel to 



its base, by a series of parallel planes. 



Let opqr be any such layer, this layer being regarded as an 



infinitely thin lamina. Let G be the center of gravity of a uni- 

 form lamina coinciding with 

 the base OPQR, and let the 

 line AG meet the lamina opqr 

 in g. Then, from the geometry 

 of similar figures, it is clear 

 that g occupies a position in 

 the lamina opqr which corre- 

 sponds exactly with that occu- 

 pied by the point G in the 

 lamina OPQR. Thus g will be 

 the center of gravity of the 

 lamina opqr. The mass of this 

 lamina may, accordingly, be 



replaced by the mass of a single particle at g. 



In the same way each of the laminas into which we are sup- 

 posing the pyramid to be divided may be 



replaced by a single particle at the point 



at which the lamina intersects the line AG. 



Thus the whole pyramid may be supposed 



replaced by a series of particles lying along 



AG. These form a rod of varying density, 



and the center of gravity of the pyramid 



will coincide with that of this rod. 



The center of gravity of the rod may be 



found by the method already explained in 



94. Consider the lamina which lies between two adjacent 



FIG. 77 



FIG. 78 



