118 CENTER OF GRAVITY 



Since the moment of the resultant is equal to the sum of the 

 moments of the separate forces, we must have 



so that x = 



Similarly y = 



These equations determine the coordinates x, y of the point in 

 which the line of action of the resultant meets the plane Oxy. 



We have, however, seen that the coordinates of the centroid o 

 masses m l at x lt y lt z v ra 2 at x 2 , y z) z 2 , etc., are 



so that the point in which a vertical through the centroid wil 

 meet the plane Oxy must be 



> m:v. 



0, 



i.e. the point must be the point x, y, in which the line of action 

 of the resultant force meets the plane Oxy. Thus 



The line of action of the resultant force of gravity is the ver- 

 tical line through the centroid of the particles. 



For this reason the centroid of a number of points, weighted 

 according to the masses of the particles which occupy these points, 

 is cal^d the center of gravity of the particles. The effect of 

 gravity acting on a rigid body is, as we have now seen, repre- 

 sented by a single force acting vertically downwards through the 

 center of gravity of the body, the amount of the force being equal 

 to the total weight of the body. The action of gravity is, accord- 

 ingly, the same as if the whole mass of the body were concentrat 

 in a single particle placed at the center of gravity. 



