472 X. STATICS OF DEFORMABLE BODIES. 



45) 



If the force acting on the liquid be gravity we have 



X = Y=0, Z=-g, 

 accordingly 



where m is the mass of the fluid displaced by the body. This is the 

 Principle of Archimedes: A body immersed in a liquid has its weight 

 diminished by the weight of the displaced liquid. 

 For the moments we have 



L 



46) 



M 



= 9 i I I Qydt = gmy, 

 = g I I I Qxdr = gmx, 



N = 0, 



where x, y denote the coordinates of the center of mass of the 

 displaced liquid. If the body is in equilibrium, by Archimedes' 

 Principle the weight and therefore the mass of the body is equal to 

 that of the displaced liquid. Consequently the resultant of the forces 

 acting on the body is equivalent to a couple whose members are 

 forces mg exerted downward at the center of mass of the body and 

 upward at the center of the mass of the displaced liquid. If the 

 couple is to vanish one of these must be vertically above the other. 

 The center of mass of the displaced liquid is called the center of 

 buoyancy of the body. 



If the floating body is slightly displaced through a small angle 

 deo from the position of equilibrium by the application of a couple, 

 the mass of the displaced fluid must remain unchanged, but the 

 position of the center of buoyancy is slightly altered. Let us take 

 the origin in the intersection of the old and new planes of flotation 

 (Fig. 151). For the new position the figure is to be tilted in the direction 

 of the arrow until the new position of the water line W'L' is horizontal. 

 The old center of buoyancy JB is now no longer under the center 

 of mass Gr and consequently, if the same portion were immersed, 



