228 VI. SYSTEMS OP VECTORS. DISTBIBUT. OF MASS. INSTANT. MOTION. 



The determination of these quantities is then, like that of centers of 

 mass, a subject belonging to the integral calculus. 



The six constants A, B, C, D, E, F together with the mass M 

 and coordinates x, ?7, 0", of the center of mass, completely characterize 

 the body for dynamical purposes, since when we know their values 

 and the instantaneous twist, the momentum or impulsive wrench is 

 completely given. The body may therefore be replaced by any other 

 having the same mass, center of mass, and moments and products 

 of inertia, and the new body will, when acted upon by the same 

 forces, describe the same motion. 



69. Centrifugal Forces. As the body moves, its different 

 parts exercise forces of inertia upon each other, so that there is a 

 resultant tending to change the instantaneous screw in the body. 

 Let us suppose the translation to vanish, and examine the kinetic 

 reactions developed by the rotation, or the centrifugal forces. The 

 instantaneous axis being again taken as the axis of Z, a particle P 



experiences the centripetal acceleration = ro 2 towards the axis, and 



the centrifugal force is Jt c = mrs? (see p. 119) directed along the 

 radius r from the axis OZ, and having the projections 



Z c = 0. 



For the moment of the centrifugal force we have 



L c = yZ c zY c = my # 2 , 



58) M c = 2X c xZc = rnxsG?) 



N c = xY c -yX c = , 



so that the coordinates of the resultant centrifugal force and couple are 

 X c = tfZmx = a* MX, 



59) c 



L c = & 



N c = . 



Thus the centrifugal force is equal and parallel to that of a 

 mass placed at the center of mass, and moving as the latter point 

 does. It vanishes when the center of mass lies in the axis. The 

 system of centrifugal forces is however, as in the case of the 



