I2 4 KINETICS OF A RIGID BODY. [226. 



as 80 X , SOy, 80 Z are independent and arbitrary, their coefficients must 

 vanish separately, and this gives the equations (6). 



226. The equations of linear momentum, (4) or (5), admit of 

 a further simplification, owing to the fundamental property of 

 the centroid. By Part II., Art. 13, the co-ordinates x, j>, z of the 

 centroid satisfy the relations 



MX ^mxy My = ^my, Mz = 2mz, 



where M= 2m is the whole mass of the body. Differentiating 

 these equations, we find 



MX = 2, mx, My = ^my y Mz = 2 

 and MX = ^mx, My = 



where x, y, z are the components of the velocity v, and x, y, 

 those of the acceleration/, of the centroid. 



The equations (4) or (5) can therefore be reduced to the form 



Mx= MxR x , My = My = R y , Mz = ~rM~z R g) (8) 



whence Mj= - Mv =R\ (9) 



at 



i.e. if the whole mass of the body be regarded as concentrated 

 at the centroid, the effective force of the centroid, or the time- 

 rate of change of its momentum, is equal to the resultant of all 

 the external forces. It follows that the centroid of a rigid body 

 'moves as if it contained the whole mass, and all the external forces 

 were applied at this point parallel to their original directions. 



227. If, in particular, the resultant R vanish (while there may 

 be a couple H acting on the body), we have by (8) and (9) 

 y=o; hence v = const. ; i.e. if the resultant force be zero, the 

 centroid moves uniformly in a straight line. 



This proposition, which can also be expressed by saying that, 

 if R = o, the momentum Mv of the centroid remains constant, 

 or, using the form (5) of the equations of motion, that the linear 

 momentum of the body in any direction is constant, is known 



