361.] 



INITIAL MOTION OF FREE BODY. 



197 



axis of rotation so as to pass through the centroid ; we shall 

 then call it the centroidal instantaneous axis I. 



359. Dynamically, the instantaneous motion of a free rigid 

 body is determined by the momenta of its particles. These 

 momenta can be reduced, for any point O as origin, to a re- 

 sultant momentum and a resultant couple, or angular momen- 

 tum, and these can be regarded as due to a certain system of 

 impulses. This reduction will at the same time lead to the 

 solution of the converse problem, viz. to determine the initial 

 motion produced by a system of impulses acting on a rigid body 

 at rest, and the change in the instantaneous motion due to such 

 a system when the body is not at rest. 



360. Translation. The velocities u of all points being equal 

 and parallel in the case of translation, the momenta mu of all 

 particles are parallel and have (see Arts. 6-8) a single resultant 



passing through the centroid G of the body. If the whole 

 mass M be regarded as concentrated at the centroid, Mu is the 

 momentum of the centroid. This momentum can be produced 

 by applying at the centroid a single impulse R=Mu. Hence 

 to impart to a free rigid body of 

 mass M a velocity of translation 

 u, it is sufficient to apply at the 

 centroid an impulse R = Mu. 



361. Rotation. Let us take 

 the instantaneous axis / as axis 

 of , and the axis of x so as to 

 pass through the centroid G 

 (Fig. 45). The momentum mar H x 

 of any particle of mass m, at / 

 the distance r from /, has the 

 components may, mwx, o; and as 



Fig. 45. 



the 



