

KINETICS OF A RIGID BODY. [237 



in other words, the principle of kinetic energy holds for the rela- 

 tive motion with respect to the centroid. 



237. Impulses. The equations determining the effect of a 

 system of impulses (see Arts. 2-5) on a rigid body are readily 

 obtained from the general equations of motion (4) and (6). 

 We shall denote the impulse of a force F by F. It will be 

 remembered that the impulse F of a force F is its time in- 

 tegral ; i.e. 



We confine ourselves to the case when t 1 t is very small and F 

 very large, in which case the action of the impulsive force F is 

 measured by its impulse F. 



If all the forces acting on a rigid body are of this nature, and 

 the impulses of X, Y, Z during the short interval t' t be 

 denoted by X, Y, Z, the integration of the equations (4) from 

 t=t to t=t f gives 



^m(x'-x) = ?.X, 2w(>'- = 2F, 2f(*'-*) = 2Z, (17) 



where x, j, z denote the velocities of the particle m at the time 

 / just before the impulse, and x\ y\ z' those at the time /' just 

 after the action of the impulse. 

 Similarly the equations (6) give 



l -x)-x(z 1 -z)] = ^(zX-xZ) t (18) 



x(y< - y) -y(x' -x)} = 2(* Y-yX). 



238. In determining the effect on a rigid body of a system 

 of such impulses, any ordinary forces acting on the body at the 

 same time are neglected because the changes of velocity pro- 

 duced by them during the very short time r are small in com- 

 parison with the changes of velocity x 1 x, y 1 y, z' z produced 

 by the impulses. For the mathematical treatment it is generally 



most convenient to define the impulse F of an impulsive force 



Jf 

 Fdt when /' / approaches o and 

 *- 



