3o6.] BODY WITH FIXED AXIS. 165 



305. Let the impulse F be produced by the inelastic impact 

 of a particle of mass m moving with a velocity u. It would not 

 be correct to put F=mu in (11); as the particle after impact 

 continues to move with the body with a certain velocity v, it 

 does not actually give up to the body its whole momentum, 

 but only the amount Fm(u v), provided that u and v have 

 the same direction. With this assumption, which evidently 

 means that the particle meets the body atTsome point of the 

 plane passing through the axis and perpendicular to u, the 

 velocity after impact is v = a>p. With the value (11) of o> this 

 gives 



whence F=muf/(f+mp 2 ), and finally, by (n), 



(12) 



As mp^ is the moment of inertia of the particle for the fixed 

 axis, this formula shows that we may substitute in (11) the 

 whole momentum mu for F if we increase the moment of 

 inertia of the body by that of the particle ; in other words, 

 that the particle may be regarded as giving up its whole 

 momentum if it be taken into account that after impact it 

 forms part of the body. 



306. It is easy to see how the considerations of the last two 

 articles can be generalized. When any number of impulses 

 act in various directions on a rigid body with a fixed axis, the 

 initial angular velocity will be determined by 



=f > (13) 



where H is the sum of the moments of all the impulses about 

 the fixed axis. 



