66, 67, 68] DYNAMICS. IMPULSIVE WRENCH. 225 



applied to the system. That is, 



dM.. dM 



49) 



dH dH 



Integrating these equations with respect to the time, 



we may, in the sense of 27, call the momentum the impulsive 

 ^vrench of the system. Physically, then, the momentum that a system 

 possesses at any instant is equal to the impulsive wrench necessary 

 to suddenly communicate to it when at rest the velocity -system that 

 it actually possesses. As a prelude to the dynamics of a rigid body 

 we must accordingly study the properties of the momentum or 

 impulsive wrench of a body possessing a given instantaneous twist- 

 velocity. 



All the systems of vectors in question may be reduced to the 

 screw type, and their respective screws are in general all different. 

 Thus we may speak of the instantaneous velocity -screw and instan- 

 taneous axis, the momentum screw, and the force -screw. As the 

 body moves, all these screws change both their pitch and position 

 in the body, describing ruled surfaces both in the body and in space. 

 The integration of the differential equations of motion 49) will enable 

 us to find these surfaces. The kinematical description of the motion 

 will be complete if we know the two ruled surfaces described in 

 space and in the body by the instantaneous axis, together with such 

 data as will give their mutual relations at each instant of time. 



68. Momentum of Rigid Body. The properties of the 

 momentum of a rigid body are conveniently investigated by the 

 consideration of the velocity -system as an instantaneous screw -motion. 

 Let F be the velocity of translation, and co of rotation. Then every 

 particle of mass m has one component of momentum parallel to the 

 axis of the instantaneous twist (which we will take for Z-axis), 

 equal to mv z = mV and the resultant for all is 



51) M 2 = ZmV=VZm = MV, 



WEBSTER, Dynamics. 15 



