MOMENT OP MOMENTUM 297 



for the internal forces occur in equal and opposite pairs which 

 contribute nothing. 



The term ^m / y z \ > which is the sum of the moments 



\ Ol/V Ctl/ I 



of momentum of the separate particles, is called the moment of 

 momentum of the system. 



Thus equation (123) expresses that 



The rate of change of the moment of momentum of any system 

 about any axis is equal to the sum of the moments of the external 

 forces about this axis. 



244. Several important consequences of this theorem follow 

 at once. 



I. If a system of bodies is acted on by no external forces, the 

 moment of momentum about every axis remains constant. 



This expresses the principle known as the conservation of angu- 

 lar momentum. 



The sun affords an instance of a body which may practically be sup- 

 posed to be acted on by no external forces. It is generally supposed that 

 the sun is gradually shrinking in size ; if this is so, we see that its velocity 

 of rotation about its axis must continually increase, in order that its 

 moment of momentum may remain constant. 



II. If all the forces acting on a system are either parallel to a 

 given line, or else intersect this line, then the moment of momentum 

 of the system about this line must remain constant. 



A peg top is acted on only by the reaction at the peg and gravity. The 

 moment of the latter about a vertical line through the peg vanishes, and 

 the moment of the former may be supposed to vanish to a close approxi- 

 mation. Hence the moment of momentum about a vertical through the 

 peg will remain constant, to a close approximation. 



III. If a rigid body is free to rotate about a fixed axis, and if 

 ft> is its angular velocity at any instant, then 



at 



where Ml? is the moment of inertia about the fixed axis, and L is 

 the sum of the moments about this axis, of all the external forces. 



