109-113] CONSERVATION OF MOMENTUM. 109 



112. Conservation of angular momentum. When the 

 sum of the moments of the external forces on a system about a 

 particular axis is zero, the sum of the moments of the kinetic 

 reactions about the same axis is zero, and therefore the moment of 

 momentum of the system about the axis is constant. 



A simple example is afforded by the motion of a particle acted 

 on by a force always passing through a point whose position 

 relative to the frame of reference is fixed. For such a case we 

 have, as in Article 50, an equation expressing that the moment of 

 the velocity about the point is constant. Thus the equation 

 pv = h for central accelerations may be regarded as an example of 

 the principle just stated. 



When all the external forces acting on a system reduce to a 

 resultant force at the centre of inertia without any couple, the 

 sum of the moments of the external forces about any axis through 

 the centre of inertia vanishes ; the rate of increase of the moment 

 of momentum about any such axis therefore vanishes, and the 

 moment of momentum of the system about any such axis is 

 constant. We note that in this statement it is implied that the 

 axis occupies a fixed position relative to the frame of reference. 



113. Equations of Impulsive Motion. As in Article 

 105, let X + X' be the sum of the resolved parts parallel to the 

 axis x of all the forces, external and internal, that act on a 

 particle m ; and, as in Article 82 suppose X and X' do not remain 

 finite at time t, but that the impulses of X and X' are finite, 

 or that X and X', defined by the equations 



/t + T Ct + $T 



Xdt = X, Lt T=0 X'dt 

 J-jT Jt-tt 



are finite. Let x and j be the resolved parts parallel to the axis 

 x of the velocity of m just after the instant t and just before this 

 instant respectively. Then we have the equation 



m(x-) = X + X'. 



In like manner the impulsive changes of velocity parallel 

 to the axes y and z will be determined by equations which may be 

 written 



