98-101] PARALLEL AXES. 103 



100. Linear Momentum. The momentum of a particle 

 has been defined to be a vector localised in the line of the velocity 

 of the particle, with the sense of the velocity, and of magnitude 

 equal to the product of the mass and the velocity. 



The resolved parts parallel to the axes of the momentum of a 

 particle of mass m at the point (x, y, z) are mx, my, mz. 



The momenta of the particles of a system can be replaced by 

 a resultant momentum localised in a line through the centre of 

 inertia, and a couple. The resultant momentum has as its resolved 

 parts parallel to the axes 



2 (mx), 2 (yny\ 2 (mz). 



Differentiating the equations such as #2m = 2(-w#), we obtain 



such equations as xZm = 2 (mx), where x stands for -j- . 



at 



We thus see that the resultant momentum is equal to the 

 momentum of a particle of mass equal to the mass of the 

 system, placed at the centre of inertia, and moving with the 

 velocity of the centre of inertia. 



The momentum just described is called the linear momentum 

 of the system. The particle at the centre of inertia just described 

 will be referred to as the &quot; particle G&quot; 



101. Kinetic Reaction. The kinetic reaction of a particle 

 has been denned as a vector localised in the line of the acceleration 

 of the particle, with the sense of the acceleration, and of magnitude 

 equal to the product of the mass and the acceleration. 



The resolved parts, parallel to the axes, of the kinetic reaction 

 of a particle of mass m at the point (x, y, z) are mx, my, mz. 



The kinetic reactions of the particles of a system can be 

 replaced by a resultant kinetic reaction localised in a line through 

 the centre of inertia, and a couple. The resultant kinetic reaction 

 has as its resolved parts parallel to the axes 

 2 (mx), 2 (my), 2 (mz). 



Differentiating the equations such as x^m = 2 (mx) we find 

 such equations as #2ra = 2 (ma;). 



We thus see that the resultant kinetic reaction is that of the 

 particle G, i.e. it is the same as that of a particle of mass equal 

 to the mass of the system placed at the centre of inertia and 

 moving with it. 



