Page:Elementary Text-book of Physics (Anthony, 1897).djvu/48

${\frac {\xi -\xi _{0}}{t-t_{0}}}\sum m=\sum {\frac {m(x-x_{0})}{t-t_{0}}};$
${\frac {\eta -\eta _{0}}{t-t_{0}}}\sum m=\sum {\frac {m(y-y_{0})}{t-t_{0}}};$	(22)
${\frac {\zeta -\zeta _{0}}{t-t_{0}}}\sum m=\sum {\frac {m(z-z_{0})}{t-t_{0}}}.$

The terms on the right are the components of momentum of the separate particles, and the equations express the law that the velocity of the centre of mass of a system of particles is equal to the resultant obtained by compounding the momenta of the separate particles and dividing it by the sum of all the masses of the system.

Representing the component velocities of the centre of mass by $U,V,W$, and those of the separate particles by $u,v,w$, the rule just given may be expressed by $U\sum m=\sum u,V\sum m=\sum v,W\sum m=\sum w$.

It the velocities of some or all of the particles vary, the velocity of the centre of mass will in general vary also. Its acceleration depends upon the accelerations of the separate particles. Letting $U$ and $U_{0}$, etc., represent the component velocities at the times $t$ and $t_{0}$, we may express the component accelerations of the centre of mass by

${\frac {U-U_{0}}{t-t_{0}}}\sum m=\sum {\frac {m(u-u_{0})}{t-t_{0}}};$
${\frac {V-V_{0}}{t-t_{0}}}\sum m=\sum {\frac {m(v-v_{0})}{t-t_{0}}};$	(23)
${\frac {W-W_{0}}{t-t_{0}}}\sum m=\sum {\frac {m(w-w_{0})}{t-t_{0}}}.$

The terms on the right represent the components of the forces which act on each particle of the system, and the equations express the law that the acceleration of the centre of mass of a system of particles is equal to the resultant of all the forces which act on the separate particles divided dy the sum of the masses of the particles. This law may be otherwise expressed by saying that the acceleration of the centre of mass is the same as that which would be given to