Page:Elementary Text-book of Physics (Anthony, 1897).djvu/50

sions for ${\tfrac {1}{2}}m_{1}v_{1}^{2}+{\tfrac {1}{2}}m_{2}v_{2}^{2}...,{\tfrac {1}{2}}m_{1}w_{1}^{2}+{\tfrac {1}{2}}m_{2}w_{2}^{2}...$ be formed in a similar manner, and added to the expression just obtained, we have on the left the sum of the kinetic energies of the particles, and on the right the expression

${\tfrac {1}{2}}(U^{2}+V^{2}+W^{2})\sum m+{\tfrac {1}{2}}m_{1}(a_{1}^{2}+b_{1}^{2}+c_{1}^{2})+{\tfrac {1}{2}}m_{2}(a_{2}^{2}+b_{2}^{2}+c_{2}^{2})+...$

The first of these terms expresses the kinetic energy of a mass equal to the sum of all the masses moving with the velocity of the centre of mass. The other terms express the kinetic energies of the separate particles moving with their velocities relative to the centre of mass. We therefore arrive at the following rule: The kinetic energy of a system of particles is equal to the kinetic energy of a mass equal to the sum of all the masses moving with the velocity of the centre of mass, plus the kinetic energies of the separate masses moving with their velocities relative to the centre of mass.

32. Work done by Forces on a System of Particles. Potential Energy.— The forces which act on the particles of a system may be classified as external and internal forces. The external forces arise from the action of bodies outside the system, the internal forces from action between parts of the system. If the resultant of all the forces which act on any one particle be considered as the force which acts on that particle, the particle will acquire kinetic energy, given by the formula $Fs={\tfrac {1}{2}}mv^{2}-{\tfrac {1}{2}}mv_{0}^{2}$, already established (§ 28). If, however, we consider the resultant of the external forces acting on the particle as producing kinetic energy and doing work against the internal forces which act on the particle, the work done by the former will be equal to the kinetic energy gained by the particle plus the work done against the latter. If the internal forces be conservative, the work done against them can be recovered when the external forces cease to act. The action of the external forces in that case gives to each particle potential energy. In case the external forces equilibrate the internal forces for each particle, the velocities of the particles remain constant, no kinetic energy is gained, and the energy given to the system by the work done is wholly potential. In any case the energy gained