Page:LewisTolmanMechanics.djvu/11

Now let us assume that the whole system is in motion with velocity v in the direction bc. Obviously, merely by making such an assumption we cannot cause the lever to turn, nevertheless we must now regard the length bc as shortened in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$ while ab has the same length as at rest. We must therefore conclude that to maintain equilibrium the force at a must be less than the force at c in the same ratio. We thus see that in a moving system unit force in the longitudinal direction is smaller than unit transverse force in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$, and therefore, by the preceding paragraph, smaller than unit force at rest in the ratio ${\frac {1-\beta ^{2}}{1}}$. It is interesting to point out, as Bumstead^[1] has already done, that the repulsion between two like electrons, as calculated from the electromagnetic theory, is diminished in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$ if they are moving perpendicular to the line joining them, and in the ratio ${\frac {1-\beta ^{2}}{1}}$ if moving parallel to the line joining them.

From the standpoint of the principle of relativity, one of the most interesting quantities in mechanics is the so-called kinetic energy, which is the increase in energy attributed to a body when it is set in motion with respect to an arbitrarily chosen point of rest. Knowing the change of the mass with velocity as given by equation I, the general equation for kinetic energy, ^[2] E' may readily be shown to be

$E'=m_{0}c^{2}\left({\frac {1}{\sqrt {1-\beta ^{2}}}}-1\right)$.

II

1. Bumstead, loc. cit.
2. Consider a body moving with the velocity v subjected to a force f in the line of its motion. Its momentum M and its kinetic energy E' will be changed by the amounts dM = fdt, dE' = fdl = fvdt. Hence dE' = vdM, or substituting mv for M, dE' = mvdv + v²dm. Eliminating m between this equation and equation I, and integrating, gives at once the above equation II.