Page:CarmichealMass.djvu/9

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S_1 bears to the unit of velocity [acceleration] on S_2 the ratio 1:1\left[1:\sqrt{1-\beta^{2}}\right] or 1:\sqrt{1-\beta^{2}}\left[1:1-\beta^{2}\right] according as the motion is parallel to I or perpendicular to I (MVLR).

Let us use F to denote force. Then from the dimensional equation

F=\frac{ML}{T^{2}},

we shall be able to draw an interesting conclusion concerning the measurement of force.

Suppose that an observer B on a system S_2 carries out some observations concerning a certain rectilinear motion, measuring the quantities M', L', T', so that he has the equation

F'=\frac{M'L'}{T'^{2}}.

Another observer A on a system S_1 (having with respect to S_2 the velocity v in the line l) measures the same force, calling it F. Required the value of F in terms of F', when the motion is parallel to l and when it is perpendicular to l, the estimate being made by A.

When the motion is perpendicular to l — that is, when the force acts in a line perpendicular to l — we have

F_{1}=\frac{ML}{T^{2}}=\frac{M'\sqrt{1-\beta^{2}}\cdot L'}{T'^{2}\left(1-\beta^{2}\right)}=\frac{F'}{\sqrt{1-\beta^{2}}}

When the motion is parallel to l we have


F_{2}
=\frac{ML}{T^{2}}
=\frac{M'\left(1-\beta^{2}\right)^{\frac{3}{2}}\cdot L\sqrt{1-\beta^{2}}}{T'^{2}\left(1-\beta^{2}\right)}=\left(1-\beta^{2}\right)F'.

These results may be stated in the following theorem:

Theorem IV. In the same systems of reference as in theorem III., let an observer on S_2 measure a given force F' in a direction perpendicular to l and in a direction parallel to l, and let F_1 and F_2 be the values of this force as measured in the first and second cases respectively by an observer on S_1. Then we have

F_{1}=\frac{F'}{\sqrt{1-\beta^{2}}},\ F_{2}=\left(1-\beta^{2}\right)F'

\scriptstyle\left(MVLRC_{1}C_{2}\right).

It is obvious that a similar use may be made of the dimensional equation of any derived unit in determining the relation which exists between this unit in two relatively moving systems of reference.