# Page:Popular Science Monthly Volume 79.djvu/257

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ATTEMPTS TO EXPLAIN GRAVITATION

according to a definite law (the attraction between two bodies being directly proportional to the product of the masses and inversely as the square of the distance, if the dimensions of the bodies are small compared with the distance between them). Having enunciated this law he proceeded to verify it by studying the motion of the moon. The moon revolves in an orbit that is nearly circular and to keep it in this orbit there must be an acceleration toward the earth equal to ${\displaystyle \scriptstyle {V^{2}/r}}$ where ${\displaystyle \scriptstyle {V}}$ is the moon's orbital velocity and ${\displaystyle \scriptstyle {r}}$ is the distance from the earth to the moon (approximately 240,000 miles). In place of ${\displaystyle \scriptstyle {V}}$ we may put ${\displaystyle \scriptstyle {2\pi r/t}}$, where ${\displaystyle \scriptstyle {t}}$ is the time of one revolution (27.3 days). Hence the acceleration toward the earth equals

${\displaystyle \scriptstyle {{\frac {V^{2}}{r}}={\frac {\left({\frac {2\pi r}{t}}\right)^{2}}{r}}={\frac {4\pi ^{2}r}{t^{2}}}={\frac {4\times 9.86\times 240,000\times 5280}{(27.3\times 86,400)^{2}}}=.0089~{\text{feet}}/{\overline {{\text{sec}}^{2}.}}}}$

The acceleration the earth should exert, if Newton's law be true, at a distance of 240,000 miles (60 times the earth's radius)${\displaystyle \scriptstyle {=32.16/60^{2}=.0089~{\text{feet}}/{\overline {{\text{sec}}^{2}}}}}$ where 32.16 ${\displaystyle \scriptstyle {{\text{feet}}/{\overline {{\text{sec}}^{2}}}}}$ is the acceleration at the surface of the earth. The verification in the case of the moon is complete. Hence we have the mathematical statement of the law: ${\displaystyle \scriptstyle {F}}$ (the force) ${\displaystyle \scriptstyle {=Mm/r^{2}\cdot G}}$ where ${\displaystyle \scriptstyle {G}}$ is a constant depending upon units only. We say nothing about the quality of the matter but only the quantity, and the distance. Notice also that there is no factor in the equation referring to the nature of the intervening medium.

It may not be out of place to call attention to the universality of the law. There are a few slight discrepancies between observed and calculated values, but as a whole it is fully attested by observation.

In referring to the magnitude of gravitational force consider first small bodies and later astronomical bodies. We know to-day that the radiation from the sun exerts a pressure. Kepler suggested this three centuries ago and one hundred and fifty years later the great mathematician Euler adopted his suggestion in accounting for the repulsion of comets' tails. So delicate is this pressure that it was not discovered until recently (1900). Albeit this pressure is very small as bodies diminish in size, we reach a limit at which it predominates over gravitation. This is due to the fact that gravitation is proportional to the mass (the cube of the linear dimension) while radiation-pressure is proportional to the surface (the square of the linear dimension).

When we consider electrons we find that the gravitational attraction between two electrons is insignificant compared with electrical attraction. The electrical force in air between two negative electrons one centimeter apart is equal to ${\displaystyle \scriptstyle {(4.5\times 10^{-10})^{2}=20\times 10^{-20}}}$ dynes, if we take the charge on an electron to be ${\displaystyle \scriptstyle {4.5\times 10^{-10}}}$ c.g.s. electrostatic units.

The gravitational attraction between two electrons at a distance of one centimeter${\displaystyle \scriptstyle {=10^{-27}\times 10^{-27}\times 6.6\times 10^{-8}=6.6\times 10^{-62}}}$ dynes,