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Gravity

Another example is gravity, which differs from the electrostatic force only in its relative weakness and the innocuous-looking fact that it only comes in one sign, namely attractive, whereas the electric force can be either attractive (for unlike charges) or repulsive (for like charges). That is, "There are no negative masses." So all these equations hold equally well for gravity, except of course that we must again shuffle constants of proportionality around to make sure we are not setting apples equal to oranges. In this case we can use some symbol, say  $\Vec{g}$,  to represent the force per unit mass at some position, as we did for  $\Vec{E} =$ force per unit charge, and talk about the "gravitational field" as if it were really there, rather than being what would be there (a force) if we placed a mass there. (Note that  $\Vec{g}$  will be measured in units of acceleration.) Then the role of "dQ/dt" in Eq. (1) is played by  M,  the total mass of the attracting body, and the constant of proportionality is  $4 \pi G$, where  G  is Newton's Universal Gravitational Constant:

$$\oSurfIntS \Vec{g} \cdot d\Vec{A} = 4 \pi G M$$ (18.5)

and

$$g(r) = {GM \over r^2}$$ (18.6)

for any spherically symmetric mass distribution of total mass  M.  Note that we have "derived" this fundamental relationship from arguments about symmetry, geometry and common sense, plus the weird notion that "lines" of gravitational force are "emitted" by masses and are "conserved" in the sense of streams of water - a pretty kinky idea, but evidently one with powerful applications. Be sure you are satisfied that this is not a "circular argument;" we really have derived Eq. (6) without using it in the development at all! Now, besides being suggestive of deeper knowledge, this trick can be used to draw amusing conclusions about interesting physical situations.