The Gravitational Field

We define the gravitational field to be the cause of the gravitational force.

We define it conveniently to be the force per unit mass

$$\vec{\bf g}(\vec{\bf r}) = - \frac{GM}{r^2}\hat{r} = \frac{\vec{\bf F}}{m}$$ (13)

The gravitational field at the surface of the earth is:

$$g(r) = \frac{F}{m} = \frac{GM_E}{R_E^2}$$ (14)

This equation can be used to find $g$, $R_E$, $M_E$, or $G$, from any of the other three, depending on which ones you think you know best. $g$ is easy. $R_E$ is actually also easy to measure independently and some classic methods were used to do so long before Columbus. $M_E$ is hard! What about $G$?

Henry Cavendish made the first direct measurement of $G$ using a torsional pendulum and some really massive balls. From this he was able to ``weigh the earth'' (find $M_E$). By measuring $\Delta\theta(r)$ ($r$ measured between the centers) it was possible to directly measure $G$. He got 6.754 (vs 6.673 currently accepted) $\times 10^{-11}$ N-m$^2$/kg$^2$. Not bad!