Physics equations/06-Uniform Circular Motion and Gravitation/A:history

Newton's law of universal gravitation
Newton published this in 1687, his knowledge of the numerical value of the gravitational constant was a crude estimate. For our purposes, it can be conveniently state as follows :
 * $$ \vec{F}_{12} =

- G {m_1 m_2 \over {\vert \vec{r}_{12} \vert}^2} \, {\hat{r}}_{12} \mbox{ where }G \approx 6.674 \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$

Solution:
 * $$ \vec{F}_{12} $$ is the force applied on object 2 due to object 1
 * $$ G $$ is the gravitational constant
 * $$ m_1 $$ and $$ m_2 $$ are respectively the masses of objects 1 and 2
 * $$ \vert \vec{r}_{12} \vert \ = \vert \vec{r}_2 - \vec{r}_1 \vert $$ is the distance between objects 1 and 2
 * $$ \hat{r}_{12} \ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf{r}_2 - \mathbf{r}_1}{\vert\mathbf{r}_2 - \mathbf{r}_1\vert} $$ is the unit vector from object 1 to 2

(Note: the minus sign is a complexity that is often ignored in simple calculations. Don't fuss with minus signs unless you have to.)

Since the magnitude of the unit vector is "one" $$(|\hat{r}_{12}|=1)$$, the unit vector vanishes when we take the magnitude of both sides of the equation to get:

$$F_{12} = G \frac{m_1 m_2}{r_{12}^2}$$.

Weight and the acceleration of gravity
The force of gravity is called weight, $$\vec w$$, If one of two masses greatly exceeds the other, it is convenient to refer to the smaller mass, (e.g.stone held held by person) as the test mass, $$m_0$$. A vastly more massive body (e.g. Earth or Moon) can be referred to as the central body, with a mass equal to $$m_C$$. It is convenient to express the magnitude of the weight ($$w=|\vec w|$$) as,

$$w = F = G \frac{m_0 m_C}{r^2}= m_0g$$,

where $$g=Gm_C/r^2$$ is called the acceleration of gravity (or gravitational acceleration). Near Earth's surface, $$\vec g = $$ is nearly uniform and equal to 9.8 m/s2. In general the gravitational acceleration is a vector field, meaning that it depends on location, g = g(r) or even location and time, g = g(r,t).

Gravity as a vector field
under construction
 * 1) define the vector field for a single massive point object
 * 2) make analogy to magnetic field as that which causes a torque on a magnet
 * 3) mention temperature and wind velocity fields in meteorology
 * 4) perhaps mention the need for vector calculus on a spherical object (problem solved by Newton, I think)

How G was actually measured
under construction: keep it brief and include an image and a reference to good wikipedia article