# 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 [1]:

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

Solution:

${\displaystyle {\vec {F}}_{12}}$ is the force applied on object 2 due to object 1
${\displaystyle G}$ is the gravitational constant
${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are respectively the masses of objects 1 and 2
${\displaystyle \vert {\vec {r}}_{12}\vert \ =\vert {\vec {r}}_{2}-{\vec {r}}_{1}\vert }$ is the distance between objects 1 and 2
${\displaystyle {\hat {r}}_{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}}$ 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" ${\displaystyle (|{\hat {r}}_{12}|=1)}$, the unit vector vanishes when we take the magnitude of both sides of the equation to get:

${\displaystyle F_{12}=G{\frac {m_{1}m_{2}}{r_{12}^{2}}}}$.

#### Weight and the acceleration of gravity

The force of gravity is called weight, ${\displaystyle {\vec {w}}}$, 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, ${\displaystyle m_{0}}$. A vastly more massive body (e.g. Earth or Moon) can be referred to as the central body, with a mass equal to ${\displaystyle m_{C}}$. It is convenient to express the magnitude of the weight (${\displaystyle w=|{\vec {w}}|}$) as,

${\displaystyle w=F=G{\frac {m_{0}m_{C}}{r^{2}}}=m_{0}g}$,

where ${\displaystyle g=Gm_{C}/r^{2}}$ is called the acceleration of gravity (or gravitational acceleration). Near Earth's surface, ${\displaystyle {\vec {g}}=}$ 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