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 :

${\overrightarrow{F}}_{12}=-G\frac{m_1{m}_2}{{|{\overrightarrow{r}}_{12}|}^2}{\hat{r}}_{12}\text{ where }G\approx 6.674\times 10^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$

Template:ClearTemplate:Hidden begin Solution:

${\overrightarrow{F}}_{12}$ 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

$|{\overrightarrow{r}}_{12}|=|{\overrightarrow{r}}_2-{\overrightarrow{r}}_1|$ is the distance between objects 1 and 2

${\hat{r}}_{12}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{{𝐫}_2-{𝐫}_1}{|{𝐫}_2-{𝐫}_1|}$ 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:

$F_{12}=G\frac{m_1{m}_2}{r_{12}^2}$. Template:Hidden end

Weight and the acceleration of gravity

The force of gravity is called weight, $\overrightarrow{w}$, 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 ($w=|\overrightarrow{w}|$) as,

$w=F=G\frac{m_0{m}_C}{r^2}=m_{0}g$,

where $g=G{m}_C/r^2$ is called the acceleration of gravity (or gravitational acceleration). Near Earth's surface, $\overrightarrow{g}=$ 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