View source for 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 [[w:gravitational constant|gravitational constant]] was a crude estimate.  For our purposes, it can be conveniently state as follows
<ref>https://en.wikibooks.org/w/index.php?title=Physics_with_Calculus/Mechanics/Newton%27s_Law_of_Gravitation_and_Weight&oldid=1571974</ref>:
:<math> \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}</math>

{{clear}}{{hidden begin|title = Click here if you are not sure what these terms mean|titlestyle = background:lightyellow;}}
'''Solution:'''
: <math> \vec{F}_{12} </math> is the force applied on object 2 due to object 1
: <math> G </math> is the gravitational constant
: <math> m_1 </math> and <math> m_2 </math> are respectively the masses of objects 1 and 2
: <math> \vert \vec{r}_{12} \vert \ = \vert \vec{r}_2 - \vec{r}_1 \vert </math> is the distance between objects 1 and 2
: <math> \hat{r}_{12} \ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf{r}_2 - \mathbf{r}_1}{\vert\mathbf{r}_2 - \mathbf{r}_1\vert} </math> 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" <math>(|\hat{r}_{12}|=1)</math>, the unit vector vanishes when we take the magnitude of both sides of the equation to get:

<math>F_{12} = G \frac{m_1 m_2}{r_{12}^2}</math>.
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====Weight and the acceleration of gravity====
The force of gravity is called '''weight''', <math>\vec w</math>,  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''', <math>m_0</math>.  A vastly more massive body (e.g. Earth or Moon) can be referred to as the '''central body''', with a mass equal to <math>m_C</math>.  It is convenient to express the magnitude of the weight (<math>w=|\vec w|</math>) as,

<math>w = F = G \frac{m_0 m_C}{r^2}= m_0g</math>,

where <math>g=Gm_C/r^2</math> is called the acceleration of gravity (or gravitational acceleration).  Near Earth's surface, <math>\vec g = </math> is nearly uniform and equal to 9.8 m/s<sup>2</sup>.  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
#define the vector field for a single massive point object
#make analogy to magnetic field as that which causes a torque on a magnet
#mention temperature and wind velocity fields in meteorology
#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

<references/>
[[Category:Physics equations]]