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 :

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

Click here if you are not sure what these terms mean

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,

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

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