In addition to creating the three laws of motion, Newton also discovered the law of gravitational attraction between two objects.
(Eq 1) $f=G\frac{m_1m_2}{r^2},~G=66.73(10^{-12})\frac{m^3}{kgs^2}$
f = attraction force
G = Universal Gravitational Constant
m1 = mass of first object
m2 = mass of second object
r = distance between objects
All objects will have a gravitational attraction towards one another. However, it requires an object with considerable mass, such as a planet, for that attraction to be noticeable.
Example
The earth has a mass of $5.972(10^{24})kg$ and the moon has a mass of $7.348(10^{22})kg$. The distance between the earth and the moon is approximately $384.4(10^6)m$. Determine the gravitational attracting between the earth and the moon.
Solution
$f=66.73(10^{-12})\frac{(5.972(10^{24}))(7.348(10^{22}))}{384.4(10^6)}=1.98(10^{20})N$
As said above all objects that have mass have a gravitational attraction to each other. This includes you and the earth. The attraction between the earth and a person results in that person’s weight. This is why astronauts weight less on the moon than they would on the earth. It is important to note though that even though your weight can change on different planets your mass will remain the same. If the mass of the object is known as well as the planet’s gravitational constant than its weight can be calculated using the equation below.
(Eq 2) $W=mg,~g=9.81/frac{m}{s^2}~or~32.2\frac{ft}{s^2}$
m = mass
g = gravitational constant
W = Weight
The gravitational constant g can change depending on the altitude do to equation 1. However, it is universally accepted that the gravitation constant on earth is 9.81$\frac{m}{s^2}$ or 32.2 $\frac{ft}{s^2}$.
For standard international units mas will always be expressed as kilograms. Imperial units however the mass can either be expressed as slugs or lbm. lbm will equal lbf on earth, however, off of earth this statement will not hold true.
Example
A cannon balls mass is 18.9 kg. How much would that cannon ball weight on earth and how much will it weight on the moon?
Solution
Weight on earth
$W=18.9(9.81) = 185.4~N$
Weight on the moon
$W=18.9(1.62) = 30.618~N$
