One of the neat things about gravity is that you can actually use it to weigh an object provided it has a companion satellite. Here is how you do it very approximately:
The force of gravity is given by Newtons' Law:
G M m
F(gravity) = ---------
r^2
where G is the constant of gravity, M is the main body mass, m is the
satellite's mass and r is the distance between their centers.
We also know that for a simple circular orbit, this force is balanced by the
centrifugal force given by:
m V^2
F(cent) = -------
r
If we set F(gravity) = F(cent) then the little m's cancel which means that the
only thing that the physics depends on are distances, speeds and the mass of
the main body ( the Earth!). A little algebra gives you:
V^2 r
M(earth) = ---------
G
Now, in a circular orbit, the average velocity, V, is just the circumference
divided by the orbit period: 2 pi r / T. So if we substitute in the
previous equation we get:
2 3
4 pi r
M(earth) = -------------
2
G T
The estimated mass of the Earth depends only on the distance from the Earth to
the Moon ( r), and the time it takes the Moon to go once around the Earth, T.
If we use some real numbers in the centimeters-grams-seconds units
: G = 6.67 x 10^-8 dynes/cm^2/gm^2; r = 3.84 x 10^10 cm and T = 27.32 days or
2.3 x 10^6 seconds, we get that the mass of the Earth in grams is about
6.0 x 10^27 grams. The actual number for the mass of the Earth is
5.9 x 10^27 grams. We are of by a small amount
because our little satellite method only works when the mass of
the satellite is much smaller than the main body. The Moon does not have
an insignificant mass compared to the Earth ( it is about 1/81 of the Earth).
This method is so powerful that it can be used to weigh the other planets in the solar system, the Sun, distant stars in binary systems, the Milky Way galaxy, and even entire clusters of galaxies! Welcome to the incredible world of modern physics!