How can I calculate the Earth's mass knowing only the distance to the Moon and the length of the lunar month?


This is one of the most exciting aspects of physics that I encountered at an early age, and that has enchanted me ever since! I still find it amazing that by knowing a distance and a time that you can figure out a mass and weigh a planet. Here's how it's done, in an approximate way. For an object of mass, m, in a circular orbit or radius, R, the force of gravity is balanced by the centrifugal force of the bodies movement in a circle at a speed of V, so from the formulae for these two forces you get:

               G M m
F(gravity) =  -------
                 2
                R

and
                        2
                     m V
F(Centrifugal) =    -------
                      R

If you balance the forces so that F(gravity) = F(centrifugal) you get after a little algebra:
                   2          G M
                  V     =    ------
                               R

Amazingly enough, the mass of the satellite, m, has canceled from both sides of the equation and this means it doesn't matter how big the satellite orbiting the planet is at all, only its distance, R, and speed, V, matter! Now, velocity in a circular orbit is just 2 x pi x R the circumference, divided by the orbital time of the satellite, T, so you get:

                 2   2
             4 pi   R           G M
             ----------    =   ------
                  2
                 T                R


and with a little algebra you can solve for the planet's mass, M, and get


                     2   3
                 4 pi   R
                -----------    =   M
                      2
                  G  T

This works if the mass of the satellite is much less than the mass of the planet you are weighing. Now, all you have to is substitute pi = 3.14159, and the period of the Moon, T, in seconds, and its distance from the center of the Earth, R, in centimeters, and then use G = 6.6 x 10^-8, and you will get an answer for the mass of the Earth in grams that is pretty close to its actual value. You can also weigh the Sun using one of the planets, or the Milky Way using the orbit of the Sun ( 8,700 parsecs and 224 million years).
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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the
NASA IMAGE/POETRY Education and Public Outreach program.