How can you measure the mass of the Earth or any other planet?

Asked by: Ricardo Kurihara

Answer

Wow, what a good question. Do you know what is so good about it? You ask about the mass of the
Earth and not about its weight. I suppose that is because you already know that as the Earth
orbits the Sun it is weightless. But it certainly is not massless.

Because the Earth has mass it exerts a force of gravity and the magnitude of this force of gravity
is determined by the mass of the Earth and the distance away from the center of the Earth another
object is. Now, you know that if you drop something it falls. But did you know that as it falls
it accelerates? That means that as it falls it falls faster and faster and faster until it hits
the ground. On earth at sea level objects that fall speed up at a rate of 9.8 m/s every second.
So, for every second something falls it falls 9.8 m/s faster with each passing second. You can see
that at this rate you are going pretty fast real soon. (9.8 m/s is a little more than 20 miles per
hour, so in three seconds you are already going 60 mph!)

If you go way high up a mountain the acceleration due to gravity is a little less than 9.8 m/s. If
you go higher still the accleleration is even less and so on. You can tell how high you are by how
fast objects that fall accelerate.

Now, here is the really neat part: all objects accelerate to the Earth at exactly the same rate
given the same distance from the Earths center. In other words, even the Moon is accelerating
towards the Earth at a rate that is appropriate for its distance from the Earth's center! If
another object, say with the mass of this desktop computer, were to be placed in the moon's orbit
it would circle the Earth at exactly the same speed as the moon. Why? Because both objects are
accelerating towards the Earth and all objects, regardless of their mass, accelerate towards the
Earth at exactly the same rate.

So here is where it all comes together: If you know how far an object is from the center of a
planet or a sun and if you know the rate of acceleration that that object is accelerating towards
that planet or that sun you can easily calculate the mass of that planet or sun because only an
object of that planet's or sun's mass can accelerate an object towards it at that rate at that
distance.

Isn't that amazing? This procedure was actually used to calculate the mass of the earth and it is
used to calculate, well, the mass of any orbited body. In fact, we did not know the mass of the
planet Mercury until we put a satellite around it.

What is that magic equation? Well, there are actually several. Here is one that can be used of
you know the time it takes an object to orbit the object you want to find the mass of (T) and if
you also know the distance between the centers of both objects (r):
M=4 pi^{2}*r^{3}/T^{2G}. 'M' is the mass of the orbited object in kilograms; 'r' is the distance between
the centers of the two ogjects in meters; 'T' is the time it takes the orbiting object to orbit the
planet or sun you want the mass of and 'G' is the Universal Gravitational Constant
(6.672 x 10^{-11} Nm^{2}/kg^{2}) and pi is 3.14. Note that all this tells you is the mass of the orbited object. This will
not tell you the mass of the orbiting object. There is no way to calculate that without finding
something to orbit it!
Answered by: Tom Young, M.S., Science Teacher, Whitehouse High School, Texas