Why is Gravity 1/r^2? Gauss's law and Gravity

Gravity is obviously a big deal in physics and in life.  But in class we are used to just one idea, which is Newton's law of universal gravity, F = -GMm/r^2.  This is usually just handed to students and we start using it to solve all sorts of problems, but here I wanted to justify WHY gravity behaves this way, with the inverse-square law of 1/r^2.  It turns out it is a result of geometry!

This video attempts to introduce the notion of Gauss's law, which is normally a big topic in electricity.  But it can be applied to gravity, as well.  Gauss's law is a formal way of defining something called flux, or the amount of stuff flowing through an area.  Mathematically, flux = (stuff)(area).  In our case, the 'stuff' is the gravitational field that all particles and massive objects creates all around itself, and will create the forces we experience when two or more objects are near each other.

We look at spheres, cylinders, and flat objects.  Gauss's law then shows us that the gravitational fields will behave differently with distance from the object.  Gravity is NOT always 1/r^2!!  This is just for spherical objects and point masses.  If the earth were cylindrical, for instance, gravity would fall off more slowly, as 1/r.  And if earth was flat, then gravity would be uniform, or constant, no matter how close or far away you were!  Check out what Gauss's law looks like and where these results come from.

Note there are how to videos for Gauss's law in electricity, specifically for electric fields, on our class blog.