3.1. Equivalence between [2] and [3] http://www.ck12.org
3.1 Equivalence between [2] and [3]
The formulas above look pretty different, but the main conceptual split is that thegin the first formula above ([2])
varies from planet to planet, but theGin [3] constant throughout the universe — in fact, it’s called theGravitational
Constant. You might think that this makes the second formula more fundamental than the first, and you would be
right. The first is actually a "special case" of the second. That is, the [3] always holds, but [2] only holds when
certain conditions are met: that is, you are at the surface of a spherical body. In this case they are equivalent, but [2]
is obviously simpler.
It is important to see the relationship between [2] and [3], since it is typical of the stuff of physics. If [3] is the more
fundamental equation, we should be able to start with it and derive [2]. First, we will make a minor simplification:
we will assume that the "object of interest" starts at the surface of the earth, and not some arbitrary height near
it (now we don’t have to deal with the deltas and hairier details without sacrificing any of the content). Then the
question we want to answer is: if we raise this object from the surface to a heighth, what will its gravitational
potential energy change by?