How Do the Equations U=mgh and U=Gm1m/R Yield Similar Results?

[Loppyfoot]

Hi, I was wondering this question:

Why does U=mgh and U=Gm1m/R come out to similar values even when their equations are completely different?

Thanks

 

[G01]

The short answer is that this is because the first equation is just a limiting case of the second.

The first equation is and approximation for the potential energy due to gravity near the Earth's surface. This equation is a limiting case of of Newton's universal law of gravity(the first equation), if we assume we are near the Earth's surface.

Let $R=R_e+h$ where Re is the radius of the Earth, and h is your height above the Earth.

Now,

$$U=GM_em\frac{1}{R_e+h}$$

Since h is really small compared to R_e(we are close to the Earth's surface), we can Taylor expand the fraction and keep only the first few terms:

$$U=GM_em\frac{1}{1+h/R_e}=GM_em\frac{1}{R_e}(1-h/R_e)$$

We get two terms, but we can drop the first, as it is a constant, and constants of potential energy don't matter. Let's drop that constant. (This is equivalent to setting the Earth's surface as 0 potential.)

So,$$U=-GM_em\frac{1}{R_e^2}(h)=-mgh$$

with $g=\frac{GM_e}{R_e^2}$

So, near the Earth's surface, both equations must give similar values, because the second is just an approximate limiting case of the first.