Gravitational potential energy confusion

[sciencegem]

Hi,
I've managed to get myself confused over the simplest thing. Intuitively I'd think that gravitational potential energy is proportional to distance as described by the near-body equation GPE=-mgh. The alternative for farther bodies (-G*M*m/R) has me confused because it is inversely proportional to distance. In other words (the way I see it) GPE grows the "higher up you go" with one equation while it shrinks perplexingly with the other. I know I'm probably just missing something so basic it's pathetic, I'm very frustrated with myself over this. Any hints on how to resolve my mental dilemma are appreciated.
Thanks!

 

[Dale]

You are probably missing the minus sign.

 

[sciencegem]

I'm not. What I mean is, say, -m*g*.01 is small whereas -G*M*m/.01 is big.

 

[HallsofIvy]

Have you taken Calculus? The "potential energy" is the integral of the force function with respect to distance (conversely, the force is the derivative of the potential energy). The integral of $x^n dx$, for any constant x, is $x^{n+1}/(n+1)$.

For movement close to the earth, where the force can be taken to be constant, the integral is just the $\int -mg dx= -mgx$, taking the "n" above to be 0 so that n+ 1= 1. But if we are talking about greater distances where we have to use the more general $-GmM/r^2$, we have n= -2 so that n+1= -1 and the integral is $Gmm/r$.

 

[256bits]

I'm not. What I mean is, say, -m*g*.01 is small whereas -G*M*m/.01 is big.

Are they?
What is g equal to in terms of G and M.
Substitute that in your GPE=-mgh equationm.

 

[Dale]

I'm not. What I mean is, say, -m*g*.01 is small whereas -G*M*m/.01 is big.

That is exactly what you are missing. It is NOT -mgh, it is +mgh. As h increases mgh increases. As r increases -GMm/r also increases (-4 is greater than -4000). They both increase as height increases.