We know that if an object has been lifted a distance h from the ground then it has a potential energy change:
ΔU=mgh
so h is proportional to ΔU.
However, we have also the gravitational potential energy law:
U=−GMmr
where the distance is inversely proportional to the potential energy.
What did I miss? Is the distance of the object proportional or inversely proportional to the potential energy?
Answer
The formula:
ΔU=mgh
is an approximation that applies when the distance h is small enough that changes in g can be ignored. As you say, the expression for U is:
U=−GMmr
So the change when moving a distance h upwards is:
ΔU=GMmr−GMmr+h
We rearrange this to get:
ΔU=GMm(1r−1r+h)=GMmhr2+rh=GMr2mh1+h/r≈GMr2mh
where the last approximation is because h≪r so 1+h/r≈1. And since GM/r2 is just the gravitational acceleration g at a distance r, we get:
ΔU=gmh
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