Monday, 8 January 2018

newtonian gravity - Is gravitational potential energy proportional or inversely proportional to distance?


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=GMmrGMmr+h


We rearrange this to get:


ΔU=GMm(1r1r+h)=GMmhr2+rh=GMr2mh1+h/rGMr2mh


where the last approximation is because hr so 1+h/r1. And since GM/r2 is just the gravitational acceleration g at a distance r, we get:



ΔU=gmh


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