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:


$$\Delta U = mgh $$


so $h$ is proportional to $\Delta U$.


However, we have also the gravitational potential energy law:


$$ U= -\frac{G M m}{r} $$


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:


$$ \Delta U = mgh $$


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:


$$ U= -\frac{G M m}{r} $$


So the change when moving a distance $h$ upwards is:


$$ \Delta U = \frac{GMm}{r} - \frac{GMm}{r + h} $$


We rearrange this to get:


$$\begin{align} \Delta U &= GMm \left( \frac{1}{r} - \frac{1}{r + h} \right) \\ &= GMm \frac{h}{r^2 + rh} \\ &= \frac{GM}{r^2} m \frac{h}{1 + h/r} \\ &\approx \frac{GM}{r^2} m h \end{align}$$


where the last approximation is because $h \ll r$ so $1 + h/r \approx 1$. And since $GM/r^2$ is just the gravitational acceleration $g$ at a distance $r$, we get:



$$ \Delta U = g m h $$


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