Thought experiments with gravitational potential energy
I have come up with two thought experiments regarding gravitational potential energy that confuses me somewhat, and that I'm hoping can be cleared up.
Experiment 1
I take an object and raise it some distance above ground level, where I leave it, so it now has some potential energy corresponding to its height. Then I dig a hole in the ground directly below the object. My first question is, if this will increase the potential energy of the object? My guess is yes, because if the object falls, it will have the ability to make a larger impact at the bottom of the hole than it would have had upon impact with the Earth at ground level.
If my guess is right, then I continue by deciding to push the object sideways, with a movement parallel to the ground. At one point in this movement, it will no longer be above the hole, and there will be a sharp decrement in the potential energy of the object. The energy will make a discontinuous jump, which I admittingly find a bit uncomfortable - but my actual second question is, when there was a decrement in the potential energy, where did it go?
Experiment 2
I lift an object a distance $h$ above ground level, increasing its potential energy, but I keep lifting it indefinitely. 483
I imagine that at one point it will be so far removed from the gravitational pull of Earth that its potential energy starts to decrease again. Indeed, according to this wiki page, the acceleration due to gravity is proportional to the inverse squared distance to the center of the Earth, so if $d$ is this distance, then $g = \frac{k}{d^2}$ for some constant $k$. If I let $r$ be the radius of the Earth, the potential energy is given by \begin{align} E &= mgh\ &= m\frac{k}{d^2}h\ &= m\frac{k}{(r+h)^2}h. \end{align} As a function of $h$, this function turns out to be increasing for $hr$, so as I am raising the object past the point $h=r$, it starts losing energy. My questions are, where is this energy going? And where is the energy that I spend raising the object going, when it is past the point $h=r$? 483