I have two questions.
1) The first one has to do with the formula for deriving Gravitational Potential Energy. I learned that, for the derivation of Gravitational Potential Energy given large distances, we have to use the mathematical analytic way to derive an expression for it at a given distance.
To do so, you need integrate F dot dr from r to infinity. However, what I don't understand are as follows:
Why do we need to take the zero point at r = infinity? Why can't I take it from any arbitrary point to get a general expression for its GPE when I integrate?
Why is the work required to push an object to that height equal to the force due to gravity times the distance? Don't I need to apply a force that overcomes the force due to gravity to even raise it to begin with? Fg * h is definitely greater in magnitude than fg, but if I'm applying work to an object equal to Fg * h in the opposite direction of where it wants to move (towards the dominant object's COM) how do I know the magnitude of that work is sufficient to do so?
If I was trying to figure out how much energy I need to give an object to raise it from one point in space to another relative to, say the Earth, I could take the change in energy from the two points. If its energy at its initial point is 2 and the energy at the point I want it to be at is 8, I need to supply 6 joules to it. But how do I reconcile that with the derivation from the above paragraph?
2) The second question has to do with zero points for potential energy. Is this allowed because, as long as the distance from each object relative to another is the same no matter where I place a zero point, everything resolves? If at point A, object 1 is 2 units from point A and object 2 is 5 units from point A (all in, say, the x axis), then I'm not cheating by taking point B to be at object 1's position and saying object 2 is now 3 units from point B, right? Wouldn't its potential energy then change here though? That's okay because it's all relative, right? But the magnitude changes. that's okay?