I was learning about inertial frames amongst other things, and I somehow came up with this question. Imagine we have a pole, made of imaginary very strong material that will never bend, placed at the equator and extending into space for a great length. I then climb this pole to certain heights, and release. The question is, at different heights, what exactly does my motion look a) in the perspective of Earth and b) in perspective of the pole.
My hypothesis is that there should be a point at which I would have enough speed (since the pole is rotating with the Earth at one rotation per day) that if I let go of the pole, I would simply enter geostationary orbit, and stay right beside the pole. So from the earth's perspective, I would stay right beside the pole, and from the pole's perspective I would just hang right beside where I let go.
Below this point however, I should drop towards earth, but the first question is whether I would actually make contact with earth, or whether at a lower altitude I would enter elliptical orbit instead? I coded this to test what happens at altitudes below the geostationary point (if it exists)(red is my path, blue is the path of the pole), and apparently it enters elliptical orbit. Then again my code isn't that great. Another point to note is that in this simulation, I go forward, past where the pole is at.
Much above the 'geostationary' point, there will be a region where you will simply have enough velocity to have escape velocity, and escape the gravitational field. I'm 98% sure on this point. Just above the 'geostat' point but below the 'escape' point (which should be root2 times the geostat point?) I'm not too sure what happens here. One conjecture is that the circular orbit deforms into other orbits, such as elliptical. Just pulling at straws, but perhaps just above and just below the 'geostat' point, are both elliptical orbits, but earth is different focus for both (as ellipse has two foci? maybe)
My main question is, how do we determine using actual physics and not computers the path? It seems to me that below the 'geostat' point I will go faster than the pole in respect to the earth, but above the 'geostat' point I will go slower than the pole in respect to the earth. (just from running dodgy simulations.) I had the idea of using angular momentum and velocity to try and attempt this question (as radius decreases, moment of Inertia decreases, angular velocity goes up? not sure if applicable) but I still had no clue how to start to attempt. Also, since gravity ONLY acts toward the centre of the planet, how exactly do we get this deviation to the front or back of the pole? This is my current model: