Forgive me if my question seems silly, but I am quite baffled. Suppose you have a satellite orbiting a horizontal swing planted into the ground and we want to find the velocity with which the satellite must be moving in order to succeed in taking a fully circular path around the swing (no failure followed by parabolic fall).
Indeed if we solve for $F = m(v^2)/R$, where $F$ can be equated to $g$, we can also solve for the velocity; all nice and good, and the nature of this method makes some intuitive sense but there is a problem (at least for me): the force that the satellite experiences is an inward force towards the center, and therefore the force described in the aforementioned equation is the centripetal force.
However, this conflicts with the method of solving the question, since at the top of the path, the centripetal Force and the gravitational force will be both pointing downwards. Therefore shouldn't the satellite actually be "doubly" compelled to accelerate to the ground? But clearly when we for for $v$ with $g$, we assume that they cancel each other out - hence they point in opposite directions. The idea that the cancel also "explains" the intuition that at the top of the circular path, the satellite will be feeling the most "weightless" it will throughout its revolution.
Additionally from a different reference frame / non centrifugal one, it makes sense that at the top, the most force will be on the satellite to turn its path downward as it is after that instant that the satellite sees its downward acceleration. I am, essentially confused with how to handle centripetal/centrifugal "fictitious" forces in this worked example.