Consider a central force F(arrow on top) = F(r)r(with hat on top). Some of the most common forces are central forces,?

OK OK first a bit about notation in plain text. Get rid of all that "arrow on top" and "hat on top" notation. We know it's a vector and a unit vector. Why? Because we know from the physics forces are vectors...duh. So all that "on top" stuff does is clutter the text without adding value. In the future you can simply say "the force vector F" or "unit vector r" and from then on even us PhDs will know it's a vector. OK?

Assuming we start with TE = 0, we put work into the target mass m or target charge q by force over distance = F(r)r = A/r^2 * r = A/r. From the conservation of energy we know that we must conserve that zero total energy. So we have QE - PE = 0 = TE when work is done. It becomes stored or potential energy.

And that work is stored as potential energy PE = - A/r, for both gravity and electricity. This shows that PE --> as r --> infinity. In other words, the closer we get to the source mass or source charge the less potential energy there is.

NOTE: - GM/r = PE for gravity for example. This is somewhat different looking than the PE = mgh you might be used to seeing. The difference is that in mgh we set PE = 0 on the surface of the planet. In GM/r we set PE --> 0 as r --> infinity. I point this out as PE needs to be defined in all cases relative to some designated zero energy point.