What is the advantage of setting PE = 0 at r = infinity instead of having, lets say, the centre of the earth to be zero?
Gravitational PE at a distance r from the earth’s centre is given by U = − GMm/r, where r > radius of the Earth, and U = 0 at r = infinity. The equation is only true when r is greater than or equal to the radius of the Earth. Newton wrote a nice theorem establishing this: for gravity, a hollow shell has no effect if you’re inside it, and if you’re outside it, all of its mass may be considered to act at the centre. So the equation involving the mass M of the whole earth only applies when you are outside the earth. So the centre of the Earth can’t be used. Using the surface (r = rE) would be possible: this would give Urelative = − GMm/r + GMm/rE However, this is not used. It is more awkward, it has a new parameter to remember, and it is specific for one particular planet. The sketch compares the usual astronomical version (GMm/r, the solid line) and the local version (mgh, dashed line). mgh is a poor approximation for altitudes that are not negligible in comparison with the radius of the
