Show that the variation of gravity with height can be accounted for approximately by the following potential function:
Where R is the radius of the earth and z the height above the surface.
The Attempt at a Solution
First, I said define z such that r = R + z. We have the potential energy function for the earth as V(r)=-GmM/r, so V(z)=-GmM/(R+z). I then expanded this around z=0 and took the first two terms in the series to get:
V=-GmM/R + GmMz/R^2
and you can factor this to get
V=GmM/R * [1-z/R]
but the force F given by the potential energy function is GmM/R^2 at the surface, so this is equal to mg, so g = GM/R^2
So the V function is then V = mgR[1-z/R]
And here I am stuck. It appears that I am still taking the centre of the planet as my reference point. Can someone help me? I'm stuck.
Somehow I need to redefine the reference point so that V = 0 for z = 0. How can I do this?