Variation of gravity with height

[KBriggs]

Homework Statement

Show that the variation of gravity with height can be accounted for approximately by the following potential function:

V(z)=mgz(1-z/R)

Where R is the radius of the Earth and z the height above the surface.

Homework Equations

r=R+z
V=-GmM/r
F=GmM/r^2

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?

 

[KBriggs]

Additional work:

I now have something very close, but not quite.

Since V is unchanged by adding a constant, I added GmM/R to V and swapped r = R+z to get

V(z) = GmM/R - GmM/(R+z), which is 0 at z = 0 as it should be.

Now I expand this around 0 and take the first three terms in the series:

V(0) + zV'(0) + z^2 V''(0)

to get:

0 + zGmM/R^2 -2z^2GmM/R^3

which can be factored and the identity g = GM/R^2 put into get

V = mgz[1-2z/R]

Why do I have an extra factor of two in my answer...?

Oops: forgot to divide by 2 factorial in the expansion...