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...

 

[tomcue]

Your approach is correct so far. To redefine the reference point, you can use the fact that the potential energy at infinity is zero. This means that at a very large distance from the Earth's surface, the potential energy is zero. So, you can subtract this value from your current potential energy function to get a new function where V=0 at z=0. This will give you the desired potential function:

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

Now, when z=0, V=0 and at the Earth's surface (z=R), V=-mgR, which matches with the potential energy at the Earth's surface. This new function takes into account the variation of gravity with height and also satisfies the condition of zero potential energy at infinity.