Relationship between work and potential energy?

[carsof]

Homework Statement

Imagine a thin tower 4000 miles tall; ı.e. as tall as the earth’s own radius Re, placed at the north pole of the earth. Suppose you start at rest at the base of the tower and climb to the top (wearing a spacesuit and carrying supplies you need, so that your total mass is M).

How much work would you have done by the time you are at the top of the tower and
at rest again? Neglect the mass of the tower compared to the mass of the earth.

Homework Equations

These are the formulas I picked up from class and from reading the textbook:
Total energy= U+K= Work by SURROUNDINGS
change in U = -Work internal
change in Ug= mgh

The Attempt at a Solution

so, the explanation said to use "W= change in U" which I see they got from change in E= change in U + change in K = Work by surroundings (because change in K is 0 in this problem). HOWEVER, the explanation also said "if we consider the system to be you plus the earth, the gravitational potential energy is Ug(r)= ..."
But, wait a second...If I and the Earth are the system, then why are we using the equation that gives work done by the SURROUNDINGS. Shouldn't we use change in U = -Work internal? (in other words, change in U should have negative sign on it).
I'm just confused on what to make my system, which equations to use, why E even equals work done by surroundings.
Can someone please explain the concept and the reasoning behind the formulas here? Thank you so much.EXPLANATION: If we consider the system to be you plus the earth,the gravitational potential energy is

Ug(r) = −G Me M / r

The initial energy is Ei = Ug,i and the final energy is Ef = Ug,f. Thus the Energy Principle gives:

W=∆Ug=Ug(2Re)−Ug(Re)= GMe M/2Re .

 

[haruspex]

I'm just confused on what to make my system

The work is done, of course, by your muscles consuming chemical energy. You can take the (weightless) muscles etc. as being external to the system consisting of your body as a mass plus the Earth as a mass.