In 1956, Frank Lloyd Wright proposed the construction of a mile-high building in Chicago. Suppose the building had been constructed. Ignoring Earth's rotation, find the change in your weight if you were to ride an elevator from the street level, where you weigh 817 N, to the top of the building.

ΔF =  0.21 N

Explanation:

For this exercise that does not ask for the change in weight we must use the law of universal gravitation

         F = G M m / r²

where r is the distance from the center of the Earth, in the lower part of the building is

        r =

for the upper part of the building h = 1 mile = 1609.34 m

        r =R_{e} + h

the weight or the force of attraction of gravity on the floor is F = 817 N, therefore the equation remains

         817 = (G M m / R_{e}²)

let's find this force for the top of the building

          F` = G M m / (R_{e} + h)²

let's take out R_{e} common factor

          F ’= (G M m / R_{e}²)       1 / (1 + h / R_{e})²2

          F ’= (G M m R_{e}²)    (1 + h / R_{e})⁻²

as the quantity h /R_{e} = 1609 / 6.37 10⁶ << 1 we can make a series space

          (1 + x)⁻² = 1 -2 x + ...

we substitute

       F ’= (GMm /R_{e}²)  (1 - 2  1609 / 6.37 10⁶)

       F ’= (GMm /R_{e}²) (1 - 2.53 10⁻⁴) = (GMm / R_{e}²) 0.99974

       F ’= 817 0.99974

        F ’= 816.79 N

weight Change

        ΔF = ΔW = 817 - 816.79

       ΔF =  0.21 N

as we see this is a very small amount