Answer:
if the weight theoretically decreases at this height, but in a fraction of 10⁻⁵, which is not appreciable in any scale, therefore, the reading of the scale in the two places is the same.
Explanation:
The weight of a person in the force with which the Earth attracts the person, therefore can be calculated using the law of universal attraction
F = G m M / r²
Where m is the mass of the person, M the masses of the earth
Let's call the person's weight at ground level as Wo and suppose the distance to the center of the Earth is Re
W₀ = G m M / Re²
In the calculation of the weight of the person on the 100th floor the only thing that changes is the distance
r = Re + 100 r₀
Where r₀ is the distance between the floors, which is approximately 2.5 m, so the distance is
r = Re + 250
We substitute
W = G m M / r²
W = G m M / (Re + 250)²
The value of Re is 6.37 10⁶ m, so we can take it out as a factor and perform a serial expansion of the remaining fraction
W = G m M / Re² (1+ 250 / Re)²
(1 + 250 / Re)⁻² = 1 + (-2) 250 / Re + (-2 (-2-1)) / 2 (250 / Re)² +….
The value of the expression is
(1 + 250 / Re)⁻² = 1 -2 250 / 6.37 10⁶ -30 (250 / 6.37)² 10⁻¹² + ...
We can see that the quadratic term is very small, which is why we despise it, we substitute in the weight equation
W = G m M / Re² (1 - 78.5 10⁻⁶)
Remains
W = Wo (1 - 7.85 10⁻⁵)
We can see that if the weight theoretically decreases at this height, but in a fraction of 10⁻⁵, which is not appreciable in any scale, therefore, the reading of the scale in the two places is the same.
