You and your friends are having a discussion about weight. He/she claims that he/she weighs less on the 100th floor of a building than he/she does on the ground floor. Is he/she correct? Support your answer with evidence.

Yes she is correct.

Using the formula:

Gravitational force, Fg = GM1M2/R^2

Where,

G = gravitational constant

M1 = mass of the Earth

M2 = mass of human

R = distance between the 100th floor and the center of the earth

Weight, which really means gravitational force, is proportional to the product of the masses of two objects acting on each other, in this case the giant earth and the tiny you, so the difference will be almost immeasurable.

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.