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 760 N, to the top of the building.
I will be happy to critique your thinking.
To find the change in your weight when riding an elevator from the street level to the top of the mile-high building, we need to consider the concept of gravitational acceleration and the change in distance from the Earth's center.
The weight of an object is directly proportional to the gravitational acceleration acting on it. The gravitational acceleration on Earth's surface is approximately 9.8 m/s^2. However, as you increase your distance from the center of the Earth, the gravitational acceleration decreases.
We can use Newton's law of universal gravitation to determine the change in gravitational acceleration as we move from the street level to the top of the building. The formula is:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force between two objects,
G is the gravitational constant (approximately 6.67430 × 10^(-11) N m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, we can treat the Earth as one object and your body as the other object. The Earth's mass remains relatively constant, so we can ignore it for now.
First, we need to find the distance from the Earth's center to the top of the mile-high building. A mile is approximately 1,609 meters, so the total distance from the Earth's center to the top of the building would be:
Distance = Earth's radius + building's height
Distance = 6,371 km + 1.609 km
Distance = 6,372.609 km
Next, we calculate the gravitational force at the street level by using the formula:
F1 = (G * m * mass_of_Earth) / (radius_of_Earth)^2
Where:
F1 is the gravitational force at the street level,
G is the gravitational constant,
m is your mass,
mass_of_Earth is the Earth's mass,
and radius_of_Earth is the radius of the Earth.
Finally, we calculate the gravitational force at the top of the building using the formula:
F2 = (G * m * mass_of_Earth) / (distance_from_Earth's_center_to_top_of_building)^2
Where:
F2 is the gravitational force at the top of the building, and
distance_from_Earth's_center_to_top_of_building is the calculated distance from the Earth's center to the top of the building.
The change in weight is then given by the difference between the two gravitational forces:
Change in weight = F2 - F1
Substituting in the values and solving for the change in weight will give you the answer.