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 600 N, to the top of the building.
To find the change in your weight when riding an elevator to the top of the mile-high building, we can make use of the universal law of gravitation. The law tells us that the gravitational force between two objects is given by:
F = (G * m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
In this case, let's assume Earth is a sphere with a radius of approximately 6,371 kilometers (6.371×10^6 meters). We can consider the mass of Earth, m1, to be concentrated at its center.
First, we need to calculate the acceleration due to gravity at the surface of Earth (street level). The acceleration due to gravity, g, is given by:
g = (G * m1) / r^2,
where m1 is the mass of Earth and r is its radius.
Plugging in the values, we get:
g = (6.67430 × 10^-11 N * m^2 / kg^2 * 5.972 × 10^24 kg) / (6.371 × 10^6 m)^2.
Calculating that, we find g ≈ 9.81 m/s^2, which corresponds to the weight of an object at the street level.
Now, let's consider the weight at the top of the mile-high building. Since we ignored Earth's rotation, we can assume the building is stationary relative to Earth's surface. In this case, the distance from the center of Earth to the top of the building would be r + 1.609 × 10^6 m (the height of the building in meters).
Using the same equation for gravitational acceleration, we can find the acceleration due to gravity at the top of the building:
g' = (G * m1) / (r + 1.609 × 10^6 m)^2.
Calculating this, we find g' ≈ 9.803 m/s^2, which corresponds to the weight of an object at the top of the building.
To find the change in weight, we subtract the weight at the street level from the weight at the top of the building:
Δw = w' - w,
where Δw is the change in weight, w' is the weight at the top, and w is the weight at the street level.
Δw = (m * g') - (m * g),
where m represents the mass of the object.
Using the given weight at the street level (600 N), we can convert it to kilograms (since weight is a force measured in newtons and mass is measured in kilograms) using the equation:
w = m * g,
600 N = m * 9.81 m/s^2.
Solving for m, we find m ≈ 61.17 kg.
Now, we substitute the values into the equation for Δw:
Δw = (61.17 kg * 9.803 m/s^2) - (61.17 kg * 9.81 m/s^2).
Calculating this, we get Δw ≈ -0.68 N.
Therefore, the change in weight when riding the elevator from the street level to the top of the mile-high building would be a decrease of approximately 0.68 N.