You weigh 516 N at sidewalk level outside the World Trade Center in New York City. Suppose that you ride from this level to the top of one of its 410 m towers. Ignoring Earth's rotation, how much less would you weigh there (because you are slightly farther from the center of Earth)

To determine how much less you would weigh at the top of the World Trade Center tower, we need to consider the change in gravitational force as you move away from the center of the Earth.

The force of gravity acting on an object is given by the equation:

F = (G * m1 * m2) / r^2

Where:
F is the gravitational force
G is the universal gravitational constant (approximated as 6.674 × 10^-11 N m^2 / kg^2)
m1 and m2 are the masses of the two objects (in this case, your mass and the Earth's mass)
r is the distance between the centers of two objects (in this case, your distance from the center of the Earth)

Let's calculate the weight (gravitational force) at the sidewalk level first:

Weight = F = m * g

Where:
m is your mass
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Given that your weight at sidewalk level is 516 N, we can rearrange the equation to solve for your mass:

516 N = m * 9.8 m/s^2

m = 516 N / 9.8 m/s^2 ≈ 52.7 kg

Now, let's calculate the weight at the top of the tower. The radius of the Earth is approximately 6,371 km (or 6,371,000 m), and the height of the tower is 410 m.

At the top of the tower, the new distance from the center of the Earth (r') will be:

r' = r + h = 6,371,000 m + 410 m = 6,371,410 m

Now we can calculate the new weight at the top of the tower using the equation for gravitational force:

Weight' = (G * m * m_Earth) / r'^2

Where:
m_Earth is the mass of the Earth (approximately 5.972 × 10^24 kg)

Substituting the known values:

Weight' = (6.674 × 10^-11 N m^2 / kg^2 * 52.7 kg * 5.972 × 10^24 kg) / (6,371,410 m)^2

Weight' ≈ 515.7 N

Therefore, your weight at the top of the World Trade Center tower would be approximately 515.7 N, which is about 0.3 N less than your weight at sidewalk level.