Do we weigh less at the skyscraper's top? Gravity and altitude

Observing the scenery from the top of a skyscraper can be an uplifting experience. Many skyscrapers offer stunning panoramic views that extend across the city and into the distance. As you ascend higher, the horizon expands, unfolding countryside and distant towns miles away. This breathtaking experience can make you feel like a bird gliding in the air, with gravity easing its grip on your body. But is this sensation deceptive, or do we indeed weigh less at the top of a skyscraper?

Fig. 1. Viewing scenery from the skyscraper can make you feel like a bird gliding in the air.

What is weight? To answer this question, let's delve into the physics of gravity and understand how it influences our weight. Weight, W, is determined by the mass of our body, m, and the strength of gravity at our altitude, g, W=mg. Mass remains constant regardless of where we are: on the ground, on the top of the mountain, or even on another planet. However, the gravitational field varies with altitude, strongest at the Earth's surface and weakening as we move away from it. Therefore, it is true that we weigh less at the top of the tall building. The question is whether this difference is significant enough to be detected by a regular scale.

Why does weight decrease with altitude? Gravity weakens as altitude rises due to the geometric properties of our three-dimensional space. The gravitational field is radial (Fig. 2, left), meaning it gets diluted as the distance from the source of gravity increases because the radial lines diverge (Fig. 2, right). The rate of this divergence is governed by the inverse square law, which states that the strength of the gravitational field is inversely proportional to the square of the distance from the source of gravity. The radial lines converge at the Earth's center. Consequently, the distance is measured from that point.

Fig. 2. The Earth's gravitational field weakens with distance from Earth, according to the inverse square law.

The gravitational field strength at a specific altitude is measured by g, which determines how fast an object will accelerate if dropped at that level. If we drop two identical stones, one near the ground and the other from the peak of Mount Everest, the stone dropped near the ground will accelerate faster because it's closer to Earth's center. Nevertheless, the difference is minimal, only 0.4%, as Everest's height is minor compared to Earth's radius. The reduction in gravity, and consequently in weight, becomes significant when the altitude is comparable to Earth's radius.

At ground level, we are one radius R, or 4,000 miles (6,400 km) from the Earth's center (Fig. 3), experiencing a gravitational acceleration of g = 9.8 m/s². If we double this distance, placing ourselves one radius R above the Earth's surface, we will experience an acceleration that is a quarter of the ground value. That, if you weighed 200 pounds (91 kg) on the ground floor of a hypothetical skyscraper, reaching 4,000 miles (6,400 km) above Earth, you would weigh only 50 pounds (23 kg) on its top. This is a dramatic weight loss, but so is the building's height, unachievable with modern technologies.

Fig. 3. At a distance of 2 radii (2R) from Earth's center, our weight is a quarter of that at the Earth's surface.

Earth's size makes skyscrapers appear small. So, just how tall are these skyscrapers? Currently, the Burj Khalifa in Dubai holds the record, reaching 2,717 feet (828 meters). When you compare this height to the Earth's radius,  the difference in weight becomes insignificant and imperceptible on a normal scale. The size of Earth dwarfs everything in its vicinity; from an airplane, even mountains appear flat. The skyscraper would have to reach above the Earth's atmosphere to exhibit a noticeable difference in weight.

At the altitude of the International Space Station (ISS), which is 250 miles (400 km), the gravity is still 90% as strong as it is on the Earth's surface. The microgravity experienced onboard is due to the balance between gravity and orbital speed. If a tower could be constructed to reach this astonishing height, our weight would be reduced by 10%. To put it in perspective, if you weigh 200 pounds (91 kg) on the Earth, you would weigh 180 pounds (82 kg) at the top of such a tower. This is a significant change, but who would want to live above a breathable atmosphere?

The microgravity experienced onboard is due to the balance between gravity and orbital speed. If a tower could be built to reach this remarkable height, our weight would decrease by 10%. For example, if you weigh 200 pounds (91 kg) on Earth, you would weigh 180 pounds (82 kg) at the top of such a tower. This is a notable difference, but who would want to live above a breathable atmosphere?