Newton’s law of universal gravitation
In 1687 Isaac Newton showed that the force that makes an apple fall also keeps the Moon in orbit. Every mass attracts every other mass with a force proportional to the product of the masses and inversely proportional to the square of the distance between their centers.
F = G × m₁ × m₂ ÷ r²
- G = 6.6743 × 10⁻¹¹ N·m²/kg², the gravitational constant
- m₁, m₂: the two masses
- r: the distance between their centers, not between their surfaces
The calculator also gives the gravitational field that the first mass creates at that distance, g = G × m₁ ÷ r², which is the acceleration any object would feel there, and compares it with Earth’s standard gravity of 9.80665 m/s².
Worked example: your weight
With the defaults, mass 1 is the Earth (5.972 × 10²⁴ kg), mass 2 is a 70 kg person and the distance is Earth’s mean radius, 6,371 km.
- F = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 70 ÷ (6.371 × 10⁶)² ≈ 687 N, or about 154.5 lbf
- g ≈ 9.82 m/s²
That force is simply the person’s weight. The result differs slightly from 70 × 9.80665 = 686.5 N because Earth’s rotation and its flattened shape change gravity a little from place to place.
Other examples
| Situation | Force or field |
|---|---|
| Two 70 kg people 1 m apart | 3.3 × 10⁻⁷ N |
| Earth and Moon (384,400 km) | 1.98 × 10²⁰ N |
| Field at the ISS altitude (about 400 km) | 8.7 m/s², 89% of the surface value |
| Field on the Moon’s surface | 1.62 m/s², about 1/6 g |
| Field on Mars’s surface | 3.7 m/s², about 0.38 g |
Why it falls off with the square of distance
Gravity spreads out in all directions, so its strength is shared over the surface of an ever larger sphere, whose area grows as r². Doubling your distance from Earth’s center, to about 6,400 km above the surface, cuts gravity to a quarter.
Weightlessness
Astronauts on the International Space Station still feel almost 90% of surface gravity. They float because they and the station are falling together around the Earth, fast enough sideways that they keep missing it. That free fall, not an absence of gravity, is what makes them weightless.
Limits
Newton’s law is extremely accurate for everyday and planetary problems. Very strong fields, such as near black holes, or very precise measurements, such as GPS timing, need Einstein’s general relativity.