Newton’s Law of Gravitation Explained
Newton’s law of universal gravitation explained: F = Gm₁m₂/r², the constant G, the inverse square law, gravitational field strength, weight, orbits and tides.
Key takeaways
- Every mass attracts every other mass with a force F = G × m₁ × m₂ ÷ r².
- Gravity follows an inverse square law: double the distance and the force drops to a quarter.
- The same law explains weight, falling objects, orbits and tides.
On this page
Newton’s law of universal gravitation says that every object with mass attracts every other object with mass. The force is F = G × m₁ × m₂ ÷ r², where m₁ and m₂ are the two masses in kilograms, r is the distance between their centres in metres and G is the gravitational constant, about 6.674 × 10⁻¹¹ N m²/kg². Because G is so small, gravity is only noticeable when at least one object is enormous, like a planet. The same law explains why apples fall, why the Moon orbits Earth and why we have tides.
The formula
F = G m₁ m₂ ÷ r²
- F = gravitational force, in newtons (N)
- G = 6.674 × 10⁻¹¹ N m²/kg²
- m₁, m₂ = the masses, in kilograms
- r = the distance between the centres of the masses, in metres
The force acts equally on both objects, in opposite directions, as Newton’s third law requires. The Earth pulls you down, and you pull the Earth up with the same force; the Earth just doesn’t notice because its mass is so large.
Worked examples
- Two people: two 70 kg people stand 1 m apart. F = 6.674 × 10⁻¹¹ × 70 × 70 ÷ 1² ≈ 3.3 × 10⁻⁷ N, far too small to feel.
- Earth and Moon: with Earth’s mass about 5.97 × 10²⁴ kg, the Moon’s about 7.34 × 10²² kg and an average distance of about 3.84 × 10⁸ m, F ≈ 2.0 × 10²⁰ N.
- Your weight: for a 1 kg mass at Earth’s surface (radius about 6.37 × 10⁶ m), F = 6.674 × 10⁻¹¹ × 5.97 × 10²⁴ × 1 ÷ (6.37 × 10⁶)² ≈ 9.8 N. That’s where g ≈ 9.8 N/kg comes from.
The inverse square law
Force depends on 1 ÷ r², so distance matters a lot:
| Distance compared with start | Force compared with start |
|---|---|
| 2 × further | ¼ |
| 3 × further | ⅑ |
| 10 × further | 1/100 |
| Half the distance | 4 × stronger |
Light and sound intensity follow the same pattern as they spread out from a source.
Gravitational field strength
The gravitational field strength at a distance r from a planet of mass M is g = G × M ÷ r², measured in newtons per kilogram (N/kg), which is the same as metres per second squared.
| Place | Approximate g (N/kg) |
|---|---|
| Earth’s surface | 9.81 |
| International Space Station orbit (about 400 km up) | 8.7 |
| Moon’s surface | 1.62 |
| Mars’s surface | 3.7 |
| Jupiter (cloud tops) | 24.8 |
Even at the height of the Space Station, gravity is still about 89% as strong as on the ground.
Mass and weight
Mass is the amount of matter in an object, measured in kilograms, and it doesn’t change from place to place. Weight is the gravitational force on that mass: W = m × g. A 70 kg person weighs about 687 N on Earth but only about 113 N on the Moon. Their mass is still 70 kg.
Why astronauts float
Astronauts on the Space Station aren’t beyond gravity; they’re in continuous free fall. The station and everyone in it are falling towards Earth, but they’re also moving sideways so fast that they keep missing it. Everything falls together, so nothing presses on the floor, and they feel weightless. Our free fall calculator shows how objects accelerate when gravity is the only force.
Orbits
For a circular orbit, gravity provides exactly the centripetal force needed to keep an object moving in a circle. Setting the two equal gives the orbital speed:
v = √(G × M ÷ r)
For the Space Station, about 6,770 km from Earth’s centre, that’s roughly 7.7 km/s, and one orbit takes about 92 minutes. Satellites further away orbit more slowly; geostationary satellites, about 36,000 km above the equator, take a day to orbit and so appear fixed in the sky.
Tides
The Moon pulls more strongly on the side of Earth facing it than on the far side. This difference stretches the oceans into two bulges, giving most coasts two high tides a day. The Sun has a similar but smaller effect. When the Sun and Moon line up at new and full moon, their effects combine to make larger spring tides; when they’re at right angles, smaller neap tides result.
A little history
Isaac Newton published his law of gravitation in 1687 in the Principia, showing that the force pulling an apple down also keeps the Moon in orbit. The first laboratory measurement of the gravitational attraction between masses was made by Henry Cavendish in 1798, which allowed G, and therefore Earth’s mass, to be calculated. In 1915, Albert Einstein’s general relativity described gravity as the curvature of space and time. Newton’s law remains accurate for almost all everyday and engineering purposes, although precise systems such as satellite navigation need relativity’s corrections.
Gravity and energy
Lifting an object against gravity stores gravitational potential energy (Ep = m × g × h near Earth’s surface), which turns into kinetic energy as it falls. See kinetic vs potential energy and our projectile motion calculator. Force is a vector with a direction as well as a size; our guide to vectors and scalars explains the difference.
Common mistakes
- Measuring r from the surface instead of from the centres of the masses.
- Forgetting to square the distance.
- Confusing G and g. G is a universal constant; g is the field strength at a particular place.
- Thinking there’s no gravity in space. Gravity reaches everywhere, weakening with distance.
Frequently asked questions
What is the value of G?
About 6.674 × 10⁻¹¹ N m²/kg². It’s one of the hardest fundamental constants to measure precisely.
Why don’t we feel the gravity between everyday objects?
Because G is tiny. The force between two people is less than a millionth of a newton.
Is the gravity on the Moon one-sixth of Earth’s?
Roughly. The Moon’s surface gravity is about 1.62 N/kg, about 16.5% of Earth’s 9.81 N/kg.
Sources
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