Newton's Law of Universal Gravitation
Every object with mass pulls on every other object with mass — an apple falls to Earth for the same reason the Moon stays in orbit around it. The pull gets weaker very quickly with distance: twice as far apart means only a quarter of the force.
The formula
F = G · M · m / r²
- F — force (N): the gravitational pull between the two objects
- G — gravitational constant (N·m²/kg²): a fixed tiny number that sets the strength of gravity everywhere
- M — mass one (kg): the mass of the first object, like a planet
- m — mass two (kg): the mass of the second object, like a satellite
- r — distance (m): the distance between the centers of the two objects
Worked example
A planet pulls on a 1 kg satellite. For this planet, G times M works out to a simple 40 in our units, and the satellite orbits at a distance of 2 meters. What is the gravitational force?
- G · M = 40 N·m²/kg
- m = 1 kg
- r = 2 m
- F = (G · M) · m / r²
- F = 40 · 1 / 2² = 40 · 1 / 4
F = 10 N
Test yourself
The same satellite now orbits twice as far away, at 4 meters, with the same G M of 40 and mass of 1 kg. What is the new force?
- 5 N
- 20 N
- Correct answer: 2.5 N
Right! Doubling the distance divides the force by 2 squared, which is 4: 40 · 1 / 4² = 40 / 16 = 2.5 N.
Keep the distance at 2 meters and G M at 40, but swap in a heavier 2 kg satellite. What is the new force?
- Correct answer: 20 N
- 10 N
- 40 N
Yes! Force scales directly with mass: 40 · 2 / 2² = 40 · 2 / 4 = 20 N.
Where you see this
The Moon staying in orbit and an apple falling from a tree are the same force at two different distances — Newton's insight was that they aren't separate phenomena at all. Satellites, the International Space Station, and every planet orbiting the Sun all stay up because gravity supplies exactly the centripetal force their curved paths need.
Common mistakes
A common misconception is that astronauts in orbit are "weightless" because gravity is too weak up there — at the ISS's altitude gravity is still about 90% as strong as at Earth's surface; astronauts feel weightless because they and the station are both in continuous free fall around the Earth, not because gravity has switched off. Another is expecting gravity to drop to zero at some finite distance — it never reaches zero, it just keeps getting weaker, following the inverse-square law forever.
How it connects
Gravitation is the specific force law that plugs into centripetal force's a = v²/r to keep orbiting bodies moving in a curve instead of flying off straight. Kepler's laws, next, describe the shapes and timing of the orbits this force law actually produces — ellipses, not circles, with a precise relationship between orbital period and distance.