The law of gravitation states that every particle of matter attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres. Written as an equation, F equals G times m1 times m2 divided by r squared, where G is the gravitational constant.
Drop your phone and it falls. Look up, and the Moon does not. For most of human history those looked like two different kinds of event — earthly things fell, heavenly things circled, and nobody expected one rule to cover both.
Newton’s claim was that they are the same event. The Moon is falling; it simply travels sideways fast enough to keep missing the Earth. One equation handles the phone, the Moon and the galaxy.
What Is the Law of Gravitation?
The law of gravitation is Newton’s statement that any two masses attract each other along the line joining them, with a force that grows in proportion to both masses and shrinks with the square of the distance between them. It applies to every pair of masses in the universe — hence universal.
That word carries the whole argument. Newton was not claiming to have found a new force; he was claiming that the force already known to make apples fall was the same force keeping planets in their orbits.
What the law actually asserts
- It is always attractive. There is no negative mass and no gravitational repulsion — unlike electric charge, gravity only ever pulls.
- It acts along the line of centres. The force on each body points straight at the other.
- It is a mutual pair. Both bodies feel the same size of force in opposite directions.
- It never reaches zero. The force weakens with distance but has no cut-off range.
One technical caveat matters for problem-solving. The formula is written for point masses, yet Newton proved that a uniform sphere pulls on outside objects exactly as though all its mass sat at its centre.
That result is why you can treat the Earth as a 5.97 × 1024 kg dot 6371 km beneath your feet and still get the right answer.
The Law of Gravitation Formula
The law of gravitation formula gives the attractive force between two masses as the product of the masses divided by the square of their separation, scaled by the gravitational constant G.
Every symbol, with its SI unit:
| Symbol | Quantity | SI unit | Notes |
|---|---|---|---|
| F | Gravitational force | newton (N) | Same magnitude on both bodies |
| G | Gravitational constant | N·m²/kg² | 6.674 × 10−11, the same everywhere |
| m1, m2 | The two masses | kilogram (kg) | Never grams — convert first |
| r | Centre-to-centre distance | metre (m) | Not surface-to-surface |
Two traps live in that table. Masses must be in kilograms, and r is measured centre to centre — a satellite 400 km up sits 6771 km from the Earth’s centre, not 400 km.
Once the units are right the arithmetic is routine, and you can check any answer against our Gravitational Force Calculator, which rearranges the same equation to solve for the force, either mass or the separation.
Where G comes from
G is not derived from anything — it is measured. The instrument that made it possible was a torsion balance designed by John Michell, in which heavy lead spheres twist a delicately suspended dumbbell through a tiny, readable angle.
Here is the detail most textbooks get wrong. Michell died before he could use it, and when Henry Cavendish finally ran the experiment in 1798 he was not chasing G at all — his paper is titled Experiments to determine the Density of the Earth, and his answer was that the Earth is 5.48 times as dense as water.
A value for G was only extracted from measurements of this kind decades later, once physicists began treating it as a constant worth knowing in its own right. Cavendish supplied the experiment; the constant came afterwards.
The currently accepted value is 6.67430 × 10−11 N·m²/kg², published in the CODATA set maintained by NIST. It is the least precisely known of the major constants, which tells you how stubbornly weak gravity is to measure.
The two forces are always equal in size and opposite in direction, and the strength collapses as 1/r².
How the Inverse-Square Rule Works
Gravity weakens with the square of distance because the influence of a mass spreads out over the surface of an ever-larger sphere. The area of that sphere is 4πr², so at twice the distance the same total influence is thinned across four times the area.
This is the same geometry that governs light from a bulb and sound from a speaker. Move twice as far from a lamp and it looks a quarter as bright — not half.
The consequences are dramatic and easy to underestimate:
- Triple the separation and the force drops to one ninth.
- Halve the separation and the force jumps to four times its old value.
- Ten times further apart means one hundredth the pull.
Notice what the rule does not say. Mass enters linearly, so doubling one mass simply doubles the force — only distance gets the squaring.
In practice this asymmetry is what makes orbits stable and tides sharp. A body slightly closer to a planet feels a noticeably stronger tug than one slightly further away, and that difference is the entire origin of tidal stretching.
Why Is Gravity So Weak Yet So Dominant?
Gravity is by far the weakest of the four fundamental interactions, but it dominates the universe at large scales because it never cancels out. Electric charge comes in two signs that neutralise each other; mass comes in only one sign, so gravity always accumulates.
The size of the gap is hard to overstate. Between two protons, the electrostatic repulsion is roughly 1036 times the gravitational attraction — a one with thirty-six zeros after it.
Put a number on it closer to home. Two 70 kg people standing a metre apart attract each other with about 3.3 × 10−7 N, some two billion times weaker than the pull the Earth exerts on either of them.
So why does gravity run the cosmos? Because a star has no negative mass to cancel its positive mass, while a star-sized ball of charge would neutralise itself almost instantly.
| Feature | Law of gravitation | Coulomb’s law |
|---|---|---|
| Equation | F = G m1m2 / r² | F = k q1q2 / r² |
| Source quantity | Mass, always positive | Charge, positive or negative |
| Direction | Attractive only | Attractive or repulsive |
| Constant | G = 6.674 × 10−11 N·m²/kg² | k = 8.99 × 109 N·m²/C² |
| Can it be shielded? | No known shielding | Yes, by a conducting shell |
| Dominates at | Planetary and cosmic scales | Atomic and molecular scales |
The structural similarity is not a coincidence anyone has fully explained, and it is worth studying both together — the mathematics of Coulomb’s law transfers to gravitation almost line for line.
Real-World Examples of the Law of Gravitation
1. Your weight on a bathroom scale
The number under your feet is the Earth pulling on you through this exact equation. Plug in a 70 kg person, the Earth’s mass and its radius and you get roughly 687 N — the reading a scale in newtons would show.
2. The Moon’s orbit
The Earth pulls the Moon with about 1.98 × 1020 N across 384,400 km. That force does not slow the Moon down; it continuously bends its path sideways, which is precisely what an orbit is.
3. Ocean tides
Tides come from the difference in the Moon’s pull across the width of the Earth. The near side is tugged harder than the far side, and the planet stretches slightly along the Earth–Moon line.
This is also why the Sun, despite pulling on Earth about 175 times harder than the Moon does in absolute terms, produces the weaker tides — it is so far away that its pull barely varies across our diameter.
4. Satellite and GPS orbits
Every satellite altitude is a solved gravitation problem. Setting the gravitational force equal to the required centripetal force fixes the speed at a given radius, which is why geostationary satellites must sit at one particular height.
5. Weighing planets you have never visited
Astronomers rarely measure a planet’s mass directly. They watch a moon orbit it, measure the period and radius, and invert the law of gravitation to get the mass — the technique that also revealed Neptune before anyone saw it.
Common Misconceptions About the Law of Gravitation
Misconception 1: there is no gravity in space
Astronauts on the International Space Station float because they are in continuous free fall, not because gravity has switched off. At the station’s altitude of roughly 400 km, g is still about 8.7 m/s² — some 89% of its value at the ground.
They fall around the Earth rather than into it, and everything nearby falls with them. That shared fall, covered further in our guide to free fall, is what removes the sensation of weight.
Misconception 2: the Earth pulls you harder than you pull the Earth
The two forces are exactly equal in magnitude. You pull the Earth with the same 687 N that it pulls you — a textbook Newton’s third law pair.
What differs is the acceleration, because a = F/m. The same force applied to 5.97 × 1024 kg produces an utterly negligible response.
Misconception 3: doubling the distance halves the force
It quarters it. This is the single most common slip in exam scripts, and it comes from reading the formula too fast and missing the square on r.
Misconception 4: mass and weight are the same thing
Mass is the amount of matter, measured in kilograms, and it never changes. Weight is the gravitational force on that mass, measured in newtons, and it changes with location — a distinction worth nailing down in our explainer on weight versus mass.
How the Law of Gravitation Sets g, Weight and Orbits
The law of gravitation is the parent equation from which surface gravity, weight and orbital motion all follow. Each one is the same formula with a different quantity made the subject.
Surface gravity
Set the gravitational force equal to mg and the small mass cancels, leaving a result that depends only on the body you are standing on.
This is why every object falls at the same rate regardless of its mass — the falling object’s own mass drops out of the equation entirely. Feed in the Earth’s mass and mean radius and you get about 9.8 m/s².
The measured value drifts from roughly 9.78 m/s² at the equator to 9.83 m/s² at the poles, because the Earth is not a perfect sphere and its rotation shaves a little off the effective value near the equator.
| Body | Mass (kg) | Mean radius (km) | Surface g (m/s²) |
|---|---|---|---|
| Moon | 7.35 × 1022 | 1737 | 1.62 |
| Mars | 6.42 × 1023 | 3390 | 3.73 |
| Earth | 5.97 × 1024 | 6371 | 9.82 |
| Jupiter | 1.898 × 1027 | 69,911 | 24.8 |
Masses and radii above are taken from the JPL Solar System Dynamics planetary parameters table. If you want to run your own body weight through these numbers, our page on weight on other planets does exactly that.
Orbits
For a circular orbit, gravity supplies the centripetal force and nothing else does. Setting GMm/r² equal to mv²/r gives the orbital speed, and the orbiting mass cancels again.
Everything about orbital mechanics falls out of this one substitution, which is why centripetal force and gravitation are almost always taught together.
Push the algebra one step further and you recover Kepler’s third law, T² = 4π²r³/(GM). Kepler found that pattern empirically in 1619; Newton explained why it had to be true.
Fire a projectile fast enough and its curved fall matches the curve of the Earth — that is orbit.
Where Newton’s law stops working
The law of gravitation is an outstanding approximation, not the final word. It breaks down for very strong fields and very high speeds, where general relativity takes over.
The classic failure is Mercury’s orbit, whose perihelion drifts by a small amount that Newton’s equation cannot account for. For every problem you will meet in school or first-year physics, though, F = Gm1m2/r² is entirely sufficient.
Worked Problems
Show Solution
Solution:
Step 1: Use the law of gravitation, F = G m1m2 / r², with G = 6.674 × 10−11 N·m²/kg².
Step 2: Substitute. F = (6.674 × 10−11 N·m²/kg²)(1000 kg)(1000 kg) / (5.0 m)²
Step 3: The numerator is 6.674 × 10−5 N·m², and the denominator is 25 m². Dividing gives 2.67 × 10−6 N.
Answer: 2.67 × 10−6 N — under three micronewtons, which is why you never notice it.
Show Solution
Solution:
Step 1: F = G M m / r², treating the Earth as a point mass at its centre.
Step 2: Substitute. F = (6.674 × 10−11)(5.972 × 1024)(70) / (6.371 × 106)²
Step 3: Numerator = 2.790 × 1016 N·m²; denominator = 4.059 × 1013 m². Dividing gives 687 N.
Answer: 687 N, which agrees with mg = 70 × 9.81 = 687 N as it must.
Show Solution
Solution:
Step 1: g = GM/r², so g is proportional to 1/r². No masses need to be substituted.
Step 2: The distance has tripled, so the field falls by a factor of 3² = 9.
Step 3: g = 9.82 m/s² ÷ 9 = 1.09 m/s².
Answer: 1.09 m/s² — note it is one ninth, not one third.
Show Solution
Solution:
Step 1: F = G m1m2 / r².
Step 2: Numerator = (6.674 × 10−11)(5.972 × 1024)(7.346 × 1022) = 2.928 × 1037 N·m².
Step 3: Denominator = (3.844 × 108 m)² = 1.478 × 1017 m². Dividing gives 1.98 × 1020 N.
Answer: 1.98 × 1020 N.
Show Solution
Solution:
Step 1: Start from F = GMm/r² and set it equal to mg. The small mass cancels, leaving g = GM/r².
Step 2: Substitute. g = (6.674 × 10−11)(6.417 × 1023) / (3.390 × 106)²
Step 3: Numerator = 4.283 × 1013 m³/s²; denominator = 1.149 × 1013 m². Dividing gives 3.73 m/s².
Answer: 3.73 m/s², about 38% of Earth’s surface gravity.
Show Solution
Solution:
Step 1: Since g is proportional to 1/r², halving g means r² must double, so r must grow by a factor of sqrt(2).
Step 2: r = 6.371 × 106 m × 1.414 = 9.010 × 106 m from the Earth’s centre.
Step 3: Altitude = r − R = 9.010 × 106 − 6.371 × 106 = 2.639 × 106 m.
Answer: about 2640 km above the surface — far higher than most students guess.
Show Solution
Solution:
Step 1: Work in orbital radius, not altitude. r = 6.371 × 106 + 4.00 × 105 = 6.771 × 106 m.
Step 2: Gravity supplies the centripetal force, so GMm/r² = mv²/r, which rearranges to v = sqrt(GM/r).
Step 3: GM = (6.674 × 10−11)(5.972 × 1024) = 3.986 × 1014 m³/s². Then v = sqrt(3.986 × 1014 / 6.771 × 106) = sqrt(5.887 × 107) = 7672 m/s.
Step 4: Period T = 2πr / v = (2π)(6.771 × 106) / 7672 = 5545 s.
Answer: v = 7.67 km/s and T = 5545 s, or about 92 minutes per orbit.
