Newton's Law of Universal Gravitation — The Inverse-Square Force

High School⏱ 10 min readClassical Mechanics

Newton's law of universal gravitation: the inverse-square force between masses, the value of G, and why weight equals mg.

Quick answer

Every pair of masses attracts each other with a force proportional to both masses and inversely proportional to the square of the distance between their centers: F = Gm₁m₂/r², with G = 6.67430×10⁻¹¹ N·m²/kg².

$F = G\frac{m_1 m_2}{r^2},\qquad G = 6.67430\times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$

Unit: Force in newtons (N); G in N·m²/kg²

Example: Two 1 kg masses 1 m apart attract each other with just 6.67×10⁻¹¹ N — utterly negligible, which is why you never feel gravity between everyday objects.

What you will learn

State Newton's law of universal gravitation and its inverse-square form
Use F = Gm₁m₂/r² to compute gravitational forces
Explain why the force falls off as the square of distance
Connect the universal law to weight and g = GM/R² near Earth

One force for apples and moons

Before Newton, the fall of an apple and the orbit of the Moon were considered unrelated phenomena. Newton's insight was that one force causes both: every mass in the universe attracts every other mass. An apple falls because the Earth pulls it; the Moon circles the Earth because the Earth pulls it too — the Moon is simply falling forever around the planet instead of into it.

The law has two parts: the force grows with each mass (doubling one mass doubles the force), and it weakens with distance in a precise inverse-square pattern.

The inverse-square law

Mathematically, the law is:

$F = G\frac{m_1 m_2}{r^2}$

$m_1$ and $m_2$ are the two masses in kilograms, $r$ is the distance between their centers in meters, and $G$ is the gravitational constant, $6.67430 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$ .

The inverse square is the crucial part: doubling the distance divides the force by four; tripling it divides the force by nine. The force never truly reaches zero — it just fades with distance — which is why gravity rules the large-scale universe despite being the weakest fundamental force.

Every pair of masses attracts each other along the line joining their centers; the force weakens as the square of the separation.

Why the force between everyday objects is negligible

The constant $G$ is astonishingly small — the smallest measured fundamental constant of nature. That smallness is the whole reason gravity is invisible in daily life between ordinary objects.

Take two $1\ \mathrm{kg}$ masses whose centers are $1\ \mathrm{m}$ apart:

$F = 6.67430\times 10^{-11}\frac{(1)(1)}{1^2} = 6.67\times 10^{-11}\ \mathrm{N}$

That is roughly the weight of a single human cell — millions of times too weak to feel. You need planetary-scale masses before $G$ 's smallness is overcome. $G$ is also notoriously hard to measure precisely; it is known to fewer digits than almost any other fundamental constant because the force is so feeble.

A measured constant

Unlike numbers like π, G is measured, not derived. Its value is known from torsion-balance experiments, and the SI defining-constant list gives it as 6.67430×10⁻¹¹ m³/kg/s² (CODATA 2018).

From the universal law to g

Your weight is simply the gravitational force the Earth exerts on you. Setting $m_1 = M_{Earth}$ and $r = R_{Earth}$ :

$W = G\frac{M m}{R^2} = m\left(\frac{GM}{R^2}\right)$

The bracketed term is the familiar $g$ . Using $M = 5.972 \times 10^{24}\ \mathrm{kg}$ and $R = 6.371 \times 10^6\ \mathrm{m}$ :

$g = \frac{(6.67430\times 10^{-11})(5.972\times 10^{24})}{(6.371\times 10^6)^2} \approx 9.82\ \mathrm{m/s^2}$

So $W = mg$ and the universal law are the same physics — $g$ is just the local, precomputed value of $GM/R^2$ at Earth's surface. Climb a mountain and $R$ grows slightly, so $g$ drops slightly.

Fields and extended bodies

Gravity acts at a distance, so physicists describe it as a gravitational field: each mass sets up a field $g = GM/r^2$ pointing toward itself, and any other mass placed there feels $F = mg$ . The field picture and the pairwise-force picture are equivalent for a single mass.

One subtle consequence: inside a hollow spherical shell, the gravitational forces from all parts of the shell cancel exactly — the field is zero everywhere inside. This is why the universal law must be applied carefully to extended bodies; for points outside a sphere, though, the whole mass can be treated as concentrated at its center.

Orbits: the law in action

Newton derived Kepler's laws of planetary motion from this force law, and the same equation flies every satellite today. A satellite in a circular orbit of radius $r$ moves at the speed where gravity provides exactly the needed centripetal force:

$G\frac{Mm}{r^2} = m\frac{v^2}{r} \implies v = \sqrt{\frac{GM}{r}}$

Lower orbits are faster: the International Space Station, about $420\ \mathrm{km}$ up, circles the Earth roughly every 90 minutes. Geostationary satellites sit at the special radius ( $\approx 42{,}164\ \mathrm{km}$ from Earth's center) where the orbital period is exactly 24 hours, so they hover over one spot on the equator.

v = \sqrt{\frac{GM}{r}}

Orbital speed from equating gravity to required centripetal force.

Variables and units

Symbol	Name	Unit	Meaning
$G$	Gravitational constant	N·m²/kg²	Newton's gravitational constant
$m₁, m₂$	Masses	kg	The two attracting masses
$r$	Center-to-center distance	m	Distance between the centers of the masses
$F$	Gravitational force	N	Magnitude of the mutual attractive force

Common mistakes

✗ Gravity is what holds your feet to the floor because you are heavy.

Weight is the Earth's gravitational force on you — it depends on Earth's mass and your distance from its center, not just your mass. On the Moon the same body weighs about one-sixth as much.

✗ Astronauts float because there is no gravity in space.

The International Space Station experiences about 90% of Earth's surface gravity. Astronauts float because they are in continuous free fall around the planet, not because gravity vanished.

✗ Gravity only pulls things down.

Gravity pulls any two masses toward each other. 'Down' is just the direction toward Earth's center — the same force, between any pair of objects, in any direction.

Real-world applications

Satellite orbits and launch windows are computed directly from the inverse-square law.
Tidal forces from the Moon and Sun follow the same law applied across a planet's diameter.
Astronomers weigh distant stars and exoplanets by the gravitational pull they exert on companions.
Spacecraft use gravitational slingshots around planets to gain speed without extra fuel.

Frequently asked questions

If gravity acts between all masses, why is Earth's pull so dominant?

Because the Earth is enormously massive and you are very close to its center. The inverse-square law rewards mass and punishes distance — the Earth's mass outweighs the tiny distance disadvantage, while everyday objects have far too little mass to produce noticeable force.

How was G measured?

Newton never measured G himself. In 1798 Henry Cavendish measured it with a torsion balance, about a century after Newton published the law. The experiment is often called 'weighing the Earth' because G plus the surface gravity gives Earth's mass.

Does a heavier object feel a stronger force but fall at the same rate?

The force itself is the same on both objects — that is Newton's third law. The difference is in the acceleration: a = F/m, so the lighter object accelerates far more. Earth pulls you down while you pull Earth up with the same force, but Earth barely notices.

Does the law fail for very small objects?

No. The inverse-square law was confirmed for falling objects by Galileo and applies at all everyday scales. What changes at atomic scales is not the law itself but its irrelevance — electromagnetic forces between charged particles dwarf gravity, so gravity is simply ignored there.