Newton’s Law of Universal Gravitation Explained
Newton’s Law of Universal Gravitation was one of the most audacious ideas in the history of science: the same force that pulls an apple from a tree also holds the Moon in its orbit and keeps every planet circling the Sun. The Moon is not falling toward Earth any differently from the apple — it is simply moving so fast sideways that as it falls, the curved Earth drops away beneath it. Understanding this single insight unlocks the mechanics of every orbit, every tide, and every spacecraft trajectory ever calculated.
1. The Law of Universal Gravitation
Newton proposed that every object with mass attracts every other object with mass. The magnitude of this attractive gravitational force depends on both masses and the distance between their centres:
G is the gravitational constant — one of the fundamental constants of nature. Its tiny value (6.674 × 10⁻¹¹) explains why gravity is the weakest of the four fundamental forces at small scales. The gravitational force between two 1 kg masses 1 metre apart is only 6.67 × 10⁻¹¹ N — utterly negligible. You only notice gravity when at least one object has planetary-scale mass.
Why gravity is unusual: Gravity is the weakest of the four fundamental forces (gravitational, electromagnetic, weak nuclear, strong nuclear) — yet it dominates at astronomical scales because it always attracts, never repels, and acts over infinite range. Electromagnetic forces are much stronger but tend to cancel out (equal amounts of positive and negative charge). Gravity has no such cancellation.
2. The Inverse-Square Law — Why r² in the Denominator?
The r² dependence is not arbitrary — it arises from the geometry of three-dimensional space. Gravitational influence spreads outward in all directions uniformly, like light from a point source. At distance r, this influence is spread over the surface of a sphere of area 4πr². The “flux” of gravitational influence through any shell around the source is fixed — so the intensity at any point decreases as 1/r² as the shell area grows.
| Distance from Earth’s centre | Gravitational field strength g | Ratio to surface value |
|---|---|---|
| 1 R_Earth (surface, ~6,371 km) | 9.81 m/s² | 1.00 (reference) |
| 2 R_Earth (~12,742 km) | 2.45 m/s² | 1/4 |
| 3 R_Earth (~19,113 km) | 1.09 m/s² | 1/9 |
| 10 R_Earth (~63,710 km) | 0.098 m/s² | 1/100 |
| 60 R_Earth (Moon’s distance) | 0.0027 m/s² | 1/3,600 |
The Moon is about 60 Earth radii away. Newton tested his law by checking that the Moon’s centripetal acceleration (0.0027 m/s²) equals g/60² = 9.81/3600 = 0.00272 m/s². It matched perfectly — the same law governing falling apples also governs the Moon’s orbit.
3. Weight vs Mass — The Critical Distinction
Mass (kg) is a fundamental property of matter — its resistance to acceleration (inertia). Mass does not change with location. A 70 kg person has the same mass on the Moon, on Mars, and in deep space.
Weight (N) is the gravitational force on that mass: W = mg. Weight changes with location because g changes. On the Moon (g ≈ 1.62 m/s²), a 70 kg person weighs only 113 N instead of the 686 N they weigh on Earth.
| Location | g (m/s²) | Weight of 70 kg person (N) | Equivalent “weight” (kg-force) |
|---|---|---|---|
| Earth’s surface | 9.81 | 686.7 N | 70 kgf |
| Moon’s surface | 1.62 | 113.4 N | 11.6 kgf |
| Mars’s surface | 3.72 | 260.4 N | 26.6 kgf |
| Jupiter’s surface | 24.79 | 1,735 N | 177 kgf |
| International Space Station (orbit) | 8.7 | 0 N (apparent) | 0 (weightlessness) |
Astronauts are NOT weightless! At the ISS’s altitude (400 km), g ≈ 8.7 m/s² — still 89% of surface gravity. Astronauts experience “weightlessness” because they and the station are both in free fall together. They feel no normal force (no floor pushing up) because there is no surface to push against. True weightlessness would require being infinitely far from all mass.
4. Gravitational Field Strength
Instead of thinking about forces between pairs of objects, it is often more useful to think about the gravitational field created by a mass. The gravitational field strength g at distance r from a mass M is the force per unit mass that any test object would experience:
At Earth’s surface: g = GM_Earth/R_Earth² = (6.674×10⁻¹¹ × 5.97×10²⁴)/(6.371×10⁶)² = 9.81 m/s² ✓
5. Circular Orbits
For a satellite in circular orbit at radius r from Earth’s centre, gravity provides exactly the centripetal force needed for circular motion. Setting them equal:
Notice: orbital speed does not depend on the satellite’s mass — it cancels out. A golf ball and the International Space Station at the same altitude orbit at exactly the same speed. This is why all debris at the same orbital altitude must be tracked — it all orbits at the same speed regardless of size.
| Object | Orbital radius | Orbital speed | Orbital period |
|---|---|---|---|
| International Space Station | 6,770 km (~400 km altitude) | 7,660 m/s (27,600 km/h) | 92 minutes |
| Moon | 384,400 km | 1,022 m/s (3,679 km/h) | 27.3 days |
| Geostationary satellite | 42,164 km | 3,075 m/s (11,070 km/h) | 24 hours |
| GPS satellite | 26,560 km | 3,874 m/s (13,950 km/h) | 12 hours |
6. Worked Examples
Given: M_Earth = 5.97 × 10²⁴ kg, M_Moon = 7.35 × 10²² kg, r = 3.84 × 10⁸ m, G = 6.674 × 10⁻¹¹ N·m²/kg²
Problem: Calculate the orbital speed of the ISS at 400 km altitude. (M_Earth = 5.97 × 10²⁴ kg, R_Earth = 6.371 × 10⁶ m)
Problem: A person has a mass of 75 kg. What is their weight on Mars? (M_Mars = 6.39 × 10²³ kg, R_Mars = 3.39 × 10⁶ m)
7. Tides — Gravity in Action
Ocean tides are caused by the variation of the Moon’s gravitational pull across Earth’s diameter. The side of Earth facing the Moon is pulled more strongly toward it than Earth’s centre; the far side is pulled less strongly. This differential force — the tidal force — stretches Earth’s oceans into a bulge on both sides, producing two high tides per day as Earth rotates beneath them.
The Sun also exerts tidal forces on Earth — about 46% as strong as the Moon’s, despite the Sun being 27 million times more massive, because it is 390 times further away and tidal forces scale as 1/r³ (not 1/r²).
Spring and Neap Tides: When the Sun and Moon align (new or full moon), their tidal forces add together → extra-high spring tides. When they are at 90° (first/third quarter moon), they partially cancel → smaller neap tides. This monthly pattern governs coastal ecosystems, shipping schedules, and extreme flood risk worldwide.
8. Common Misconceptions
“Gravity stops at some distance from Earth.” Gravity has infinite range — it decreases as 1/r² but never reaches exactly zero. Even at the distance of the nearest star (4 light-years), Earth exerts a tiny but non-zero gravitational pull. Every mass in the universe gravitationally influences every other mass, however weakly.
“There is no gravity in space.” The ISS orbits in a region where g ≈ 8.7 m/s² — almost as strong as on Earth’s surface. Astronauts experience weightlessness not because there is no gravity, but because they and the station are both in free fall. They are not floating — they are falling, continuously, around the Earth.
“The Moon and Earth don’t pull on each other equally.” By Newton’s Third Law, the force Earth exerts on the Moon is exactly equal and opposite to the force the Moon exerts on Earth. The Moon pulls Earth toward it with exactly 1.98 × 10²⁰ N — the same force Earth pulls the Moon. Earth “barely moves” because its mass is 81 times greater, giving it 81 times less acceleration.
9. Frequently Asked Questions
Conclusion
Newton’s Law of Universal Gravitation — F = Gm₁m₂/r² — was the first successful attempt to describe a force acting at astronomical distances with mathematical precision. It unified terrestrial and celestial mechanics into one framework and gave humanity the ability to predict planetary positions, design satellite orbits, and understand tides.
The inverse-square law arises geometrically from three-dimensional space. The tiny value of G explains why we only notice gravity at planetary scales. And the crucial equivalence between gravitational mass (what determines force in F = Gm₁m₂/r²) and inertial mass (what determines resistance to acceleration in F = ma) means that all objects fall with the same acceleration — a profound fact that eventually led Einstein to his general theory of relativity.