Astronomy 2e · Orbits and Gravity
Newton’s Universal Law of Gravitation
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In 30 seconds
Every object with Mass The amount of matter in an object, in kilograms; independent of location. Full entry → attracts every other object with mass. That one stripped-down statement is Newton’s universal law of gravitation. Newton expressed the attraction as a force that (1) grows when either mass grows and (2) weakens as the square of the distance between the objects’ centers grows:
F = G m₁m₂ / r²
where F is the Gravitational force The mutual attraction between any two masses, given by F = G m₁m₂/r². Full entry → (in newtons), m₁ and m₂ are the two masses (in kilograms), r is the distance between their centers (in meters), and G is the gravitational constant, ≈ 6.674 × 10⁻¹¹ N·m²/kg² (a commonly taught reference value). The word “universal” carries the real punch: Newton claimed the identical rule governs a falling apple and the orbiting Moon, uniting “earthly” physics with “heavenly” motion for the first time. He tested the claim with a famous calculation — the Moon orbits at roughly 60 Earth radii, so if gravity falls off as 1/r², Earth’s pull on the Moon should be 1/60² = 1/3600 of its strength at the surface. The Moon’s observed acceleration toward Earth matches g/3600 ≈ 0.0027 m/s² using commonly taught values. Two phenomena that look completely different — an apple’s fall and the Moon’s circuit — turned out to be one phenomenon.
Why this matters
Gravity is the force that assembles and organizes the cosmos. It holds planets in orbit, raises ocean tides, binds the Sun’s family of worlds, and glues galaxies together. Understanding this law gives you the tools to explain all of that from a single equation.
- Everyday physics: Your Weight The gravitational force on a mass, W = mg, in newtons. Full entry → is literally the gravitational force Earth exerts on you. Knowing the difference between mass and weight explains why you “weigh less” on the Moon but are not less massive.
- Spaceflight: Every launch, orbit, and landing is a gravity problem. Astronauts float in orbit not because gravity is absent but because they are falling — the same law predicts that (see Motions of Satellites and Spacecraft).
- Later chapters: Tides, planetary formation, stellar evolution, and even galaxies all lean on the Inverse-square law A relationship in which a quantity falls off as 1/r² with distance. Full entry → introduced here.
- Exams: Inverse-square reasoning and the mass/weight distinction are among the most-tested ideas in introductory astronomy, and favorite sources of trap answers.
The college version
Core Concepts
The inverse-square law
Gravity follows a 1/r² pattern: double the distance and the force drops to one-quarter; triple it and the force drops to one-ninth. Two consequences follow. First, gravity weakens fast — which is why a planet’s pull on you is negligible compared with Earth’s, even though Jupiter is far more massive than Earth. Second, gravity never quite reaches zero: 1/r² approaches zero but never equals it, so every mass in the universe exerts at least a tiny tug on every other. That faint but endless reach is why a spacecraft with enough speed can escape Earth but never leaves gravity behind entirely.
Mass versus weight
Mass is the amount of matter in an object, measured in kilograms; it does not depend on location. Weight is the gravitational force on that mass, W = mg, measured in newtons, where g is the local acceleration of gravity. On Earth’s surface g ≈ 9.8 m/s²; on the Moon g ≈ 1.6 m/s² (commonly taught reference values). A 60-kg student therefore weighs about 590 N on Earth but only about 96 N on the Moon — same mass, different weight. A bathroom scale measures your weight (force); it only reads out “kilograms” because it divides by Earth’s g.
Where g comes from: g = GM/r²
Set the weight formula equal to the gravity law: mg = GmM/r², where M is the planet’s mass and r is your distance from its center. Cancel m and you get g = GM/r² — the local gravity of any world. Plug in Earth’s mass and radius and you recover ≈ 9.8 m/s². The formula also shows g shrinks with altitude, but slowly: at the ISS altitude of roughly 400 km (r ≈ 6,800 km versus 6,371 km at the surface — commonly taught values), g is still about 90% of its surface value. Astronauts float not because gravity vanished but because they are in continuous Free fall Motion under gravity alone, with no other force supporting you. Full entry →.
The constant G and Cavendish’s experiment
G is staggeringly small, which makes gravity the weakest of the four fundamental forces: two 1-kg masses held 1 m apart attract each other with only about 6.7 × 10⁻¹¹ N — roughly the weight of a single bacterium. That is why gravity only dominates when at least one mass is enormous. Henry Cavendish first measured G in 1798 using a torsion balance with lead spheres (a commonly taught historical account); knowing G let scientists literally “weigh the Earth” by combining it with the measured value of g.
Limits of the model
Newton’s law is stunningly accurate at everyday and solar-system scales, but it is a model with boundaries. Near very strong gravity (black holes, neutron stars) or at speeds approaching light, Einstein’s general relativity is required. The classic warning sign is Mercury: its orbit slowly rotates (precesses) slightly more than Newtonian gravity alone predicts — an extra ~43 arcseconds per century (commonly taught) explained by general relativity. For launching rockets and tracking planets, though, Newton’s law remains the practical workhorse.
Common Confusions
| Do Not Confuse | With | Difference |
|---|---|---|
| Weight | Mass | Mass (kg) is how much matter you have; weight (N) is the force gravity exerts on it. Your mass is unchanged on the Moon; your weight is not. |
| g (≈ 9.8 m/s² on Earth) | G (≈ 6.674 × 10⁻¹¹) | g is a local acceleration that depends on which planet you stand on; G is the same everywhere in the universe. |
| “Zero gravity” in orbit | Free fall | Astronauts float because they are continuously falling around Earth, where gravity is still ~90% of the surface value. |
| Doubling the distance halves the force | The force quarters | The law is inverse square: 2× distance → ¼ force; 3× distance → 1/9 force. |
| Gravity as one-directional | A mutual attraction | Forces come in equal-and-opposite pairs: the Moon pulls Earth just as hard as Earth pulls the Moon. |
Eli explains
The same idea, in plain words
Explain it like I’m 10
Imagine every thing in the universe has an invisible rope tied to every other thing. Heavy things pull harder, and things far apart pull much, much weaker — if you double the distance, the pull gets four times weaker. The same rope that pulls an apple to the ground is the rope that keeps the Moon circling Earth. That’s why we call the rule “universal”: it works the same way everywhere.
Worked example
Newton’s own “Moon test,” replayed step by step:
- Assume universality: Earth’s pull on the Moon obeys the same 1/r² rule as its pull on an apple.
- Apply the inverse square: the Moon is about 60 Earth radii away, so its acceleration should be 1/60² = 1/3600 of the surface value.
- Compute the prediction: 9.8 m/s² ÷ 3600 ≈ 0.0027 m/s².
- Compare with observation: the Moon’s centripetal acceleration is v²/r. Using commonly taught values — orbital speed ≈ 1 km/s and distance ≈ 384,000 km — gives (1000 m/s)² / (3.84 × 10⁸ m) ≈ 0.0026 m/s².
- Conclude: the two numbers agree within rounding. One law, two phenomena — the apple and the Moon fall for the same reason.
Key takeaways
- The law: F = G m₁m₂/r² — memorize it and be able to define every symbol (F in newtons; m₁, m₂ in kg; r center-to-center in m; G the universal constant).
- Inverse square: distance ×2 → force ÷4; distance ×3 → force ÷9. Never ÷2.
- Universal means mutual: every mass attracts every other mass, and the forces come in equal-and-opposite pairs (Newton’s third law).
- Mass ≠ weight: mass is matter (kg), weight is force (N) = mg; weight changes with g, mass does not.
- g = GM/r²: surface g ≈ 9.8 m/s²; at ISS altitude gravity is still ~90% of the surface value.
- G ≈ 6.674 × 10⁻¹¹ N·m²/kg², measured by Cavendish; gravity is by far the weakest fundamental force.
- The Moon test: 60 Earth radii → 1/3600 of surface g ≈ 0.0027 m/s² — the classic demonstration of universality.
- Model limits: general relativity supersedes Newton’s law in strong gravity or at near-light speeds (Mercury’s perihelion precession is the classic clue).
Check yourself
5 review questions from the chapter. Try each one, then open the answer.
Write Newton’s law of universal gravitation and define every symbol.
Show answer
F = G m₁m₂/r², where F is the gravitational force in newtons, m₁ and m₂ are the two masses in kilograms, r is the center-to-center distance in meters, and G ≈ 6.674 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant.
Two asteroids 1,000 km apart attract each other with force F. What happens to F if they move to 2,000 km apart? To 3,000 km?
Show answer
At 2,000 km (2× distance), F drops to F/4. At 3,000 km (3× distance), F drops to F/9 — the inverse-square rule.
An astronaut’s mass is 80 kg. What is her weight on Earth (g ≈ 9.8 m/s²)? What would it be on the Moon (g ≈ 1.6 m/s²)?
Show answer
On Earth: W = mg = 80 × 9.8 ≈ 784 N. On the Moon: W ≈ 80 × 1.6 ≈ 128 N. Her mass stays 80 kg in both places.
Why do astronauts on the ISS float, even though gravity there is still about 90% as strong as at Earth’s surface?
Show answer
Because they are in free fall: the ISS and everything in it are falling toward Earth continuously while their sideways motion keeps them missing the ground. “Weightlessness” is falling, not the absence of gravity.
Why is the law called “universal,” and what did Cavendish’s experiment actually measure?
Show answer
It is universal because it applies to all masses everywhere — the same law explains the apple, the Moon, and the planets. Cavendish measured G, the gravitational constant, with a torsion balance, which made it possible to compute Earth’s mass.
Study tools & related lessonsKey vocabulary · Related
Key vocabulary
- Gravitational force
- The mutual attraction between any two masses, given by F = G m₁m₂/r².
- Gravitational constant (G)
- The universal number (≈ 6.674 × 10⁻¹¹ N·m²/kg²) that sets the strength of gravity.
- Inverse-square law
- A relationship in which a quantity falls off as 1/r² with distance.
- Mass
- The amount of matter in an object, in kilograms; independent of location.
- Weight
- The gravitational force on a mass, W = mg, in newtons.
- Free fall
- Motion under gravity alone, with no other force supporting you.
- Centripetal acceleration
- The acceleration toward the center needed to keep an object moving in a circle.
Sources & references
This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.
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