Gravitational Force Calculator: Newton's Law of Universal Gravitation

What Is Newton's Law of Universal Gravitation?

In 1687, Isaac Newton published his Philosophiæ Naturalis Principia Mathematica — one of the most consequential books in the history of science. Inside it, he described a single mathematical law that governs how every object in the universe pulls on every other object: the law of universal gravitation.

The core idea is elegantly simple. Any two masses attract each other. The strength of that attraction depends on how massive each object is and how far apart they are. Make the masses bigger, the force grows. Move them farther apart, the force shrinks — fast.

This law predicts how apples fall, how the Moon orbits Earth, how planets orbit the Sun, and how satellites stay in their paths. It remained the definitive theory of gravity for over 200 years, until Einstein's general relativity refined it for extreme conditions. For most everyday and engineering calculations, Newton's formula is still exactly what you need.

The F = Gm₁m₂/r² Formula, Explained

The gravitational force formula is:

F = Gm₁m₂ / r²

Each variable has a specific meaning and unit:

• F — gravitational force, measured in Newtons (N). This is the attractive pull between the two objects.
• G — the universal gravitational constant: 6.674 × 10⁻¹¹ N·m²/kg². It's the same everywhere in the universe.
• m₁ — mass of the first object, in kilograms (kg).
• m₂ — mass of the second object, in kilograms (kg).
• r — the distance between the centers of the two masses, in meters (m). Note: it's the center-to-center distance, not the surface-to-surface gap.

Worked Example: Earth and the Moon

Earth's mass is 5.972 × 10²⁴ kg. The Moon's mass is 7.342 × 10²² kg. The average center-to-center distance is 384,400 km = 3.844 × 10⁸ m.

F = (6.674 × 10⁻¹¹) × (5.972 × 10²⁴) × (7.342 × 10²²) / (3.844 × 10⁸)²
F ≈ 1.982 × 10²⁰ N

That's roughly 200 quintillion Newtons — the continuous force keeping the Moon locked in its orbit.

4 Key Facts About the Gravitational Constant G

The gravitational constant is one of the most fundamental — and historically tricky — numbers in physics.

1. Its value: G = 6.674 × 10⁻¹¹ N·m²/kg² (NIST 2018 CODATA recommended value). It is extraordinarily small, which is why gravity feels weak at human scales.
2. First measured in 1798: Henry Cavendish used a torsion balance — two lead balls on a horizontal rod suspended by a thin wire — to detect the tiny gravitational pull between them and derive G. It was the first laboratory measurement of gravity between ordinary objects.
3. Hardest constant to pin down: G is the least precisely known of the fundamental constants. Different high-precision experiments have produced values that disagree at the fifth significant figure — an active area of physics research.
4. Universal: G applies equally whether you're calculating the pull between two ball bearings on a lab bench or the gravitational interaction between two galaxies 100 million light-years apart.

Surface Gravity Across the Solar System

Surface gravity is what you'd feel standing on the surface of each body — calculated as g = GM/r², where M is the planet's mass and r is its radius. Earth's surface gravity is defined as 1g = 9.807 m/s².

A few things stand out here. Venus and Uranus have nearly identical surface gravity despite being very different planets — their size and density happen to produce the same result. Jupiter's massive gravitational pull at 2.528g would make a 70 kg person feel like they weigh about 177 kg. And the Moon's 0.165g is why astronauts on the Apollo missions could jump several feet off the surface in a bulky suit.

The Inverse-Square Law: Why Distance Matters So Much

The r² term in the denominator creates what physicists call an inverse-square relationship. This has dramatic consequences:

This is why the ISS, orbiting at about 408 km above Earth's surface, still experiences roughly 89% of Earth's surface gravity(the center-to-center distance only increases from 6,371 km to 6,779 km — a modest 6.4% increase). The ISS isn't weightless because gravity is absent; it's in continuous free fall, perpetually curving around Earth at 7.66 km/s.

5 Real-World Applications of Newton's Gravity Formula

1. Satellite orbital mechanics:Every GPS satellite, weather satellite, and communication satellite uses Newtonian gravity to stay in a precise orbit. Geostationary satellites orbit at exactly 35,786 km altitude where their orbital period matches Earth's 24-hour rotation.
2. Spacecraft trajectory planning:NASA's Jet Propulsion Laboratory uses gravitational calculations to plan interplanetary trajectories. Gravity assists — where a spacecraft swings around a planet to gain or shed speed — rely entirely on the F = Gm₁m₂/r² formula.
3. Tidal forces:Ocean tides result from the slight difference in the Moon's gravitational pull on the near side of Earth vs. the far side. The tidal force scales as 1/r³, which is why the Moon (closer but less massive) produces larger tides than the Sun.
4. Structural engineering: Any structure must account for the gravitational loads on its materials. Bridges, skyscrapers, and dams are all designed against the constant downward pull that Newton quantified.
5. Planetary science:Astronomers calculate the mass of planets, moons, and even black holes they can't directly observe by measuring how strongly they deflect nearby objects. The mass of Neptune was predicted from Uranus's orbital perturbations before Neptune was directly observed in 1846.

Gravity on the ISS: Why Astronauts Float

A common misconception is that astronauts on the International Space Station are weightless because there's no gravity in space. At 408 km altitude, gravity is only about 11% weaker than at Earth's surface — far from absent.

The ISS and everything inside it are in continuous free fall. The station falls toward Earth at the same rate the astronauts do, so relative to the station, the astronauts appear to float. This is called microgravity— it's freefall, not the absence of gravity.

The ISS orbits at roughly 7.66 km/s (about 27,600 km/h). At that speed, the curve of the Earth drops away beneath the station at exactly the rate the station falls, creating a continuous orbit. Newton described this himself in the Principia using a thought experiment about a cannonball fired so fast it orbits the Earth — the ISS is that cannonball.

Frequently Asked Questions

What is Newton's law of universal gravitation?

Newton's law of universal gravitation states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula is F = Gm₁m₂/r², where G is the gravitational constant 6.674 × 10⁻¹¹ N·m²/kg².

What is the gravitational constant G?

The gravitational constant G is 6.674 × 10⁻¹¹ N·m²/kg², as defined by NIST (2018 CODATA). It was first estimated by Henry Cavendish in 1798 using a torsion balance experiment. G sets the absolute strength of gravitational attraction between any two masses in the universe.

How does gravity change with distance?

Gravitational force follows an inverse-square law: double the distance and the force drops to one-quarter. Triple the distance and it falls to one-ninth. This relationship (force ∝ 1/r²) explains why planets farther from the Sun orbit more slowly and why escaping Earth's gravity requires less energy at higher altitudes.

What is surface gravity on different planets?

Surface gravity varies significantly across the solar system. Earth is 9.807 m/s² (1g). The Moon is 1.62 m/s² (0.165g), Mars is 3.72 m/s² (0.379g), and Jupiter is 24.79 m/s² (2.528g). A 70 kg person weighs about 686 N on Earth, 113 N on the Moon, and 1,735 N on Jupiter.

How is Newton's gravity formula used in real life?

Newton's gravitational formula is used to calculate satellite orbits, plan spacecraft trajectories, predict tidal forces, and design geostationary communication satellites. NASA uses it for every interplanetary mission. The ISS orbits at roughly 408 km altitude, where Earth's gravitational pull is still about 89% of its surface value.

What is the difference between gravitational force and gravitational acceleration?

Gravitational force (F) is the actual pull between two masses, measured in Newtons. Gravitational acceleration (g) is the acceleration a free-falling object experiences, measured in m/s². They relate via Newton's second law: F = mg. On Earth's surface, g ≈ 9.807 m/s². The value varies slightly by latitude and altitude.