Newton's Law of Gravitation Calculator
Solve F = Gm₁m₂/r² for the gravitational force between any two masses. Two 3D diagrams compare a weak, distant attraction to a strong, close one, and charts show how force falls off with distance and how it scales across everyday and astronomical mass pairs.
Reviewed by the ToolNestr Editorial Team — July 2026
Enter both masses and the distance between them
Weak vs. strong attraction
1. Distant masses (weak pull)
Two masses far apart — the inverse-square law means force drops off fast with distance.
2. Close masses (strong pull)
Same two masses, much closer together — force grows dramatically as distance shrinks.
Gravitational force graphs
How it works
The core idea in one line: every object with mass pulls on every other object with mass, with a strength that grows with both masses but collapses rapidly with distance — which is why gravity only becomes noticeable when at least one of the objects involved is planet-sized.
F = G·m₁·m₂ / r²
gravitational force — G = 6.674×10⁻¹¹ N·m²/kg², m₁ and m₂ = the two masses, r = distance between centres
Newton's law of universal gravitation states that the attractive force between any two masses depends on the product of those masses (bigger masses pull harder) divided by the square of the distance between their centres (farther apart means dramatically weaker). The gravitational constant G fixes the overall scale of this attraction, and its tiny value (6.674×10⁻¹¹) is exactly why gravity between two everyday-sized objects is far too weak to ever feel — it only becomes significant once one of the masses involved is enormous, like a planet or star.
Worked example 1 — a person standing on Earth
Given: Earth (m₁ = 5.97×10²⁴ kg) and a 70 kg person (m₂) standing on the surface, r = 6.37×10⁶ m (Earth's radius).
This confirms that weight is nothing more than gravitational force applied to Earth's surface — F = mg is just Newton's law of gravitation evaluated at Earth's radius.
Worked example 2 — the Moon's pull on Earth vs. the Sun's pull on Earth
Given: Moon: m = 7.35×10²² kg, r = 3.84×10⁸ m from Earth. Sun: m = 1.99×10³⁰ kg, r = 1.496×10¹¹ m from Earth (Earth's mass = 5.97×10²⁴ kg in both cases).
The Sun's enormous mass (about 27 million times the Moon's) more than compensates for the inverse-square penalty of being much farther away — this is exactly why Earth orbits the Sun, not the Moon.
Gravitational force across everyday and astronomical scales
Same formula, an enormous range of results depending on mass and distance.
| Pair | Force |
|---|---|
| Two 1 kg objects, 1 m apart | 6.67×10⁻¹¹ N |
| 70 kg person on Earth's surface ★ | ≈687 N |
| Moon's pull on Earth | ≈1.98×10²⁰ N |
| Sun's pull on Earth | ≈3.54×10²² N |
★ Reference row (worked example 1). Even a single-kilogram pair produces a force smaller than the weight of a bacterium — gravity between everyday objects is real but utterly negligible next to planetary-scale attraction.
Where gravitational force actually matters
🛰️ Satellite orbits
Every satellite orbit is a balance between gravitational pull (pulling inward) and orbital velocity (carrying the satellite forward) — engineers use this exact formula to calculate the orbital speed and altitude needed for stable orbits.
🌊 Ocean tides
The Moon's gravitational pull is strong enough to noticeably stretch Earth's oceans, creating tidal bulges. The Sun contributes too, and when Sun, Earth, and Moon align (full or new moon), their combined pull produces the especially strong spring tides.
🔭 Astrophysics and orbital mechanics
Calculating the gravitational force between stars, planets, and moons is the foundation for predicting orbits, detecting exoplanets from their gravitational wobble, and understanding how galaxies hold together.
🚀 Spacecraft trajectory planning
Mission planners calculate gravitational forces from multiple bodies at once to design fuel-efficient trajectories, including gravity-assist maneuvers that use a planet's gravity to accelerate a spacecraft without using extra fuel.
Common misconceptions
"Objects in orbit are weightless because gravity has stopped acting on them."
Gravity is still fully acting on orbiting objects — an astronaut on the International Space Station experiences about 90% of Earth's surface gravity. The 'weightlessness' is actually continuous free fall: the station and everything in it are falling toward Earth at the same rate, so nothing pushes against anything else.
"Heavier objects fall faster because gravity pulls harder on them."
Gravity does pull harder on a more massive object (larger F), but that object also has more inertia resisting acceleration — the two effects cancel exactly, so in the absence of air resistance, all objects fall at the same acceleration regardless of mass.
"Gravitational force only matters for planets and stars, not everyday objects."
Every object with mass exerts a gravitational pull, including people and furniture — it's simply too small to notice at everyday masses and distances. Extremely sensitive torsion-balance experiments, like Cavendish's original 1798 measurement, can detect gravity between lab-scale objects.
"Doubling the distance between two masses halves the gravitational force."
Gravity follows an inverse-square law, not an inverse law — doubling the distance reduces the force to one-quarter (1/2² = 1/4), not one-half. This is a common and important distinction from simpler inverse relationships.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 1 — Chapter 13, "Gravitation" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 13, Gravitation.
- • NASA/NIST — standard reference masses for Earth, Moon, and Sun.
F = Gm₁m₂/r², G = 6.67430×10⁻¹¹ N·m²/kg² (CODATA value). Results are rounded for display.
How to use this calculator
Enter both masses
Type each mass in kilograms, or use the Earth/Moon/Sun preset buttons for common astronomical bodies.
Enter the distance
Distance between the centres of mass, in metres — for surface calculations, use the body's radius.
Read the force
Shown in newtons and scientific notation, so both tiny and enormous results stay readable.
Related tools
Frequently asked questions
What is Newton's law of universal gravitation?
Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: F = Gm₁m₂/r².
What is the gravitational constant G?
G = 6.67430×10⁻¹¹ N·m²/kg². It is a fundamental constant that sets the overall strength of gravity, first measured by Henry Cavendish in 1798 using a torsion balance.
Why does gravity feel so weak in everyday life?
Gravity is by far the weakest of the four fundamental forces — the electromagnetic force between two protons is about 10³⁶ times stronger. Gravity only dominates at large scales because, unlike charge, mass has no negative version to cancel it out.
How does distance affect gravitational force?
Gravitational force follows the inverse square law — doubling the distance cuts the force to one-quarter, tripling the distance cuts it to one-ninth. This is why astronauts feel noticeably lighter even a few hundred kilometres above Earth's surface.
Can gravitational force ever be repulsive?
In Newtonian gravity, no — gravity is always attractive. General relativity does allow repulsive effects through dark energy on cosmological scales, but on everyday and planetary scales gravity is always attractive.