Gravitational Force Calculator
Calculate the gravitational attraction between two masses with Newton's law of universal gravitation, from their masses and the distance between them.
How to use this calculator
- 1Enter the two masses in kilograms — scientific notation like 5.972e24 is accepted.
- 2Enter the distance between their centres and choose its unit.
- 3Read the force, and note the acceleration each mass actually experiences.
How it works
Newton's law of universal gravitation
F = G × m₁ × m₂ ÷ r² G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant) m₁, m₂ = the two masses (kg) r = distance between their centres (m) the force is attractive and acts along the line joining them
Newton's insight was that the same force pulling an apple to the ground holds the Moon in orbit — gravity is universal. The force grows with each mass and falls off with the square of the distance, so doubling the separation quarters the force. It acts equally on both bodies in opposite directions, which is why you and the Earth attract each other with identical force even though only you noticeably move.
Worked example
A 70 kg person on Earth's surface (Earth's mass 5.972 × 10²⁴ kg, radius 6,371 km): F = 6.674×10⁻¹¹ × 5.972×10²⁴ × 70 ÷ (6.371×10⁶)² ≈ 687 N. That is the person's weight — about 154 lbf — and it equals their mass times 9.81 m/s², as it must.
Gravitational Force Calculator: the complete guide
One law for the apple and the Moon
Before Newton, the fall of objects on Earth and the motion of the heavens were thought to obey different rules. His law of universal gravitation unified them: every mass attracts every other mass with a force set by the same equation, whether it is an apple and the Earth or the Earth and the Sun. This was one of the great unifications in the history of science, and it held as the complete theory of gravity for over two centuries until Einstein refined it.
The word 'universal' is literal. The equation applies to any two masses anywhere — the pull between two people standing next to each other is real, just unimaginably small (a fraction of a millionth of a newton). It is only when at least one mass is planetary that the force becomes something we feel.
The inverse-square law
Force falls off with the square of distance, and that squaring has dramatic consequences. Move twice as far apart and the force drops to a quarter; ten times as far and it drops to a hundredth. This is why gravity, despite being universal and infinite in range, becomes negligible quickly with distance, and why a spacecraft only a few Earth radii out feels a small fraction of surface gravity.
The inverse-square form is not arbitrary — it reflects the geometry of space. A gravitational influence spreading out from a point is diluted over the surface of an expanding sphere, and that surface area grows as the square of the radius. The same geometry gives us the inverse-square laws for light intensity and electric force.
Equal and opposite, unequal in effect
Newton's third law guarantees the force is identical on both masses: you pull the Earth up exactly as hard as it pulls you down, around 687 newtons each way. What differs is the effect. Acceleration is force divided by mass, so the same force gives you a brisk 9.8 m/s² while giving the Earth, six thousand billion billion times more massive, an acceleration too small to measure.
This resolves a common puzzle — if the forces are equal, why does only the smaller object move? Both move, but the more massive one responds imperceptibly. It is also why the calculator reports the acceleration of each mass separately: the shared force tells you the interaction, but the accelerations tell you what actually happens.
Frequently asked questions
What is Newton's law of universal gravitation?
It states that any two masses attract each other with a force equal to G × m₁ × m₂ ÷ r², where G is the gravitational constant and r is the distance between their centres. The force grows with the masses and falls off with the square of the distance.
Why does gravity get weaker so quickly with distance?
Because it follows an inverse-square law: force is proportional to 1 ÷ distance². Doubling the distance quarters the force. This reflects the geometry of an influence spreading over the surface of an expanding sphere, whose area grows as the square of the radius.
If I pull the Earth as hard as it pulls me, why don't I move it?
You do — imperceptibly. The force is equal on both, but acceleration is force divided by mass. Your small mass gives you a large acceleration; the Earth's enormous mass gives it an acceleration far too tiny to detect. Both respond to the same force.
What is the value of the gravitational constant G?
6.674 × 10⁻¹¹ N·m²/kg². It is one of the least precisely known fundamental constants because gravity is so weak that measuring the tiny attraction between laboratory masses is extraordinarily difficult.