Newton's law of universal gravitation is one of the shortest equations in physics and one of the most frequently mis-applied. Every pair of masses in the universe attracts every other pair with a force proportional to the product of the masses and inversely proportional to the square of the distance between their centres. The gravitational force calculator above applies that law directly, using the 2022 CODATA value of the gravitational constant, and reports not just the force but the three numbers people usually actually want next: the acceleration each body experiences, the gravitational potential energy stored in the configuration, and the force expressed as an everyday weight so you can feel the scale of it.
Arb Digital builds free calculators that do the physics rather than paraphrase it. This page is deliberately explicit about the two things that trip people up — the difference between the force and the acceleration it causes, and the fact that the distance in the equation is measured between centres, not between surfaces. Get either of those wrong and the answer will be confidently, invisibly incorrect. This is a physics solver, not a unit converter: if you already have a force in newtons and want it in pounds-force or dynes, the force converter is the right page, and it does not attempt to calculate anything.
What This Gravitational Force Calculator Does
You enter two masses and the distance separating their centres. Each field has its own unit selector, so you can mix kilograms with astronomical units or Earth masses with kilometres without doing any preliminary conversion by hand. Everything is converted to SI internally, the force is computed in newtons, and the result is displayed with an appropriate metric prefix or in scientific notation when the magnitude runs away from human scale — which, in gravitation, it usually does.
The supporting grid earns its space. The first two items show the acceleration of each body, obtained by dividing the mutual force by that body's own mass. Those two numbers are almost never equal, and the gap between them is the clearest possible demonstration of why the Moon orbits Earth rather than the other way round. The third item is the gravitational potential energy of the two-body configuration, which is negative by convention because zero energy is defined as the state where the two bodies are infinitely far apart. The fourth converts the force into the mass that would weigh the same amount at Earth's surface, which turns an abstract number of newtons into something you can picture.
Two preset buttons load configurations worth seeing. The Earth–Moon preset returns a force close to 2 × 1020 newtons, which is the actual tether holding the Moon in orbit. The two-1,000-kilogram preset places a pair of small cars one metre apart and returns a force of about 67 micronewtons — roughly the weight of a grain of sand. Both are computed by exactly the same equation, and the ratio between them is a good illustration of why gravity is simultaneously the weakest of the four fundamental interactions and the one that shapes galaxies.
How to Use It
- Enter the first mass and pick its unit. Scientific notation is accepted in the number field, so 5.972e24 works as well as typing the digits out. Earth masses and solar masses are offered because astronomical problems are usually stated that way.
- Enter the second mass. Order does not matter — the force is symmetric, and swapping the two masses changes nothing except which acceleration appears in which grid slot.
- Enter the centre-to-centre distance. This is the field that decides whether your answer is right. For an object resting on a planet's surface, use the planet's radius. For an orbit, use the orbital radius from the centre of the primary, not the altitude above its surface.
- Read the force, then read the accelerations. The hero number is the force each body exerts on the other. The two accelerations tell you what that force actually does to each of them.
- Follow the number somewhere useful. A force plus a distance gives work; a mass plus an acceleration gives motion. The free fall calculator takes the acceleration and turns it into fall times and impact speeds.
The Formula: How Gravitational Force Is Calculated
The whole calculation is a single expression: F = G·m₁·m₂ / r². Here F is the magnitude of the attractive force in newtons, m₁ and m₂ are the two masses in kilograms, r is the centre-to-centre separation in metres, and G is the Newtonian constant of gravitation. This tool uses G = 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻², the value published by NIST as the 2022 CODATA recommended value for the Newtonian constant of gravitation. That constant carries a relative uncertainty of about 2.2 parts in 100,000 — extraordinarily poor by the standards of modern metrology, and the reason gravitational calculations are rarely quoted to more than five significant figures.
Work one through by hand. Take a 70 kg person standing at Earth's surface, with Earth's mass about 5.972 × 10²⁴ kg and its mean radius about 6,371 km. Multiply the masses: 5.972 × 10²⁴ × 70 = 4.180 × 10²⁶. Multiply by G: 4.180 × 10²⁶ × 6.674 30 × 10⁻¹¹ = 2.790 × 10¹⁶. Square the distance: (6.371 × 10⁶)² = 4.059 × 10¹³. Divide: 2.790 × 10¹⁶ ÷ 4.059 × 10¹³ ≈ 687 N. Divide that force by the 70 kg mass and you get 9.82 m/s², within a fraction of a percent of the standard acceleration of gravity, defined by CODATA as exactly 9.806 65 m s⁻². The small discrepancy is real physics, not rounding: mean radius is an average over an oblate planet, and the standard value already folds in a centrifugal correction for Earth's rotation.
Gravitational potential energy for a two-body pair is U = −G·m₁·m₂ / r — note the single power of r and the minus sign. It is negative because the reference point of zero energy is the two bodies separated by an infinite distance. Bringing them closer releases energy, so the stored energy of any bound configuration is below zero. The familiar classroom formula U = mgh is a local approximation of this, valid only over height changes small enough that g does not measurably change.
Why the Inverse Square Makes Distance Errors So Expensive
A mistake in either mass propagates linearly: enter a mass ten percent too high and the force comes out ten percent too high. A mistake in the distance does not behave that way. Because r is squared, a ten percent error in separation produces roughly a twenty-one percent error in force, and doubling the distance cuts the force to a quarter rather than a half.
The most common version of this error is entering an altitude where a radius belongs. The International Space Station orbits at roughly 420 km altitude. Enter 420 km as the distance and the calculator will faithfully report a gravitational force about 260 times larger than reality, because the true centre-to-centre distance is 6,371 + 420 = 6,791 km. The equation cannot detect the mistake — it has no idea what a planet is. It sees a distance and squares it.
Force, Weight and Mass Are Three Different Quantities
In casual speech these words are interchangeable, and in a physics calculation they are not. Mass is the amount of matter in a body, measured in kilograms, and it is the same everywhere in the universe. Weight is the gravitational force acting on that mass, measured in newtons, and it changes with location. A 70 kg person has a weight of about 687 N on Earth and about 114 N on the Moon, while their mass is 70 kg in both places.
The confusion is baked into everyday units. A bathroom scale marked in kilograms is not measuring mass directly; it measures the force pushing on its sensor and divides by an assumed value of g to display a mass. That is why the fourth grid item on this page exists — it takes whatever force the calculator produced and divides by 9.806 65 to express it as an equivalent Earth-surface weight. If you need to move between mass units rather than compute forces, the weight converter handles that conversion and nothing more, and the energy converter does the equivalent job for the potential energy figure.
Where the Point-Mass Assumption Holds and Where It Breaks
Newton's law as written applies to point masses. Real objects have extent, so a natural question is whether the equation is legitimate for planets, people or buildings. The answer comes from Newton's shell theorem: a spherically symmetric body attracts external objects exactly as if all its mass were concentrated at its centre. That is not an approximation — it is an exact result for perfect spherical symmetry, and it is why treating Earth as a point mass at its core gives the right surface gravity to several decimal places.
The theorem has a second half that is less well known and more surprising. Inside a uniform spherical shell, the net gravitational force is exactly zero everywhere, not just at the centre. If you could descend into a hollow inside a planet, the shell of material above you would contribute nothing at all, and only the mass beneath your feet would pull on you. This is why the calculator's answers stop being meaningful when the distance you enter is smaller than the radius of either body: you are no longer outside the mass distribution, and the simple inverse-square form no longer applies. The tool flags this case rather than pretending otherwise.
How This Connects to Orbits and Energy
Set the gravitational force equal to the centripetal force required to hold a body in a circular path and a great deal of orbital mechanics falls out in one step. G·M·m/r² = m·v²/r rearranges to v = √(G·M/r), which says orbital speed depends only on the mass of the primary and the orbital radius — the orbiting body's own mass cancels entirely. Push the same algebra one step further and you arrive at Kepler's third law, which the Kepler's third law calculator handles for both the solar-system convenience form and the general two-body form.
The energy view is just as useful. Setting kinetic energy equal to the magnitude of gravitational potential energy gives the escape condition, v = √(2GM/r) — precisely a factor of √2 larger than circular orbital speed at the same radius, which is why the escape velocity calculator and orbital speed differ by about 41 percent for any body. The kinetic energy calculator covers the other half of that energy balance, and the centripetal force calculator handles the circular-motion side directly.
Reading Very Large and Very Small Results
Gravitational forces span an absurd dynamic range — from hundreds of nanonewtons between two people to more than 10²² newtons between the Sun and Earth — so the calculator switches presentation to match. Below a threshold it uses metric prefixes down to nanonewtons; above it, scientific notation with a fixed number of significant figures. It never rounds a nonzero force to zero, because a displayed zero would be a false statement about the physics. If you are working with these figures in a spreadsheet or a lab report, the scientific notation converter is useful for reformatting them consistently.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using altitude instead of orbital radius — the distance in the equation runs from centre to centre. Add the primary's radius to any altitude before entering it.
- Confusing the force with the acceleration — both masses feel the same force, but a lighter body accelerates far more. Divide the force by the specific mass you care about.
- Treating potential energy as positive — U is negative for any bound pair. A positive value means you have dropped the minus sign, not that the system is repulsive.
- Mixing grams with kilometres without converting — the equation is only correct in coherent SI units. Use the unit selectors rather than eyeballing the conversion.
- Applying the formula inside a body — below the surface, the shell theorem means only the mass beneath you contributes, and the inverse-square form no longer holds.
Related Free Tools From Arb Digital
Gravitation connects to most of mechanics. Use the Kepler's third law calculator for orbital periods, the escape velocity calculator for the speed needed to break free, and the free fall calculator for drop times under constant acceleration. For the energy side try the kinetic energy calculator, and for circular motion the centripetal force calculator. Pure unit work belongs in the force converter or the energy converter. The full free online tools hub lists everything else.
Frequently Asked Questions
It uses 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻², the 2022 CODATA recommended value published by NIST. That constant has a relative uncertainty of about 2.2 parts in 100,000, which is why gravitational results are rarely meaningful beyond five significant figures.
Between centres. Newton's law measures separation from one centre of mass to the other. For an object on a planet's surface that distance is the planet's radius, and for an orbiting object it is the planet's radius plus the altitude.
Because zero energy is defined as the two bodies being infinitely far apart. Any bound configuration has less energy than that, so its potential energy is below zero. The magnitude tells you how much energy separating them completely would require.
Yes. The forces are equal in magnitude and opposite in direction, which is Newton's third law. What differs is the resulting acceleration, because the same force acting on a much larger mass produces a much smaller acceleration.
No. Below the surface the shell theorem applies: the mass in the shell above you contributes nothing, and only the mass beneath you pulls. The inverse-square form is valid only outside a spherically symmetric body.
Because 9.806 65 m s⁻² is a defined standard value rather than a measurement of any particular place. Real surface gravity varies with latitude, altitude and local density, and the standard figure also includes a correction for Earth's rotation.
This page computes a force from masses and a distance using Newton's law. A force converter only rescales an existing force between newtons, pounds-force, dynes and similar units, and performs no physics at all.
This tool is provided for educational and study use. It applies Newtonian gravitation to the values you enter and is not intended for engineering design, navigation or any application where an incorrect result could cause harm.
