Gravitational Force Calculator Online Free

Drop a ball and it falls. Watch the Moon and it stays in its orbit, month after month, without any rope holding it up. Both things happen for the same reason: every mass pulls on every other mass. The pull is usually far too small to notice between everyday objects, and it becomes enormous when one of the masses is a planet. A gravitational force calculator turns that idea into a number, so you can see exactly how strong the attraction is between any two masses at any distance, instead of just knowing that gravity exists in general.

That is a useful thing to be able to do, because gravity is easy to talk about vaguely and surprisingly hard to feel in numbers. People know that the Earth pulls them down, but few could say how many newtons of force that pull amounts to for a one-kilogram bag of sugar, or why the same bag would weigh differently on the Moon. People know that the Sun holds the Earth in orbit, but the size of that force, about three and a half times ten to the twenty-second power newtons, is not something anyone can picture without calculating it. The calculator at the top of this page does that arithmetic for you, and this article explains the physics behind it, so the answer means something when you read it.

What It Is

This gravitational force calculator applies Newton's law of universal gravitation, the law Isaac Newton published in 1687 that first treated the fall of an apple and the motion of the Moon as the same phenomenon. The law says that the gravitational force between two masses is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centres. In symbols, F equals G times m1 times m2 divided by r squared, written F = G·m1·m2/r². Here F is the gravitational force measured in newtons, m1 and m2 are the two masses measured in kilograms, r is the distance between the centres of the masses measured in metres, and G is the gravitational constant, a number that sets the overall strength of gravity in our universe. The gravitational constant G is 6.674 × 10⁻¹¹ N·m²/kg², and the calculator uses exactly that value. It is a very small number, and that smallness is the whole reason gravity feels weak between ordinary objects. Two one-kilogram masses held one metre apart attract each other with a force of only 6.674 × 10⁻¹¹ newtons, a force so tiny that no household scale could detect it and no human hand could feel it. It took Henry Cavendish's famously delicate torsion-balance experiment in 1797, more than a century after Newton, to measure gravity between laboratory masses at all. The constant is small, but the masses of planets are huge, and that is why the same law that gives a negligible force between two bags of sugar gives a force of nearly ten newtons on a one-kilogram mass at the Earth's surface, and a force of about 1.98 × 10²⁰ newtons between the Earth and the Moon. The force the calculator reports is the magnitude of the mutual attraction. That word mutual matters. The Earth pulls on you, and you pull on the Earth with exactly the same force, in the opposite direction, as Newton's third law requires. The Earth does not noticeably move towards you, because its mass is so much larger that the same force gives it a far smaller acceleration, but the forces themselves are equal. The calculator also reports the acceleration of the second mass, found by dividing the force by that mass, because acceleration is often what people really want to know: at the Earth's surface it comes out at about 9.82 metres per second squared, the familiar acceleration due to gravity, usually written as g.

How to Use the Calculator

Using the tool takes less than a minute. Follow these steps:

1. Enter the first mass m1 in kilograms. For a planet, use its full mass, for example 5.972 × 10²⁴ kilograms for the Earth.
2. Enter the second mass m2 in kilograms. For your own weight on Earth, this is your mass in kilograms, not your weight in newtons.
3. Enter the distance r in metres between the centres of the two masses. For an object on the Earth's surface, this is the Earth's radius, about 6.371 × 10⁶ metres, not zero and not your height above the ground.
4. Press "Calculate Force". The tool shows the gravitational force in newtons, the full-precision value, and the acceleration of the second mass.
5. Read the result with its units. A force without its unit is just a number, so always keep the newtons attached when you quote it.

Scientific notation is accepted, and for astronomy it is essential. Typing 5972000000000000000000000 is an invitation to miscount zeros, while 5.972e24 is short and exact enough for this purpose. The tool accepts that form directly. Masses and distance must be greater than zero; a zero distance would mean the two centres occupy the same point, and dividing by zero squared has no physical meaning, so the tool asks for a positive distance instead of inventing an answer. One point about distance causes more errors than any other, so it is worth stating plainly. The r in the formula is the distance between centres, not the distance between surfaces and not the height above a surface. If you stand on the Earth, your distance from the Earth's centre is the Earth's radius plus your height, and your height is negligible beside a radius of over six million metres, which is why the surface value works so well. If you are working out the force between the Earth and a satellite, though, you must add the satellite's altitude to the Earth's radius. A satellite 400 kilometres up is not 400,000 metres from the Earth's centre; it is about 6,771,000 metres away, and using the altitude alone would overstate the force considerably. The same care applies to the Moon: the Earth-Moon distance used in the example below is the centre-to-centre distance.

Worked Examples

Example 1: Two 1 kg masses 1 m apart, the live test. This is the simplest possible test of the law, and the one used to check the tool. Both masses are one kilogram, and the distance is one metre. The formula gives F = 6.674 × 10⁻¹¹ × 1 × 1 / 1², which is 6.674 × 10⁻¹¹ newtons. The calculator reports exactly that. It is a good number to remember, because it is the clearest possible demonstration of why gravity between everyday objects is ignored in daily life. The force exists, the law applies, and it is roughly a hundred billion times smaller than the weight of a paperclip. The universe is consistent; it is simply that G is small and laboratory masses are small, so their product is extremely small. Example 2: A 1 kg mass at the Earth's surface, the live test. Now replace the first mass with the Earth, 5.972 × 10²⁴ kilograms, keep the second mass at one kilogram, and set the distance to the Earth's mean radius, 6.371 × 10⁶ metres. The calculation gives F = 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ × 1 / (6.371 × 10⁶)², which works out to about 9.82 newtons. That is the weight of a one-kilogram mass on Earth, and dividing by the one kilogram gives an acceleration of 9.82 metres per second squared, matching the standard value of g to two decimal places. The small difference from the often-quoted 9.81 comes from the rounded values of the Earth's mass and radius used here; real surface gravity also varies slightly with latitude and altitude, because the Earth is not a perfect sphere and its radius is larger at the equator. For a browser calculator, 9.82 newtons is the honest answer to the inputs given, and the article states the inputs so you can reproduce it.

Example 3: The Earth and the Moon. With the Earth's mass, the Moon's mass of about 7.348 × 10²² kilograms, and a centre-to-centre distance of about 3.844 × 10⁸ metres, the force comes out at roughly 1.98 × 10²⁰ newtons. That number is almost beyond intuition, yet it is the force that keeps the Moon in orbit and drives the ocean tides. It is also a good check on the inverse square law: the Moon is about sixty Earth radii away, so the Earth's pull on a kilogram at the Moon's distance is about one thirty-six-hundredth of its pull at the surface, roughly 0.0027 metres per second squared. That was, in essence, Newton's own check. He compared the Moon's centripetal acceleration with the acceleration of a falling apple, found the ratio close to the square of the distance ratio, and concluded that the same gravity reaches all the way to the Moon. Example 4: Doubling the distance. Take two ten-kilogram masses two metres apart. The force is 6.674 × 10⁻¹¹ × 10 × 10 / 2², which is about 1.67 × 10⁻⁹ newtons. Now move them to four metres apart, double the distance. The masses are unchanged, but the distance squared goes from four to sixteen, so the force falls to a quarter of its previous value, about 4.17 × 10⁻¹⁰ newtons. This is the inverse square law in action, and it is why gravity weakens so quickly with distance and why spacecraft escaping a planet need most of their effort early, when they are still close. Halving the distance, by contrast, multiplies the force by four. The relationship is symmetric and unforgiving, and it governs everything from satellite orbits to the tides.

The Inverse Square Law, Explained

The inverse square part of the law deserves its own explanation, because it is the feature people most often get wrong. It is tempting to think that doubling the distance halves the force, by analogy with sharing something between two people. Gravity does not work like that. The force spreads, in effect, over the surface of a sphere centred on the mass, and the surface area of a sphere grows with the square of its radius. At twice the distance, the same influence is spread over four times the area, so its strength at any one point is a quarter. At three times the distance, it is spread over nine times the area, so the strength is a ninth. At ten times the distance, it is a hundredth. This is the same geometry that makes a light bulb look dimmer as you walk away from it, and the parallel is exact: both gravity in Newton's picture and light intensity follow inverse square laws for the same reason. The practical consequence is that gravity has unlimited range in principle but rapidly becomes negligible in practice. The Sun's pull on the Earth is strong because the Sun is enormously massive, not because the Earth is close by astronomical standards; at the distance of Neptune, the Sun's pull is about nine hundred times weaker than at Earth, yet still enough to hold a planet in orbit over billions of years because there is almost nothing else to disturb it.

Weight, Mass, and Gravity on Other Worlds

One of the commonest confusions in physics is between mass and weight, and this calculator is a good place to clear it up. Mass is a property of an object: how much matter it contains, measured in kilograms, the same wherever the object goes. Weight is a force: the gravitational pull on that mass at a particular place, measured in newtons, and it changes with location. Your mass is the same on Earth and on the Moon, but your weight on the Moon is about one sixth of your weight on Earth, because the Moon is much less massive than the Earth and somewhat smaller, and both factors enter the formula. The calculator makes this concrete. Put your mass in as m2, the Moon's mass and radius in as m1 and r, and the force it reports is your weight on the Moon, in newtons. The same reasoning explains why astronauts on the International Space Station float even though they are only about 400 kilometres above the Earth, where gravity is still roughly 87 percent of its surface strength. They are not beyond gravity; they are in continuous free fall around the Earth, falling towards it at exactly the rate their forward motion carries them around it. The calculator can show the first half of that story: at 6.771 × 10⁶ metres from the Earth's centre, a one-kilogram mass still feels about 8.7 newtons of pull. The orbital part needs velocity as well, which is why a satellite must move fast as well as high. Gravity alone never turns off; orbit is falling without ever landing.

Limits: What This Tool Does and Does Not Do

This tool uses Newton's law in its point-mass form. It treats each mass as if all of it were concentrated at its centre, which is exactly correct for a spherically symmetric body outside that body, and a good approximation for most everyday and astronomical purposes. Inside a planet, the law needs a different treatment, because only the mass beneath you pulls you inwards while the shell above you pulls in all directions and cancels out; the tool does not model that, and you should not use it for a mass buried inside the Earth. It also uses Newtonian gravity, not Einstein's general relativity. For almost everything on Earth, in the solar system, and in school and college physics, Newton's law is accurate to far better than the precision of the inputs you will type. Relativistic corrections matter for the precise orbit of Mercury, for GPS satellites, and for extreme objects such as black holes, but they are far below the rounding in values like the Earth's mass and radius as given here. If you need relativistic precision, you need a different tool and far better data. Finally, the tool reports a force, not a prediction about motion. Knowing the force between the Earth and the Moon does not by itself tell you the Moon's orbit; that also needs the Moon's velocity and direction. Knowing your weight does not tell you whether you will fall; that depends on what is supporting you. The calculator answers one question, the size of the gravitational attraction, and answers it exactly for the inputs given. That is its job, and the article has tried to show how much physics hangs off that one number.

Privacy: Your Numbers Stay in Your Browser

Every number you enter into this gravitational force calculator stays in your browser. Nothing is uploaded to a server, nothing is stored, and there is no account to create. The calculation runs entirely on your own device. That matters even for a physics tool, because the numbers people enter can be part of homework, coursework, a design calculation, or a piece of work they would rather keep private. Closing the page removes your inputs, and no one else sees them. This browser-only design is the same principle behind the other free tools on Your Smart Kit: a simple tool that does one job well, keeps your data private, and gives an answer you can check.

Frequently Asked Questions

What is the gravitational force formula? The formula is F = G·m1·m2/r², Newton's law of universal gravitation. F is the force in newtons, m1 and m2 are the masses in kilograms, r is the centre-to-centre distance in metres, and G is the gravitational constant, 6.674 × 10⁻¹¹ N·m²/kg². The force grows with either mass and falls with the square of the distance. What is the value of G? G, the gravitational constant, is 6.674 × 10⁻¹¹ N·m²/kg², the value this calculator uses. It is one of the least precisely known fundamental constants, because gravity between laboratory masses is so weak that measuring it is extraordinarily delicate, but the value given here is accurate to the precision any browser calculation needs. Why is gravity so weak between everyday objects? Because G is extremely small. Two one-kilogram masses a metre apart attract each other with only 6.674 × 10⁻¹¹ newtons. Gravity only becomes noticeable when at least one mass is enormous, like a planet, because the force is proportional to the product of the masses. Electromagnetism, by contrast, is vastly stronger between small charges, which is why a small magnet can lift a nail against the pull of the whole Earth. What is the difference between mass and weight? Mass, in kilograms, is how much matter an object contains and does not change with location. Weight, in newtons, is the gravitational force on that mass at a particular place and does change with location. This calculator reports a force, so when the second mass is you and the first mass is a planet, the result is your weight on that planet. Does the distance mean surface to surface or centre to centre? Centre to centre, always, in this formula. For an object on a planet's surface, that is effectively the planet's radius. Using the height above the surface instead would give a wildly wrong answer, because the distance from the centre is what the law requires. Is the tool really free, and do I need an account? Yes, it is free, and no account is needed. There is no sign-up, no download, and no data uploaded. Open the page, enter two masses and a distance, press "Calculate Force", and read the answer. You can use it as often as you like, for homework, revision, checking work, or curiosity about the universe.

Related Tools on Your Smart Kit

If you found the gravitational force calculator useful, you may also like the other free calculators on Your Smart Kit, including the geometry calculator for working with shapes and angles, the unit converter for changing between measurement systems, and the mean median mode calculator for summarising a set of numbers. Each runs in your browser, free, with no sign-up, and each is built on the same principle: a simple tool that does one job well, keeps your data private, and gives an answer you can check. Try the gravitational force calculator at the top of this page with your own numbers, and see how a law written in 1687 still predicts, to the newton, the pull you feel every time you stand up.