About this gravitational force calculator
Gravity is the most familiar force in daily life and the most far-reaching force in the universe, and a gravitational force calculator lets you apply Newton's law to any pair of masses, from two people in a room to the Earth and the Moon. Enter the two masses and the distance between their centres, and the tool returns the attractive force between them, together with the acceleration that force would give to the first mass. The results are a lesson in scale: the force between everyday objects is almost unimaginably small, while the force between planets is almost unimaginably large, and the same equation, with the same constant, produces both answers. That unity is why Newton's law remains one of the great achievements of science.
Newton's law of universal gravitation states that the force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between them: F = G times m1 times m2 divided by r squared. G, the gravitational constant, is about 6.674 times 10 to the minus 11 in SI units, a number so small that it explains at a glance why gravity between ordinary objects is negligible. The inverse-square dependence is equally important: double the distance and the force falls to a quarter, triple it and the force falls to a ninth, which is why the Moon, though massive, pulls on us far more gently than Earth does, and why the tides are dominated by the nearer Moon rather than the far more massive Sun.
Using the tool follows the formula directly. Enter the first mass, the second mass, and the distance between their centres, not between their surfaces, which is the detail that matters most when one of the bodies is a planet. The presets cover the cases people actually wonder about: your own weight on Earth, your weight on the Moon, the immense pull between Earth and the Moon, and the tiny attraction between two one-tonne masses ten metres apart. Results show the force in newtons, the acceleration of the first mass, and a plain-language comparison, and the history keeps each calculation so you can watch the same mass produce wildly different forces on different worlds.
The Earth and Moon example shows the equation at planetary scale. With Earth's mass at about 5.972 times 10 to the 24 kilograms, the Moon's at about 7.348 times 10 to the 22 kilograms, and a centre-to-centre distance of about 384.4 million metres, the force comes out at about 1.98 times 10 to the 20 newtons, a number so large that comparisons fail: it is roughly the weight of a hundred trillion fully loaded cargo ships. Your own weight on Earth is the same law at human scale: a 70-kilogram person at Earth's surface, 6,371 kilometres from its centre, feels a force of about 687 newtons, which divided by 70 kilograms gives the familiar surface gravity of about 9.81 metres per second squared. On the Moon, with a smaller mass and radius, the same person weighs only about 114 newtons, which is why Apollo astronauts could bound across the surface in long, slow leaps.
The two-tonne example is the most instructive of all, because it answers a question almost everyone asks: if gravity is universal, why don't we feel pulled towards furniture? Two masses of 1,000 kilograms each, ten metres apart, attract each other with a force of about 6.674 times 10 to the minus 7 newtons, roughly the weight of a grain of sand. That is far below anything human senses can detect, and it is why Henry Cavendish needed an exquisitely sensitive torsion balance in 1798 to measure gravity between laboratory masses at all. Gravity only becomes noticeable when at least one mass is planetary, which is why we live our lives in Earth's grip while remaining blissfully unaware of the pull of the desk beside us.
Beyond the classroom, Newton's law remains the working tool of spaceflight. Mission planners use it, refined by the full precision of orbital mechanics, to plot transfers between planets, to park satellites in stable orbits, and to use gravity assists, in which a spacecraft steals a tiny amount of a planet's orbital momentum to gain speed without fuel. The same law explains the tides, since the Moon pulls harder on the ocean facing it than on Earth's centre, and harder on the centre than on the far ocean, stretching the water into two bulges. It explains why planets orbit in ellipses, why galaxies hold together, and, with Einstein's later refinement, why clocks on GPS satellites must be corrected for both their speed and their altitude before your phone can locate you.
A few cautions will keep your results meaningful. Always use centre-to-centre distances: for a person on Earth that means adding Earth's radius, not using zero or your height above the ground. Remember that the force is mutual and equal: Earth pulls you with 687 newtons and you pull Earth with exactly the same force, though Earth's acceleration from it is imperceptibly small because its mass is so large. For objects near a planet's surface, the surface-gravity shortcut, weight equals mass times g, gives the same answer faster, but the full formula is what you need for orbits, for other worlds, and for any pair of masses in space. And note the limits: at extreme densities and speeds, Einstein's general relativity supersedes Newton, but for planets, moons, satellites, and spacecraft, Newton's law remains accurate to far better than any measurement you will make with this tool.
A second worked example, your weight on Mars, shows how surface gravity depends on both mass and radius. Mars has a mass of about 6.417 times 10 to the 23 kilograms and a radius of about 3,390 kilometres, so a 70-kilogram person on its surface feels a force of about 259 newtons, roughly 38 percent of their Earth weight, giving a surface gravity of about 3.71 metres per second squared. That is why Mars feels so different from the Moon despite being a planet: it is much more massive than the Moon but also much larger, and the two effects partly cancel. The same calculation explains why a small, dense world can have surprisingly strong gravity, and why a large, fluffy planet can have a surface gravity lower than Earth's despite outweighing it enormously.
The story of the gravitational constant is a story of patience. Newton published the law in 1687 but never knew the value of G, because it requires measuring the attraction between masses small enough to handle in a laboratory, and that attraction is vanishingly weak. Henry Cavendish finally measured it in 1798 with a torsion balance so sensitive that it could detect the pull of lead spheres weighing a few hundred pounds, an experiment sometimes called weighing the Earth, because combining G with Earth's radius and surface gravity gives Earth's mass. Modern measurements use far more sophisticated versions of the same idea, and G remains the least precisely known of the fundamental constants, a reminder that the weakest force is also the hardest to pin down.
A final thought experiment is worth trying in the tool: set the distance to twice Earth's radius and watch the force fall to a quarter, then halve the distance and watch it quadruple. That inverse-square behaviour is the single most important feature of gravity, and feeling it through numbers is more convincing than reading about it.