Gravitational Force Calculator - Newton's Law of Gravitation

Newton's law of gravitation, from a sofa to the Sun

The Earth and the Moon pull on each other with 1.982 × 10²⁰ N, and two people sitting a meter apart pull on each other with 3.270 × 10^-7 N. Both numbers come from the same line, F = G m₁ m₂ / r², and this calculator runs it for any two masses and any distance, in kilograms or pounds and in meters, kilometers, feet or miles. A second mode turns the law around and shows how strongly eleven bodies pull at their surface, from Pluto to the Sun, and what a bathroom scale from home would read there.

Quick start. Pick "Weight and surface gravity", type 150, choose pounds and Mars: the scale reads 57.05 lb. For the pull between two bodies pick Newton's law and type the masses and the distance between their centers; scientific notation such as 5.972e24 is fine.

Eight cases, from two people on a sofa to the Sun and Earth

Case 1: the Earth and the Moon

Situation: 5.9722e24 kg and 7.346e22 kg, 384,400 km apart.
Result: 1.982 × 10²⁰ N. The Moon accelerates toward the Earth at 0.0027 m/s², the Earth toward the Moon at 3.318 × 10^-5 m/s².

Same force on both, very different response: that is the whole meaning of "each body pulls the other equally".

Case 2: two people a meter apart

Situation: 70 kg and 70 kg, 1 m.
Result: 3.270 × 10^-7 N, or 7.352 × 10^-8 lbf.

Real, but far too small to feel. Gravity only becomes noticeable when one of the masses is planet-sized.

Case 3: a person standing on the Earth

Situation: 5.9722e24 kg and 70 kg, 6,371,000 m.
Result: 687.42 N, which is the person's weight worked out from the law instead of from g.

Standard gravity gives 686.47 N for the same 70 kg. The 0.14 % gap is the Earth's spin and shape, which the plain law leaves out.

Case 4: the same person on the space station

Situation: as case 3, but at 6,771 km, the Earth's radius plus about 400 km of altitude.
Result: 608.60 N, 88.5 % of the pull on the ground.

Astronauts float because they and the station fall around the Earth together, not because gravity is gone.

Case 5: the Sun and the Earth

Situation: 1.9884e30 kg and 5.9722e24 kg, 1.496e11 m.
Result: 3.541 × 10²² N, about 179 times the Earth-Moon pull.

The distance is 389 times larger, but the Sun is so heavy that it still wins by a wide margin.

Case 6: a 150 lb reading taken to Mars

Situation: surface mode, 150 lb, Mars.
Result: the scale reads 57.05 lb, 38.0 % of Earth. A 1 ft hop on Earth becomes 2.63 ft.

At the Martian equator the value drops to 3.69 m/s² because of the spin.

Case 7: the same reading on Jupiter

Situation: surface mode, 150 lb, Jupiter.
Result: 396.47 lb on the mean value of 25.92 m/s², but 353.64 lb at the equator, where the just-under-ten-hour spin takes the value down to 23.12 m/s².

Jupiter has no solid surface, so these figures apply where its atmosphere is at 1 bar.

Case 8: a drop on the Moon

Situation: surface mode, 150 lb, Moon.
Result: the scale reads 24.78 lb; an object takes 1.11 s to fall 1 m, against 0.45 s on Earth, and a 1 ft hop becomes 6.05 ft.

The hop figure assumes the same push off the ground, without a pressure suit weighing you down.

Running your own case in either mode

Calculate - surface gravity for a planet, moon or star, or Newton's law for two masses you choose.
Mass and unit (surface mode) - what your scale shows at home, in pounds or kilograms. Mass does not change between worlds; the pull on it does.
Planet, moon or star - one body for the details, or all eleven as a table.
First and second mass (Newton mode) - any size. Commas may group thousands (6,371,000) and a period marks decimals; a decimal comma is refused rather than guessed at.
Distance between centers - center to center. For anything standing on a planet that is the planet's radius, and for anything above it the radius plus the height.
Units - one mass unit for both bodies and one distance unit. Then read the force, the acceleration of each body and what happens at twice and half the distance.

Surface gravity from Pluto to the Sun

Mean surface gravity is taken from NASA's planetary fact sheets, and each value matches NASA's own mass and mean radius run through Newton's law, which is why both modes of this calculator agree. The equator column is shown where the planet's spin takes off a visible share. The last column is what a scale reading 150 lb at home would show there.

Body	Mean g (m/s²)	At the equator	vs Earth	150 lb reads
Sun	274.0	-	2,794.0 %	4,191.03 lb
Jupiter	25.92	23.12	264.3 %	396.47 lb
Neptune	11.27	11.00	114.9 %	172.38 lb
Saturn	11.19	8.96	114.1 %	171.16 lb
Earth	9.807	9.780	100.0 %	150.00 lb
Uranus	9.01	8.69	91.9 %	137.81 lb
Venus	8.87	-	90.4 %	135.67 lb
Mars	3.73	3.69	38.0 %	57.05 lb
Mercury	3.70	-	37.7 %	56.59 lb
Moon	1.62	-	16.5 %	24.78 lb
Pluto	0.62	-	6.3 %	9.48 lb

Saturn is the odd one out. Its mean value, 11.19, puts it above Earth, while at its equator the spin brings it down to 8.96, below Earth, so the same 150 lb reads 137.05 lb there. Whether Saturn "has stronger gravity than Earth" depends on where you ask.

How fast the pull fades with distance

The distance is squared, so doubling it quarters the force. Starting from case 3, a 70 kg person at the Earth's surface, and moving outward:

Distance from the center	Force on 70 kg	Share of the surface pull
Half the radius (formula no longer applies inside)	2,749.68 N	400 %
1 radius, the surface	687.42 N	100 %
6,771 km, about the station's orbit	608.60 N	88.5 %
2 radii	171.86 N	25 %
3 radii	76.38 N	11.1 %
10 radii	6.87 N	1 %

The first row is there as a warning, not a result. Inside a planet only the mass below you pulls you toward the center, so the force falls toward zero on the way down instead of quadrupling. The calculator flags distances under 1,000 km when one of the masses is the size of a moon or planet.

Questions about gravity the law answers

What is the gravitational constant G?

The number that turns masses and distance into newtons: 6.67430 × 10^-11 m³ per kg per s², as recommended by CODATA and published by NIST. It is one of the least precisely known constants, with a relative uncertainty of about 22 parts per million, because gravity between lab-sized masses is so weak.

Why does the Moon accelerate more if the force is the same?

Acceleration is force divided by mass. The Moon has about 1.2 % of the Earth's mass, so the same 1.982 × 10²⁰ N moves it roughly 81 times faster. Both bodies circle a shared point that sits inside the Earth.

Why does the law give 687.42 N when 70 kg times 9.80665 is 686.47?

Standard gravity is a defined reference value, while the law here uses the Earth's mass and mean radius and ignores its spin. NASA lists the Earth's mean surface gravity as 9.820 m/s², and the effective value runs from 9.780 at the equator to 9.832 at the poles, so both figures are fair; they answer slightly different questions.

Is gravity the same everywhere on Earth?

No. It is weaker at the equator than at the poles, by about 0.5 % between 9.780 and 9.832 m/s², because the spin pushes outward most at the equator and the planet bulges there. Altitude matters too, which the distance table above shows on a larger scale.

How much weaker is gravity on the space station?

At about 400 km up it is still 88.5 % of the pull on the ground. The crew float because the station is in free fall around the planet, moving sideways fast enough to keep missing it.

What does "surface gravity" mean on a gas giant?

Jupiter, Saturn, Uranus, Neptune and the Sun have no ground to stand on, so the figure is given where the pressure is 1 bar, about sea-level pressure on Earth. For the fast-spinning giants the equatorial value is noticeably lower than the mean.

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