Gravitational Force Calculator

gravitational force = G × m₁ × m₂ / r²

G

The gravitational constant, 6.6743 × 10⁻¹¹ N·m²/kg². It is the least precisely measured constant in physics — the recommended value carries a relative uncertainty of 2.2 × 10⁻⁵, four orders of magnitude or more worse than the next least well known constant — so the last digits of any answer on this page are not meaningful, and no input you can type will change that

m₁

The mass of the first object, in kilograms. The field also takes grams, tonnes and pounds, and it has to be a mass rather than a weight: two 70 kg people are simply two quantities of matter here. It must be positive, because something with no mass exerts no gravitational force

m₂

The mass of the second object, in the same units and under the same rule. The two masses enter the formula multiplied together, so they are interchangeable — swapping them changes nothing, which is why the page needs no way to say which one is which

r

The distance between the two centres of mass, in metres, not the gap between their surfaces. A satellite 400 km above the ground is 6771 km from the centre of the Earth, and it is the second number the equation wants. The field also takes kilometres, centimetres, feet and miles, and it must be positive

F

The gravitational force the two objects exert on each other, in newtons and in pounds-force, in scientific notation. It is one force and not two: the Earth pulls you with 687 N and you pull the Earth with 687 N in the opposite direction. The second output row is the same number converted, not a second result

Use this page when you have two masses and the distance between them and want the pull: the force holding the Moon in its orbit, the attraction between a satellite and the planet it circles, whether the gravitational force between two laboratory masses is measurable at all (it is not — it is around 10⁻⁷ N, and that is why the experiment took until 1798), or what the surface gravity of a body would be if you fed its own mass and its own radius into this same equation. That last one is what the table below is: eight bodies with their surface gravity worked out this way, which is the number that decides how much you would weigh on each of them. Two things are worth checking before you trust an answer. First, r is a centre-to-centre distance: putting the altitude where the radius belongs overstates the force severely, by a factor of about 290 at 400 km, while ignoring the altitude and using the body's radius alone misses it by a little over ten percent, a gap that grows with altitude. Second, the formula treats each object as a point, which is exact for spheres and a good approximation for anything small compared with r — but two objects nearly touching are outside it entirely.

1. Two 70 kg people, one metre apart

   1. Masses: 70 kg and 70 kg, and the distance between their centres is 1 m
   2. Product of the masses: 70 × 70 = 4900 kg²
   3. Square of the distance: 1² = 1 m²
   4. Force: 6.6743 × 10⁻¹¹ × 4900 ÷ 1 = 3.2704 × 10⁻⁷ N
   5. In pounds-force: 3.2704 × 10⁻⁷ ÷ 4.4482 = 7.352 × 10⁻⁸ lbf

   A third of a micronewton, which is the answer that makes the whole subject land: two people standing next to each other are pulling with about the force that a few tens of micrograms of matter would weigh, and nothing in daily life reveals it. Notice how the arithmetic never leaves the exponent — 4900 is only four digits, and the factor that shrinks the answer is the 10⁻¹¹. That is the sense in which gravity is weak: not that the constant is small compared with other numbers, but that it is small compared with anything a human being can produce as mass.

2. The Earth and the Moon

   1. Masses: 5.972 × 10²⁴ kg for the Earth, 7.342 × 10²² kg for the Moon
   2. Distance between centres: 384400 km = 3.844 × 10⁸ m
   3. Product of the masses: 5.972 × 10²⁴ × 7.342 × 10²² = 4.3846 × 10⁴⁷ kg²
   4. Square of the distance: (3.844 × 10⁸)² = 1.4776 × 10¹⁷ m²
   5. Force: 6.6743 × 10⁻¹¹ × 4.3846 × 10⁴⁷ ÷ 1.4776 × 10¹⁷ = 1.980 × 10²⁰ N
   6. In pounds-force: 1.980 × 10²⁰ ÷ 4.4482 = 4.452 × 10¹⁹ lbf

   1.98 × 10²⁰ N is roughly 200 exanewtons, and it is the number that makes people think gravity is strong — it is the force that keeps a body 384000 km away turning a corner every month. Compare it with the previous example and the whole lesson is in the exponents: the masses grew by a factor of about 10⁴⁴, the distance by about 3.8 × 10⁸ (so 1.5 × 10¹⁷ once squared), and the force by 10⁴⁴ ÷ 10¹⁷ = 10²⁷. Nothing about the law changed between the two calculations; only how much matter was on either side of it.

3. The Earth and a 70 kg person standing on it

   1. The distance the formula wants is the Earth's mean radius: 6371 km = 6.371 × 10⁶ m, not the distance to the ground under your feet
   2. Product of the masses: 5.972 × 10²⁴ × 70 = 4.1804 × 10²⁶ kg²
   3. Square of the distance: (6.371 × 10⁶)² = 4.05896 × 10¹³ m²
   4. Force: 6.6743 × 10⁻¹¹ × 4.1804 × 10²⁶ ÷ 4.05896 × 10¹³ = 687.398 N
   5. In pounds-force: 687.398 ÷ 4.4482 = 154.533 lbf

   This is the same 687 N your bathroom scale reports as 70 kg, and it is worth being clear that the scale is measuring a force and dividing by 9.8 to print a mass. It is also the one row of this page that lands in ordinary notation: the answer sits between 10⁻³ and 10⁶, so the page leaves it as 687.398 rather than rewriting it as 6.874 × 10². And the figure is a little low on purpose — it uses the mean radius, so a person at the equator is 21 km further from the centre than a person at the pole and this single number is an average of the two.

The largest limitation is not in the model but in the constant. G is known to about four significant figures — the CODATA recommended value is 6.67430 with an uncertainty of 15 in the last two digits, a relative uncertainty of 2.2 × 10⁻⁵ — and it is the least well measured constant in physics, worse than the next least well known constant by four orders of magnitude or more. Every answer on this page therefore has a fourth digit that is decoration: 1.980 × 10²⁰ N is really 1.98 × 10²⁰ N, and typing more precise masses will not improve it. Beyond that, the equation treats both objects as points, which is exact only for spheres; it ignores every other force, so it says nothing about whether the two bodies are actually in orbit or falling towards each other; it ignores general relativity, which matters for Mercury's perihelion and for anything near a black hole; and it assumes the masses are constant, which fails for a rocket or an evaporating comet. At the small end the model is also being used outside its intended range: two objects a millimetre apart are not well described as points, and the attraction between two atoms is chemistry rather than this equation, which is why the distance field stops at 1 mm. Finally, the surface gravities in the table use mean radii and assume the body is a perfect sphere with no rotation, so the Earth's row comes out at 9.82 m/s² rather than the defined standard value of 9.80665 m/s².

What is the gravitational force formula?

F = G × m₁ × m₂ / r², where G is the gravitational constant 6.6743 × 10⁻¹¹ N·m²/kg², the two m terms are the masses in kilograms, and r is the distance between their centres in metres. Two 70 kg people a metre apart give 6.6743 × 10⁻¹¹ × 4900 ÷ 1 = 3.270 × 10⁻⁷ N. The r² in the denominator is what makes it an inverse-square law: move the two objects twice as far apart and the force falls to a quarter, not to a half.

Why is the gravitational force between two people so small?

Because G itself is small: 6.6743 × 10⁻¹¹, which is eleven orders of magnitude below 1. The masses are the only thing working in the other direction, and 70 kg apiece is not much mass. Put the same equation to work with a planet on one side and the exponent flips: the Earth and the Moon attract each other with 1.98 × 10²⁰ N. Nothing about the law is different between those two cases — gravity is weak, and it takes a planet's worth of matter to make it look otherwise, which is why the force that holds the solar system together is the same force that cannot pull two people together.

Is r the distance between the surfaces or between the centres?

Between the centres, always. For a person standing on the Earth, r is the Earth's radius, 6371 km, and not zero. For a satellite 400 km up, r is 6771 km and not 400 km — the altitude is a surface distance and has to have the radius added to it. This is the single most common way of getting the answer wrong, and the size of the error is worth seeing: typing the 400 km altitude where the radius belongs makes the force about 290 times too large, while leaving the altitude out altogether and using 6371 km still leaves it about 12 percent too large.

What is surface gravity and how is it worked out here?

Surface gravity is the acceleration a body gives to something standing on it, g = GM/R², and it is the same gravitational force formula with the second mass set to 1 and r set to the body's own radius. The table on this page does exactly that for eight bodies: the Moon comes out at 1.62 m/s², Earth at 9.82 m/s² and the Sun at 274 m/s². That number is what decides how much you would weigh there — a 70 kg person needs 687 N on Earth, 113 N on the Moon and about 19 kN on the Sun, which is why the question of landing on the Sun is answered before the temperature comes up.

How accurate is this calculation?

To about four significant figures, and the limit is the constant rather than the arithmetic. G is the least precisely known constant in physics: the recommended value is 6.67430 with an uncertainty of 15 in the last two digits, which is a relative uncertainty of 2.2 × 10⁻⁵, four orders of magnitude or more worse than the next least well known constant. So 1.980 × 10²⁰ N should be read as 1.98 × 10²⁰ N, and entering the masses to ten digits will not add a fifth meaningful figure. The input masses and distances are usually the looser end anyway — the Earth's mass is known to about one part in 10⁵, and the figure used here is the mean radius rather than the radius at your latitude.

Does the heavier object pull harder, or do they pull equally?

Equally. The two masses are multiplied together in the formula, so the force on the Earth from you and the force on you from the Earth are the same 687 N in opposite directions — there is no sense in which the Earth pulls harder. What differs is what that force does: 687 N accelerates you at 9.82 m/s² and accelerates the Earth at 1.15 × 10⁻²² m/s², which is why you fall and the planet does not visibly move. The same reasoning is why this page needs only one distance and one answer, rather than a pair of forces.

• Newton's Universal Law of Gravitation (College Physics 2e, §6.5) — the inverse-square law, the value of G, and the Cavendish experiment that first measured it — OpenStax
• Newton's law of universal gravitation — centre-to-centre distance, the shell theorem for spherical bodies, and the range over which the point-mass idealisation holds — Wikipedia
• CODATA Value: Newtonian constant of gravitation — the recommended 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² and its relative standard uncertainty of 2.2 × 10⁻⁵ — National Institute of Standards and Technology