Newton's Law of Gravitation and Field Strength Calculator

Q: What is the difference between G and g?

G is a universal constant, 6.674 30 × 10−11 m³ kg−1 s−2, and it is the same number everywhere in the universe. g is a local field strength, GM/r², and it depends entirely on which body you are near and how far from its centre — 9.8 m/s² near the Earth's surface, 8.68 at the Space Station, 1.6 on the Moon, 274 at the Sun's photosphere. G is a constant of nature; g is a property of a place. The mnemonic that works is the capital letter: big G is big in scope and small in value, little g is the opposite.

F = Gm₁m₂/r² and g = GM/r² from the same inputs, with the choice of G made explicit — the published 6.674 30 × 10⁻¹¹, or the three-figure 6.67 your textbook prints — and what that choice costs in the answer.

Last updated 11 Nov 2026

Gravitational force, and the field strength behind it

F = Gm₁m₂/r², g = GM/r²

Which value of GAll three are the same measurement at different precisions, so this is a rounding choice and not a disagreement between sources. It is on the page because G is by far the worst-known of the constants a school course uses: its relative uncertainty is 2.2 parts in a hundred thousand, against about one part in 10¹⁰ for the Planck constant and exactly zero for the speed of light, which is now a defined number. Switching between the options moves the answer in the third or fourth significant figure, which is usually below the precision of everything else in the question — and that is worth seeing rather than being told.

First massIn kilograms. The default is the Earth, and it is worth knowing where that figure comes from: it is not a measurement. The measured quantity is GM = 3.986 004 × 10¹⁴ m³/s², known to seven figures from satellite motion, and the mass is that divided by G — which drags G’s uncertainty along with it. That is why published tables give GM and why the IAU resolution behind the figure recommends no value for G at all.

Second massIn kilograms. The default is a person. The two masses are interchangeable — swap them and the force is identical, which is Newton’s third law showing up inside the formula rather than being added to it. The accelerations are not interchangeable at all, and the page prints both.

Distance between the centresIn metres, centre to centre. The default is the Earth’s nominal equatorial radius, 6378.1 km, so the default answer is roughly the weight of a 70 kg person. Two traps. This is centre-to-centre and not surface-to-surface, which only coincides for point masses. And for a uniform sphere the whole mass may be treated as being at the centre — a theorem, not an approximation, and the reason the formula works at all for planets.

685.892NExample

The Earth (5.9722e24 kg) and a 70 kg person at 6 378 100 m, with the CODATA G

F = Gm₁m₂/r², and the same equation divided by a mass

F = Gm₁m₂/r² · g = F/m = GM/r² · GM = gR² · M = GM/G

G: the Newtonian constant of gravitation, 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻² ± 0.000 15 × 10⁻¹¹ (CODATA 2022). Universal, and the least well known constant in a school syllabus by a wide margin — 2.2 parts in a hundred thousand, against zero for c and for h, both of which are now exact by definition.
r: the distance between the centres, in metres, and squared. Squared means an inverse-square law: double the separation and the force drops to a quarter, triple it and to a ninth. It also means the force never quite reaches zero, which is why “infinity” appears in the potential-energy definition.
the sphere theorem: a uniform sphere attracts an outside body exactly as though all its mass sat at its centre. This is a theorem and not an approximation, which is the only reason the formula is usable for planets at all — Cambridge 9702 states it as outcome 13.2.1. Inside a uniform sphere it fails completely: the field there grows linearly with distance from the centre and is zero at the middle.
g = GM/r²: the gravitational field strength: force per unit mass, in N/kg, which is dimensionally the same as m/s². Dividing the force by the mass it acts on is why everything falls at the same rate — and the identity of field strength with free-fall acceleration is not a coincidence but the content of the equivalence principle.
GM = gR²: the rearrangement worth remembering, because it lets you get a body’s GM from its surface gravity and radius without ever knowing G or its mass. That is how the gravitational parameter of a small body is usually quoted, and the escape velocity page takes GM as its input for exactly this reason.

Worked example

The Earth (5.9722e24 kg) and a 70 kg person at 6 378 100 m, with the CODATA G
THE FORCE. F = Gm₁m₂/r² = 6.674 30 × 10⁻¹¹ × 5.9722 × 10²⁴ × 70 / (6.378 1 × 10⁶)². The numerator is 2.790 × 10¹⁶ and r² is 4.068 0 × 10¹³, giving 685.892 N — which is, as it should be, roughly the weight of a 70 kg person.
THE FIELD STRENGTH, WHICH IS THE SAME SUM WITHOUT THE PERSON. g = GM/r² = 685.892 / 70 = 9.798 45 m/s². Note that this is not 9.806 65. Standard gravity is a defined conventional value and this is a point-mass calculation at the Earth's equatorial radius, and the two are not the same kind of thing.
WHY THEY DIFFER, WHICH IS THE INTERESTING PART. Two reasons, neither of them in the formula. The Earth rotates, so a body resting at the equator uses part of the gravitational pull as its centripetal acceleration: ω²R = (7.292 115 × 10⁻⁵)² × 6.378 1 × 10⁶ = 0.0339 m/s². And the Earth is neither a sphere nor uniform, so GM/r² is not its real surface field. Using the nominal polar radius of 6356.8 km instead gives 9.864 m/s², a swing of 0.67 per cent from the equatorial figure for the same planet.
THE THIRD LAW, INSIDE THE FORMULA. The same 685.892 N acts on the Earth. Its acceleration is F/m₁ = 1.15 × 10⁻²² m/s², against 9.798 m/s² for the person — a ratio of 8.5 × 10²², which is the mass ratio. Equal forces, wildly unequal accelerations: that is the whole of why “the Earth falls towards you too” is true and useless.
AND WHAT ROUNDING G COSTS. With G = 6.67 × 10⁻¹¹, the force becomes 685.448 N — 0.064 per cent low. The published uncertainty on G is worth only 0.0045 per cent of the answer, so the textbook's rounding is about fourteen times larger than the entire uncertainty in humanity's knowledge of G. If you have a mass quoted to three figures, you cannot see either; quote your answer to the precision of your weakest input and say which G you used.

Where G sits among the constants a school course uses

This table is the context that makes G interesting rather than just another letter. Four of the constants in a school course have **no uncertainty at all**, because the 2019 revision of the SI defined the kilogram, the ampere, the kelvin and the mole by fixing them — c, h, e and N_A are now exact numbers, and the metre has been defined from c since 1983. Standard gravity has no uncertainty either, for a different reason: it was never a measurement, but a conventional value adopted for trade and weights, so no place on Earth is obliged to agree with it. G is the exception, and it is the only constant on this list whose value a new experiment could still move. The 2022 CODATA adjustment left it exactly where 2018 did, uncertainty included, because the published measurements still differ from one another by more than their own stated errors — which is why the relative uncertainty is 2.2 × 10⁻⁵ and not the 10⁻⁶ the individual experiments claim.
Constant	Value	Standard uncertainty	Relative uncertainty	Status
Speed of light in vacuum, c	299 792 458 m/s	0	0	exact — the metre is defined from it
Planck constant, h	6.626 070 15e-34 J s	0	0	exact since the 2019 SI revision
Elementary charge, e	1.602 176 634e-19 C	0	0	exact since the 2019 SI revision
Avogadro constant, N_A	6.022 140 76e23 /mol	0	0	exact since the 2019 SI revision
Standard acceleration of gravity, g_n	9.806 65 m/s²	0 (exact)	0	a defined conventional value, not a measurement
Earth’s GM (IAU 2015 nominal)	3.986 004e14 m³/s²	nominal	—	a conversion factor fixed by convention
Newtonian constant of gravitation, G	6.674 30e-11 m³/kg/s²	0.000 15e-11	2.2e-5	still a measurement, and the experiments disagree

One equation, two readings, and the one constant still worth measuring

F = Gm₁m₂/r² and g = GM/r² are the same statement, and the second is the first divided by a mass. That division is the whole reason objects of different masses fall at the same rate: double the mass of the falling body and gravity pulls twice as hard, but there is also twice as much inertia to accelerate, so the acceleration is unchanged. Field strength is force per unit mass, measured in N/kg, and N/kg is dimensionally identical to m/s² — which is why the same symbol g does duty for both and why the two meanings are almost never distinguished in practice. The deeper reason they are identical is the equivalence principle, and it is worth knowing that this is an experimental fact about the universe rather than an algebraic consequence of the formula.

The distance is centre to centre, and it is squared. Both of those cause trouble. Centre to centre means that for a person standing on the Earth the separation is 6378 km, not zero and not their height; the reason you may do this at all is a theorem, not an approximation — a uniform sphere attracts an external body exactly as if all its mass were concentrated at its centre, which Cambridge 9702 states as outcome 13.2.1. Inside the sphere the theorem fails outright: the field there is proportional to the distance from the centre and is zero at the middle, which is a standard follow-up question. The square means an inverse-square law: twice as far gives a quarter of the force, three times gives a ninth, and the force approaches zero only at infinite distance. That last point is why the gravitational potential energy convention uses infinity as its zero, and why choosing between mgh and −GMm/r is a judgement about how far you have moved rather than a matter of taste.

The two forces are equal and the two accelerations are not. On the default inputs the Earth pulls the person with 685.9 N and the person pulls the Earth with 685.9 N — exactly, with no approximation, because the formula is symmetric in m₁ and m₂. Newton’s third law is not an extra rule bolted on; it is visible in the algebra. The accelerations then differ by the mass ratio: 9.80 m/s² for the person against about 1.1 × 10⁻²² m/s² for the Earth. “The Earth falls up to meet you” is true and is one part in 10²³.

G is the one constant in a school syllabus that is still a measurement. Since the 2019 revision of the SI, the speed of light, the Planck constant, the elementary charge and the Avogadro constant are all exact by definition — their uncertainties are zero, not small. Standard gravity, 9.806 65 m/s², is exact too, for a quite different reason: it is a conventional value adopted for weights and measures and was never a measurement of any particular place. G is the outlier. CODATA 2022 gives 6.674 30 × 10⁻¹¹ with a standard uncertainty of 0.000 15 × 10⁻¹¹, a relative uncertainty of 2.2 parts in a hundred thousand, and the 2022 adjustment left both figures exactly where the 2018 one did — because the published experiments still disagree with one another by more than their own stated errors. Gravity is extraordinarily weak between masses you can put on a bench: two 70 kg people a metre apart attract with about 3 × 10⁻⁷ N, which is why the measurement is so hard.

Which is why this site tabulates GM and not M. The Earth’s mass is not measured; GM is, from the motion of satellites, to seven or more significant figures. The mass is GM divided by G, so it inherits G’s uncertainty and is about three orders of magnitude less certain than the product it came from. The IAU’s 2015 resolution behind the figures on these pages makes this explicit: it recommends nominal values for GM and declines to recommend one for G, asking instead that any publication needing a mass in kilograms state which estimate of G it used. The default first mass on this page, 5.9722 × 10²⁴ kg, is that division done with the CODATA value — and it is printed here as a derived number rather than as a fact about the Earth.

Frequently asked questions

What is the difference between G and g?

G is a universal constant, 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻², and it is the same number everywhere in the universe. g is a local field strength, GM/r², and it depends entirely on which body you are near and how far from its centre — 9.8 m/s² near the Earth’s surface, 8.68 at the Space Station, 1.6 on the Moon, 274 at the Sun’s photosphere. G is a constant of nature; g is a property of a place. The mnemonic that works is the capital letter: big G is big in scope and small in value, little g is the opposite.

Why can I treat a planet as a point mass at its centre?

Because of a theorem Newton proved: a uniform sphere attracts an external body exactly as though all its mass were at its centre. It is exact rather than approximate for a spherically symmetric body, which is why F = Gm₁m₂/r² works for planets at all. Two caveats. It only holds outside the sphere — inside a uniform one the field grows linearly from zero at the centre, so a tunnel through the Earth is a quite different problem. And real planets are not uniform spheres: the Earth is flattened at the poles and lumpy, which is exactly why its measured surface gravity does not match GM/r² and why satellite orbits have to be computed with correction terms.

Why is my calculated g at the Earth’s surface 9.798 and not 9.81?

Three separate reasons, and none of them is an arithmetic error. Which radius: the nominal equatorial radius gives 9.798 m/s² and the polar radius 9.864, a spread of 0.67 per cent for the same planet, so there is no single correct answer. Rotation: a body resting on the spinning Earth needs part of the pull for its centripetal acceleration, about 0.034 m/s² at the equator and nothing at the poles. And 9.806 65 is standard gravity, exact by definition as a conventional value for weights and measures rather than a measurement of anywhere, so no location has to match it. Measured surface gravity does vary by a few tenths of a per cent with latitude; this page does not print the published equator and pole figures because they could not be sourced for it.

Which value of G should I use in an exam?

The one on the paper’s data sheet, every time, even if it is cruder than the published figure. Most papers print 6.67 × 10⁻¹¹ and some print 6.674 × 10⁻¹¹, and the difference is 0.064 per cent in your answer — about fourteen times larger than the entire published uncertainty on G, and almost always below the precision of the other numbers in the question. What matters for marks is consistency and sensible significant figures: round at the end, quote to the precision of the weakest input, and if the question gave you a mass to three figures do not report a force to six.

Related calculators

References

Fundamental Physical Constants — Complete Listing, 2022 CODATA adjustment, NIST Standard Reference Database 121 (read 11 November 2026 from the machine-readable ASCII listing). Three lines are used across this batch and all three are quoted as printed: the Newtonian constant of gravitation is 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻² with a standard uncertainty of 0.000 15 × 10⁻¹¹ — a relative uncertainty of 2.2 × 10⁻⁵, which makes G the worst-known constant on this page by four orders of magnitude; the standard acceleration of gravity is 9.806 65 m s⁻² (exact), exact because it is a defined conventional value and not a measurement of anywhere; and the standard atmosphere is 101 325 Pa (exact). The 2022 adjustment is the current one and it did not move G: the value and its uncertainty are unchanged from 2018.
IAU 2015 Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties, XXIXth General Assembly of the International Astronomical Union (read 11 November 2026). Four numbers on this site come from its table: the nominal terrestrial equatorial radius 6.3781 × 10⁶ m, the nominal terrestrial polar radius 6.3568 × 10⁶ m, the nominal terrestrial mass parameter (GM) 3.986 004 × 10¹⁴ m³ s⁻², and the jovian and solar pairs. Two things about it are worth a student’s attention. The values are conversion factors, not measurements — they are fixed numbers close to the best estimates, so that everyone converting to SI gets the same answer. And the resolution gives no value for G: it notes that G is among the least precisely determined constants and asks that a publication needing a mass in kilograms state which estimate of G it used. That is exactly why this site tabulates GM and not M.
Cambridge International AS & A Level Physics 9702, syllabus for use in 2025, 2026 and 2027, Version 1 (cambridgeinternational.org, read 11 November 2026). Quoted by outcome number so that the difference from the Indian syllabus is checkable rather than asserted. Topic 13 is the gravitation half of this page. 13.1.1 defines gravitational field as “force per unit mass”; 13.2.1 is the sphere theorem, “for a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre”; 13.2.2 is F = Gm₁m₂/r²; 13.3.1 asks the candidate to derive g = GM/r² from the law of gravitation and the definition of the field, which is the division this page prints as a row; and 13.3.3 asks them to “understand why g is approximately constant for small changes in height near the Earth’s surface”.