Gravitational Acceleration Calculator

Definition of gravitational field strength

Gravitational field strength at a point is the acceleration a small test body would have there. For a point mass, or for a sphere outside its surface, Newton wrote it as g = GM/r², where G is the gravitational constant, M is the source mass, and r is the distance from the center. The SI unit is m/s², the same unit as a kinematic acceleration.

This formula differs from the force F = GMm/r² by cancellation of the test mass m. That is why g does not depend on whether you drop a key or a feather. The distance dependence is inverse-square: twice the r gives four times less g. This is a property of the field, not of the motion that the formulas h = ½gt² and t = √(2h/g) describe later.

History: Newton, Cavendish, and the constant G

Isaac Newton stated the law of universal gravitation in the Principia (1687): two bodies attract each other with a force proportional to the product of the masses and inversely proportional to the square of the distance. The field acceleration is that force divided by the test mass. For more than a century G stayed hidden inside the product GM, because pendulums and orbits measure GM, not G alone.

Henry Cavendish measured the attraction of lead spheres with a torsion balance in 1798 and thereby made a numerical G possible. The value recommended by CODATA and published by NIST is 6.67430×10⁻¹¹ m³/(kg·s²), with a relative uncertainty of order 2×10⁻⁵. The 2018 and 2022 adjustments give the same number, because no newer measurement changed the result. The calculator inserts it when the G field is blank.

Deriving g = GM/r² from the law of gravitation

Step 1. The law of gravitation: F = G·M·m / r². Step 2. Newton's second law for the test body: F = m·a, so a = F/m. Step 3. The mass m cancels and a = G·M/r² remains. That acceleration is written g when the source is a planet or a star. Step 4. Circular-orbit speed follows when g equals the centripetal acceleration v²/r, so v = √(GM/r) = √(g·r).

For Earth: G·M = 6.67430×10⁻¹¹ · 5.972×10²⁴ = 3.9859×10¹⁴ m³/s². At r = 6,371,000 m the square is 4.0590×10¹³ m², and the quotient is 9.82 m/s². The square root of GM/r is 7,910 m/s. None of these steps rounds the product before the division; rounding to two decimal places applies only to the g that is shown.

Assumptions: a point, a sphere, and distance from the center

The formula is exact for a point mass and, outside the surface, for a spherically symmetric sphere. The distance r is measured from the center, not from the surface. Altitude above the ground must be added to the radius: 400 km above the surface at a radius of 6371 km is r = 6771 km, not 400 km. Entering the altitude alone in the r field would produce a field that is far too strong.

Inside a uniform sphere a different expression holds: g grows linearly from zero at the center to the surface value. This calculator does not use it. It also does not compute tides, flattening, or the centrifugal effect of daily rotation. At Earth's surface those effects shift local g by a few thousandths of a meter per second squared relative to a pure GM/R², which is why 9.82 m/s² and the conventional gₙ = 9.80665 m/s² are not the same number.

Earth, the ISS, and the difference from 9.81

Standard gravity gₙ = 9.80665 m/s² was adopted by the 3rd General Conference on Weights and Measures in 1901 as a conventional value. Real g on the surface ranges from about 9.780 m/s² at the equator to about 9.832 m/s² at the poles. The quotient GM/R² for the mean radius is a different number: 9.82 m/s² with the CODATA G.

In an orbit at an altitude of 400 km, g = 8.69 m/s². That is still 0.887 gₙ, so saying there is “no gravity” on the station describes weightlessness relative to the cabin, not the disappearance of the field. A summit 8,849 m above the mean radius lowers the result only to 9.79 m/s². A mountain does not switch the field off.

The Moon, Mars, and the Sun's field on Earth's orbit

The same formula, different M and r. The Moon at 7.35×10²² kg and 1,737,000 m gives 1.63 m/s². Mars at 6.39×10²³ kg and 3,390,000 m gives 3.71 m/s². The Sun at 1.989×10³⁰ kg and a distance of 1.496×10¹¹ m gives 5.932×10⁻³ m/s². That acceleration is small, but it acts continuously and bends Earth's yearly orbit.

The calculator does not turn this g into a force, a fall time, or an escape speed as the primary result. It shows circular-orbit speed beside the result, because that speed is √(GM/r). The force between two masses has its own calculator, F = GMm/r². Escape from a given radius is covered by the second cosmic velocity calculator. This calculator reports the field itself.

Sources

The definition, the derivation, and the constant come from Newton's law of gravitation, Cavendish's measurement, and metrology recommendations: I. Newton, Philosophiæ Naturalis Principia Mathematica, 1687; H. Cavendish, Experiments to determine the density of the Earth, Philosophical Transactions of the Royal Society, 1798; CODATA 2022, Newtonian constant of gravitation, physics.nist.gov/constants (G = 6.67430×10⁻¹¹, identical to the 2018 adjustment); BIPM, The International System of Units (SI), 9th edition, and the 1901 resolution of the 3rd CGPM on gₙ; D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, the chapter on universal gravitation; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 7; NASA National Space Science Data Center, Planetary Fact Sheet, as the source of the orders of magnitude for masses and radii in the examples, not as the formula. This page is for educational purposes.