How this instrument works
Release a hammer and a feather together in vacuum and both quicken at one identical rate, because g belongs to whatever is doing the pulling and never to what falls. Newton's force law carries two masses; divide through by whichever one is dropping and it disappears, leaving G·M ⁄ r² — a number attached to a place rather than to an object. Read that number as newtons per kilogram and it tells you weight; read it as metres per second squared and it tells you fall. One quantity, one set of digits, two ways of caring about it.
Since g shifts from place to place, measuring it grew into a discipline of its own. Its unit honours Galileo: a gal is one centimetre per second squared, and field crews work in milligals, roughly a millionth of Earth's pull. An absolute gravimeter drops a corner-cube mirror down an evacuated tube and times its descent by laser interferometry against an atomic clock, resolving a microgal — sensitive enough to register having been carried up one flight of stairs. Surveys flown across sedimentary basins find buried salt domes by their deficit, and since 2002 GRACE satellite pairs have weighed vanishing groundwater and thinning ice from orbit, purely by how much a passing field sags.
Three conditions sit beneath that quotient. Mass must be spherically symmetric, so a single point at middle can stand in for all of it — fine for a planet, poor for a rubble-pile asteroid. Nothing may spin: rotation hands Earth's equator about 0.034 m/s² of centrifugal relief, and a 21 km equatorial bulge holds sea level further out there, which together lift a measured 9.7803 m/s² at low latitude to 9.8322 m/s² near either pole. Gravity must also stay weak, easily satisfied on any rocky world. One consequence worth carrying: climbing thins g by roughly 3.1 microgal per centimetre, so a barometric height error of ten metres is already enough to ruin a survey.
- Put your figure into Mass of the body, switching that field to tonnes if kilograms would mean typing an exponent by hand.
- Fill Distance from its centre in metres or kilometres, measured outward from middle of the body rather than up from its surface.
- Read Surface gravity in m/s². Six digits are carried, because useful comparisons between two sites often live in a third decimal.
- Flip that output to ft/s² for imperial work, or to g₀ to see your answer expressed as a multiple of standard gravity.
Worked example — Earth, taken at its mean radius
Earth's mass, 5.9722 × 10²⁴ kg, goes into Mass of the body, and 6,371,000 m — its mean radius — into Distance from its centre. Multiplying gives G·M = 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ = 3.986026 × 10¹⁴ m³/s². Squaring gives r² = 4.0589641 × 10¹³ m². Divide one by another and Surface gravity reads 9.82030229339 m/s².
That sits 0.14% above 9.80665 m/s², and every bit of that gap is honest physics rather than rounding. Standard gravity was fixed by convention at the third General Conference on Weights and Measures in 1901, approximating sea level near 45° latitude; our planet meanwhile spins, and stretches 21 km wider across its equator than pole to pole, so no single radius ever described a sphere it was not. Feed the Moon's 7.346 × 10²² kg and 1,737,400 m into these same two fields and Surface gravity returns 1.62 m/s². Mars gives 3.73, Jupiter's cloud tops near 25, and photosphere of our Sun about 274.
Questions
Why does this return 9.82 rather than 9.81?
Because G·M ⁄ r² describes a perfect, motionless sphere, and our planet is neither. Spin removes up to 0.034 m/s² of apparent weight at low latitude, while an equatorial bulge of 21 km puts sea level further from middle there than at a pole. Real sea-level readings therefore run from 9.7803 m/s² on the equator to 9.8322 m/s² in polar regions, with 9.82 falling naturally inside that band. Standard gravity, 9.80665 m/s², is not a measurement at all but an agreed convention — and it is what engineering tables, the kilogram-force and aircraft load factors are built upon.
If gravity is nearly full strength in orbit, why do astronauts float?
Gravity really is almost undiminished up there. Put 5.9722 × 10²⁴ kg and 6,771,000 m — a 400 km orbit — into these two fields and Surface gravity returns 8.69 m/s², roughly 89% of what presses you into a chair right now. Crews float because station and occupants fall around our planet together at matching rates, leaving nothing to push between them. Weightlessness reports how something is moving, not how much pull surrounds it; genuinely negligible g would demand an r vastly larger than any orbit.
Is lowercase g the same thing as capital G?
No — they differ in size, unit and meaning. Capital G is Newton's gravitational constant, 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻², shared by every pair of masses anywhere. Lowercase g is a local field strength in m/s² that varies with where you stand: 9.82 at Earth's surface, 1.62 on our Moon, about 274 at solar photosphere. Capital G is an input to this formula, held fixed; lowercase g is what emerges from it. Mixing them up by even a single case produces answers wrong by thirty-five orders of magnitude.
Should g be entered or reported as negative?
Surface gravity here is always positive, since this instrument reports magnitude and direction is understood to point toward middle of the attracting body. Signs belong to kinematics instead. Pick an upward-positive vertical axis and acceleration becomes −9.82 m/s²; pick downward-positive and it becomes +9.82 m/s². Either choice works so long as displacement, velocity and acceleration all obey it — and quietly mixing both is a classic route to a projectile that appears to land before launch.
How much does g really vary across Earth's surface?
About half a percent between extremes, which sounds negligible until you weigh something valuable. Sea level spans 9.7803 m/s² at low latitude to 9.8322 m/s² at a pole. Height removes a further 3.1 microgal per centimetre climbed, near 0.003 m/s² atop a 1,000 m plateau. Local rock density shifts things again — dense basalt reads high, thick sediment reads low, which is precisely what gravity prospecting sells. A two-pan balance comparing masses is immune, but any scale that senses force must be recalibrated wherever it is installed.
What separates mass from weight in these fields?
Mass counts how much matter is present and never changes; weight is a force, m·g, which follows g wherever you carry it. An 80 kg person remains 80 kg on Mars, yet weight drops from 786 N here to 298 N there. Bathroom scales sense force and quietly divide by an assumed 9.80665, which makes them, strictly speaking, force instruments wearing mass labels. SI keeps a firm line between both: kilograms for mass, newtons for weight, and this page supplies whichever g that conversion needs.