Getting the Number Right
The first time I had to use this for actual field work, I learned pretty fast that just plugging Earth's mass and radius into a calculator doesn't get you what you need. Everyone learns g = 9.80665 m/s² in high school and calls it a day. That number is a convention, not a measurement. It's the standard gravity value defined by the 3rd CGPM in 1901, based on a theoretical smooth ellipsoid at sea level and 45 degrees latitude. If you're doing anything that requires real precision, that convention will drift far enough from reality to matter. Here is the actual starting point. Newton's second law says F = ma. Universal gravitation says F = GMm/r². Set them equal and cancel the test mass m, and you are left with g = GM/r². That is The Acceleration Of Gravity Formula Decoding The Constant That Governs Our Fall in its most stripped-down form. G is the gravitational constant, approximately 6.67430 × 10¹¹ m³ kg¹ s². M is Earth's mass, about 5.9722 × 10² kg. r is the distance from Earth's center of mass to your point of interest. Do the arithmetic with those numbers and you land right around 9.81 m/s² for a spherical Earth at the surface. That is the textbook answer. It is also insufficient for anything beyond introductory physics.
The Acceleration Of Gravity Formula Decoding The Constant That Governs Our Fall
The real problem is that Earth is not a sphere and it is not uniform. It bulges at the equator and flattens at the poles. The crust has density variations. The core is denser than the mantle. And the planet is spinning, which introduces a centrifugal term that reduces the apparent gravitational pull depending on where you are. All of that means g varies from place to place on the surface by roughly half a percent. That sounds small until you are calibrating instruments or running orbit simulations and your error budget is measured in parts per million. I remember a project where we were setting up a microgravity sensor array at a research station and the local gravity value from the standard tables was off by about 0.012 m/s² from what our instrument read. That seems absurdly wrong for something supposed to be constant. The issue was that the station sat on a basalt formation with a subsurface density anomaly that the WGS84 ellipsoid model completely ignores. The formula g = GM/r² assumes a point mass or a perfect sphere with uniform density. Neither describes the local geology. We ended up using a free-air correction based on the station's latitude and elevation, then layering on a Bouguer correction for the rock between the station and the reference ellipsoid. Only then did the numbers match. The practical formula most people actually need is not g = GM/r². It is the Somerfeld or WGS84 gravity formula, which accounts for latitude. The International Gravity Formula 1967 looks like this:
g() = 9.780327 (1 + 0.0053024 sin² - 0.0000058 sin²2) m/s² Here is the geographic latitude. At the equator this gives roughly 9.780 m/s². At the poles it gives roughly 9.832 m/s². The centrifugal effect of Earth's rotation accounts for about two-thirds of that difference. The remaining one-third comes from the equatorial bulge, which puts you farther from Earth's center at the equator than at the poles. Both effects are real and both reduce your measured g at low latitudes. Once you have the latitudinal value, you apply altitude corrections. A free-air correction subtracts approximately 0.3086 milligals per meter of elevation gain. One milligauss is 10 m/s², so that works out to about 3.086 × 10 m/s² per meter. At the top of Everest, roughly 8,849 meters above sea level, that correction alone drops g by about 0.027 m/s². You then add a Bouguer slab correction if you are working over terrain with known density, which adds roughly 0.1119 milligals per meter times the density ratio. For typical crustal rock at 2670 kg/m³, that adds about 0.008 m/s² back. The net result at high altitude is still a reduction in g, but not as large as the free-air term alone would suggest.
Get the Full Details
There is a counter-intuitive thing here that trips people up constantly. Increasing your distance from Earth's center reduces gravity, yes, but Earth's mass also increases as you go deeper because you are including more of the planet beneath you. Inside a uniform sphere, gravity decreases linearly with depth. Earth is not uniform, so the relationship is more complicated, but the general principle holds: at the core, gravity is actually higher than at the surface, around 10.7 m/s², because the immense mass surrounding you pulls in all directions and the net effect depends on the radial density profile. This is not something you will find in most introductory textbooks, and it matters if you ever work with seismic modeling or planetary interior physics. Another practical limitation nobody warns you about is the sensitivity of g to local topography. The Bouguer correction assumes an infinite horizontal slab of rock, which is never true. Mountains add mass that pulls sideways and upward. Valleys remove mass. In rugged terrain, a full terrain correction using a digital elevation model is necessary. I once spent three days manually computing terrain corrections for a valley site because the default Bouguer anomaly was throwing off a mineral exploration survey by enough to suggest a false void. The workaround was using a prism-based terrain correction algorithm that treats each grid cell as a finite rectangular prism rather than an infinite sheet. That brought the anomaly into line with the drill core data. If you are doing this for a school project, the textbook formula g = GM/r² with r equal to Earth's mean radius of 6,371 kilometers is perfectly adequate. If you are doing precision work, you need the full WGS84 gravity model or, better yet, a local gravimetric survey. There is no single number for gravity on Earth. The best you can do is pick a model appropriate to your required accuracy and understand what assumptions it makes. The constant is not constant. That is the whole point of decoding it.
For quick reference values, standard gravity is exactly 9.80665 m/s² by definition. Mean surface gravity is approximately 9.81 m/s². The range across the actual surface is roughly 9.76 to 9.83 m/s² depending on latitude, elevation, and local geology. When you see g listed as a single number in a handbook, check the footnote. It is almost certainly the standard value, not a measurement from any specific location.
