Gravitational Field Strength — The Complete Physics Guide
Every object with mass warps the space around it — creating a gravitational field that pulls every other mass toward it. Gravitational field strength (g) quantifies exactly how strong that pull is at any given point: the force per unit mass a test object would feel there. On Earth's surface, this value — approximately 9.81 N/kg — is so familiar it goes by another name entirely: the acceleration due to gravity, the same 9.81 m/s² that appears throughout kinematics and projectile motion problems.
That g can be expressed equally validly in N/kg (a field strength) or m/s² (an acceleration) is not a coincidence — it's a direct consequence of Newton's second law, and reflects one of the deepest and strangest facts in all of physics: gravitational mass and inertial mass appear to be exactly, perfectly identical, a mystery Einstein would later elevate into the founding principle of general relativity.
What is Gravitational Field Strength?
Gravitational field strength (g) at a point is defined as the gravitational force per unit mass that would act on a small test mass placed there: g = F/m. Since Newton's Law of Universal Gravitation gives F = GMm/r² for the force between two masses, dividing through by the test mass m leaves g = GM/r² — the field depends only on the source mass M and the distance r, exactly paralleling how electric field strength depends only on source charge and distance, not on whatever test charge or test mass happens to be present.
Field lines for gravity always point inward, toward the source mass — unlike electric fields, which can point either outward (positive charges) or inward (negative charges), gravity is exclusively attractive. There is no such thing as "negative mass" that would repel, at least not in any form yet observed in nature.
The Formula Explained
g is the gravitational field strength. G is the universal gravitational constant — one of the least precisely known fundamental constants in physics, owing to gravity's extreme weakness making it notoriously difficult to measure in a laboratory. M is the mass of the body generating the field (a planet, moon, or star). r is the distance from the centre of that body — a detail students frequently overlook, since for a body like a planet, r must be measured from the centre of mass, not from the surface.
How to Use This Calculator
Use "M & r" — the most common use case — to find the surface gravity of any planet, moon, or star given its mass and radius; quick-select buttons provide real values for Earth, the Moon, Mars, Jupiter, and the Sun. Use "g & r" to find the mass of a body from a measured field strength and known radius. Use "g & M" to find how far from a body's centre you'd need to be to experience a specific field strength.
Worked Example 1 — Earth's Surface Gravity
Problem: Calculate g at Earth's surface, given M = 5.972×10²⁴ kg and r = 6.371×10⁶ m.
g = GM/r² = (6.674×10⁻¹¹)(5.972×10²⁴) / (6.371×10⁶)²
g = 9.82 N/kg — matching the familiar value of 9.81 m/s² used throughout kinematics
Worked Example 2 — Gravity on the Moon
Problem: Find surface gravity on the Moon (M = 7.342×10²² kg, r = 1.737×10⁶ m) and compare it to Earth's.
g_Moon = (6.674×10⁻¹¹)(7.342×10²²) / (1.737×10⁶)²
g_Moon = 1.62 N/kg — about 16.5% of Earth's gravity, explaining the distinctive bouncing gait of Apollo astronauts
Worked Example 3 — Weighing a Planet
Problem: A newly discovered planet has surface gravity 15.0 N/kg and radius 5.0×10⁶ m. Find its mass.
M = gr²/G = (15.0)(5.0×10⁶)² / (6.674×10⁻¹¹)
M = 5.62×10²⁴ kg — this is exactly how astronomers determine planetary masses from measured surface gravity or orbital data, without ever needing to place the planet on a scale
Why g Decreases with Altitude
Because g follows an inverse-square law, gravitational field strength decreases measurably even over relatively modest altitude changes — g at the cruising altitude of a commercial airliner (around 10 km) is about 0.3% weaker than at sea level, and g in low Earth orbit (around 400 km, where the International Space Station orbits) is still roughly 90% of its surface value. Astronauts aboard the ISS aren't in "zero gravity" as popular language suggests — they experience near-Earth gravity but are in continuous free fall around the planet, which is what produces the sensation of weightlessness, not an absence of gravitational field.
This distinction matters: true weightlessness (zero gravitational field) only occurs at enormous distances from any mass, while the "weightlessness" felt by orbiting astronauts is really the experience of continuous free fall — g is still very much present and pulling the station (and everything in it) toward Earth at every instant, just as it pulls a skydiver during the moments before their parachute opens, producing an identical physical sensation despite the very different circumstances.
Common Mistakes
Measuring r from the surface instead of the centre: the formula requires distance from the body's centre of mass. For a point on the surface, r equals the planet's radius; for a satellite in orbit, r equals the planet's radius plus the orbital altitude.
Confusing g with G: lowercase g is the local gravitational field strength (varies by location and body), while uppercase G is the universal gravitational constant (the same everywhere in the universe). Mixing the two up is one of the most common notational errors in introductory mechanics.
Superposition — Fields from Multiple Bodies
When more than one massive body is present, the total gravitational field at any point is the vector sum of the fields from each individual body — the same superposition principle that applies to electric fields and forces. This is why calculating the precise trajectory of a spacecraft travelling between Earth and the Moon requires accounting for both bodies' gravitational fields simultaneously, and why the tides on Earth arise from the combined (and constantly shifting) gravitational influence of both the Moon and the Sun.
Superposition also explains why the gravitational field strength exactly at the centre of a spherically symmetric body is zero — by symmetry, the pull from mass on every side cancels out perfectly at the exact centre, even though the total mass of the body is nonzero. This is a genuinely useful fact in geophysics, where the field strength at different depths within the Earth doesn't simply follow g = GM/r² using Earth's full mass, but instead depends on the mass enclosed within that depth, following a modified relationship derived from Gauss's law applied to gravity.
Beyond Newton — Gravity as Curved Spacetime
Newton's law of gravitation, and the field strength formula derived from it, describes gravity with extraordinary accuracy for almost every practical purpose — sending spacecraft to other planets, predicting tides, and calculating orbital mechanics. But it isn't the deepest description available. Einstein's general theory of relativity (1915) reveals that gravity isn't really a force propagating through space at all, but a consequence of massive objects curving the very fabric of spacetime itself — objects move along the straightest possible paths through this curved geometry, which appear to us as gravitational attraction.
For everyday calculations — and even for most of astrophysics — Newton's simpler g = GM/r² remains an excellent approximation, differing from Einstein's more complete picture only in extreme conditions: very strong gravitational fields (near a black hole or neutron star), very high precision requirements (GPS satellites must correct for relativistic effects to remain accurate), or very large cosmic scales. This calculator uses the Newtonian formula, appropriate and highly accurate for the vast majority of real-world gravity problems encountered in physics education and engineering.