How It Works
Gravitational acceleration is the acceleration any free-falling object experiences at distance r from a massive body of mass M: g = GM / r², where G = 6.6726 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant. The formula comes directly from Newton's law of gravitation divided by the test mass — that's why all objects fall with the same g regardless of their own mass (Galileo's principle of equivalence). This calculator solves the equation for any one of three unknowns — acceleration, radius, or central mass — given the other two.
Example Problem
Calculate Earth's surface gravitational acceleration. Earth's mass is M = 5.972 × 10²⁴ kg and its mean radius is r = 6.371 × 10⁶ m.
- Write the gravitational acceleration formula: g = G × M / r².
- Compute the numerator: G × M = (6.6726 × 10⁻¹¹) × (5.972 × 10²⁴) ≈ 3.986 × 10¹⁴ m³/s².
- Compute the denominator: r² = (6.371 × 10⁶)² ≈ 4.058 × 10¹³ m².
- Divide: g = 3.986 × 10¹⁴ / 4.058 × 10¹³ ≈ 9.82 m/s².
- Compare to the standard reference value g₀ = 9.80665 m/s² — the tiny excess comes from rounding Earth's radius and mass; local variations from latitude, altitude, and density anomalies cause the actual surface g to range from about 9.78 to 9.83 m/s².
Key Concepts
Gravitational acceleration g depends only on the source mass M and the distance r — never on the falling object's own mass. The inverse-square fall-off is steep: at 400 km altitude (ISS), g drops to ≈ 8.7 m/s², about 89% of surface value, which is why astronauts inside the ISS experience apparent weightlessness as the station continuously free-falls around Earth. Other planets give very different values: the Moon's surface gravity is 1.62 m/s² (about 1/6 of Earth's), Mars 3.71 m/s², Jupiter 24.79 m/s², and the Sun a crushing 274 m/s². Use this calculator to find g on any body or to invert the formula and estimate a planet's mass from its measured surface gravity.
Applications
- Aerospace engineering: computing local g for spacecraft trajectory simulations and reentry calculations.
- Geophysics: measuring tiny variations in surface g to map subsurface density (oil, mineral, and groundwater exploration).
- Astronomy: estimating planetary masses from observed surface gravity or from satellite orbital data.
- Physics education: demonstrating that g = GM/r² produces the familiar 9.81 m/s² from first principles.
- Lunar and Martian mission planning: setting equipment weight budgets based on each body's surface gravity.
Common Mistakes
- Confusing the universal constant G (6.6726 × 10⁻¹¹ N·m²/kg², constant everywhere) with the local gravitational acceleration g (≈ 9.81 m/s² at Earth's surface, varies with location).
- Using altitude instead of distance from the planet's center for r — the formula needs the full center-to-center distance, not the height above the surface.
- Assuming g is constant with altitude — it actually decreases as 1/r², so satellites in high orbits experience much less gravity than at sea level.
- Forgetting that on a planet's surface, r is the planet's radius — not zero — because gravitational acceleration is computed at the center-of-mass distance.
- Mixing mass and length units (e.g., kilograms with kilometers) without converting to SI — the formula returns m/s² only when M is in kg and r is in meters.
Frequently Asked Questions
How do you calculate gravitational acceleration?
Apply the formula g = G × M / r², where G = 6.6726 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant, M is the mass of the central body in kilograms, and r is the distance from that body's center in meters. The result g comes out in m/s².
What is the formula for gravitational acceleration?
g = G × M / r², where G is the universal gravitational constant, M is the mass of the body creating the gravity, and r is the distance from its center. This is the local gravitational acceleration felt at any point r away from a mass M.
Why is gravity 9.81 m/s² on Earth?
Substituting Earth's mass (5.972 × 10²⁴ kg) and mean radius (6.371 × 10⁶ m) into g = GM/r² gives approximately 9.82 m/s². The standard reference value 9.80665 m/s² is a defined average that accounts for Earth's oblateness, rotation, and local density variations.
Does gravitational acceleration depend on the falling object's mass?
No. The formula g = GM/r² shows g depends only on the central body's mass M and the distance r — not on the falling object. This is why a feather and a hammer fall at the same rate in vacuum on the Moon, a result demonstrated by Apollo 15 astronauts in 1971.
How does gravitational acceleration change with altitude?
It falls off as the inverse square of the distance from the planet's center. At the ISS's 400 km altitude (r ≈ 6,771 km), g ≈ 8.7 m/s² — about 89% of the surface value. At geostationary altitude (~36,000 km), g drops to about 0.22 m/s².
What is gravitational acceleration on other planets?
Moon: 1.62 m/s² (about 1/6 of Earth's). Mars: 3.71 m/s². Venus: 8.87 m/s². Jupiter (cloud tops): 24.79 m/s². Sun: 274 m/s². Use this calculator with each body's mass and radius to verify these values.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
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