Gravitational Force Calculator

Calculate the gravitational attraction between any two masses using F = Gm₁m₂/r². Covers planets, moons, and everyday objects.

🌍 F = Gm₁m₂/r²⚡ Gravity🛰️ Orbital
Mass 1 (m₁)
Unit
Mass 2 (m₂)
Unit
Distance (r) between centers
Unit
⚠️ Enter valid positive numbers. G = 6.674×10⁻¹¹ N·m²/kg².

What Is Newton's Law of Gravitation?

Newton's Law of Universal Gravitation states that every mass in the universe attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers: F = Gm₁m₂/r².

The gravitational constant G = 6.674 × 10⁻¹¹ N·m²/kg² is one of the fundamental constants of nature, first measured by Henry Cavendish in 1798 using a torsion balance experiment. It is extraordinarily small, meaning gravity is by far the weakest of the four fundamental forces — yet it dominates the universe at large scales because it acts over infinite distances and is always attractive (unlike electromagnetism, which has both charges).

The inverse-square relationship (∝ 1/r²) is crucial. Doubling the distance reduces force to 1/4; tripling it reduces to 1/9. This explains why gravity weakens rapidly with altitude, why planets closer to the Sun orbit faster (Kepler's Laws follow directly from this), and why a satellite in low Earth orbit experiences much stronger gravity than one in geostationary orbit.

While Newton's law is extremely accurate for most purposes, it is superseded by Einstein's General Theory of Relativity for extreme cases: very strong fields (near black holes), very precise measurements (GPS requires relativistic corrections), and cosmological scales. For all everyday, engineering, and most astrophysical calculations, F = Gm₁m₂/r² is indispensable.

Formula Reference

Quantity Formula Notes
Gravitational Force F = G·m₁·m₂ / r² G = 6.674×10⁻¹¹ N·m²/kg²
Surface gravity g = GM/R² M = planet mass, R = radius
Earth's surface g g = 9.8 m/s² M_Earth = 5.97×10²⁴ kg, R = 6.37×10⁶ m
Escape velocity v_e = √(2GM/r) Minimum speed to escape gravity
Orbital velocity v_o = √(GM/r) Circular orbit at radius r
Gravitational PE PE = −GMm/r Negative; zero at infinity

3 Worked Examples

Example 1
Earth–Person — Confirming g

Verify Earth's surface gravity using F = Gm₁m₂/r² for a 70 kg person.

  • M_Earth = 5.97×10²⁴ kg, R = 6.37×10⁶ m
  • F = (6.674×10⁻¹¹ × 5.97×10²⁴ × 70) / (6.37×10⁶)²
  • F = (2.789×10¹⁶) / (4.058×10¹³) = 687 N
  • Check: F = mg = 70 × 9.8 = 686 N ✓
✓ Gravitational force = 687 N (matches mg)
Example 2
Moon's Gravity — Surface g

Calculate the Moon's surface gravitational acceleration (M = 7.35×10²² kg, R = 1.74×10⁶ m).

  • g_Moon = GM/R² = (6.674×10⁻¹¹ × 7.35×10²²) / (1.74×10⁶)²
  • Numerator: 4.906×10¹²
  • Denominator: 3.028×10¹²
  • g_Moon = 1.62 m/s² (16.5% of Earth's g)
✓ Moon surface gravity = 1.62 m/s²
Example 3
Two Mountains — Unexpected Attraction

Two adjacent mountains each 10⁹ kg are separated by 500 m center-to-center. What is their mutual attraction?

  • F = G×m₁×m₂/r² = 6.674×10⁻¹¹ × 10⁹ × 10⁹ / 500²
  • F = 6.674×10⁻¹¹ × 10¹⁸ / 250,000
  • F = 6.674×10⁷ / 2.5×10⁵ = 267 N
  • About 27 kg-force — measurable but small
✓ Gravitational attraction = 267 N (≈27 kg-force)

Real-World Applications

🛰️
Satellite Orbits
GPS, communications, and weather satellites orbit at specific altitudes where gravitational force provides exactly the centripetal force needed: GMm/r² = mv²/r → orbital velocity.
🌊
Tidal Forces
The Moon's gravity differs between Earth's near and far sides by ~10⁻⁷ m/s² — a tiny differential that stretches Earth's oceans, creating tides twice daily. The Sun contributes ~46% of tidal effect.
🔭
Dark Matter Detection
Galaxy rotation curves don't match visible mass. The extra gravitational force needed implies dark matter: F = Gm₁m₂/r² with unseen mass. Its detection is one of physics' greatest open problems.
🚀
Gravity Assists
Spacecraft like Voyager use planetary gravity to gain speed through hyperbolic fly-bys. The planet's gravity does work on the spacecraft, changing its velocity vector and speed in the heliocentric frame.
⏱️
Gravitational Time Dilation
GPS satellites run fast by 45 μs/day due to weaker gravity at altitude. Einstein's GR predicts clocks run faster in weaker gravitational fields. Without corrections, GPS would drift 10 km/day.

Common Mistakes

⚠️
Forgetting G is incredibly small

G = 6.674×10⁻¹¹. Without planetary masses (10²⁴ kg), gravitational forces between everyday objects are unmeasurably tiny. Two 100 kg people 1 m apart feel only 6.7×10⁻⁷ N — far too small to notice.

⚠️
Using diameter instead of radius

r in F = Gm₁m₂/r² is the distance between centers, not between surfaces. Earth's radius is 6,371 km from center — not the number to use for surface separation problems.

⚠️
Confusing g with G

G is the universal constant (6.674×10⁻¹¹ N·m²/kg²). g is the surface gravitational acceleration (9.8 m/s² on Earth, varies by planet). They're related by g = GM/R² but are very different quantities.

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Squaring before converting units

Always convert r to meters before squaring. If r = 400 km = 4×10⁵ m, then r² = 1.6×10¹¹ m² — not 400² = 160,000 (in km²).

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Applying it to uniform spherical shells incorrectly

For a uniform sphere, the shell theorem states: external point feels all mass concentrated at center; internal point feels only mass inside its radius. The formula F = Gm₁m₂/r² is exact only for point masses or from outside uniform spheres.

Frequently Asked Questions

What is the formula for gravitational force? ▾
Newton's law of universal gravitation gives F = Gm₁m₂/r², where G is 6.674 × 10⁻¹¹ N·m²/kg². The force acts along the line joining the two centres of mass and is always attractive. Note that r is the distance between centres, not between surfaces.
Why is gravity so weak compared with other forces? ▾
The gravitational constant is tiny — about 10⁻¹¹ — so two 1 kg masses one metre apart attract with only 6.7 × 10⁻¹¹ N. Gravity dominates at astronomical scales only because it is always attractive and never cancels, whereas electric charges come in both signs and neutralise each other.
What is the difference between G and g? ▾
G is the universal gravitational constant, 6.674 × 10⁻¹¹ N·m²/kg², the same everywhere in the universe. Lowercase g is the local gravitational acceleration, 9.81 m/s² at Earth's surface, and it varies with location and altitude. They are related by g = GM/r² for the body you are standing on.
How does gravitational force change with distance? ▾
It follows an inverse square law: double the distance and the force drops to a quarter, triple it and the force drops to a ninth. This is why the ISS at 400 km altitude still experiences about 89% of surface gravity — it is only 6% further from Earth's centre than the ground is.
If the ISS has almost full gravity, why do astronauts float? ▾
They are in free fall, not in zero gravity. The station and everything in it are accelerating toward Earth at the same rate while moving sideways fast enough to keep missing it. With no relative acceleration between astronaut and floor, there is nothing to press against, which reads as weightlessness.
Do both objects feel the same force? ▾
Yes, exactly — Newton's third law guarantees it. Earth pulls on you with the same magnitude of force you pull on Earth. The accelerations differ enormously because a = F/m, and Earth's mass is around 10²³ times yours, so its acceleration toward you is unmeasurably small.
How do I calculate surface gravity on another planet? ▾
Use g = GM/R², with M the planet's mass and R its radius. Mars, with M = 6.42 × 10²³ kg and R = 3.39 × 10⁶ m, gives 3.72 m/s², about 38% of Earth's. Radius matters as much as mass: Mars has 11% of Earth's mass but only 53% of its radius.
Does the formula work for objects that are not point masses? ▾
For a spherically symmetric body it works exactly, treating all the mass as concentrated at the centre — a result Newton proved with the shell theorem. For irregular shapes, or when you are very close to a large extended object, you must integrate over the mass distribution instead.

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Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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