Gravitational Force Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 20, 2026The Weakest Force, Felt by Everything
Gravity is by far the weakest of the four fundamental forces, yet it's the one that shapes planetary orbits, keeps your feet on the ground, and holds galaxies together — because unlike the other forces, it never cancels out and always adds up over enough mass. Newton's law of universal gravitation, published in 1687, was the first equation to describe this pull mathematically, and it still gives accurate results for anything short of extreme gravitational fields or relativistic speeds.
The Formula
G is the gravitational constant, 6.674×10&supminus;¹¹ N·m²/kg², m1 and m2 are the two masses, and r is the distance between their centers. The tiny value of G is exactly why gravitational force between everyday objects is imperceptible — it takes planet-scale mass to produce forces we actually notice.
Where This Force Is the Whole Story
- Orbital mechanics — satellite orbits, planetary motion, and spacecraft trajectories are all computed from this exact inverse-square relationship.
- Tidal forces — the difference in gravitational pull across an extended body, like the Earth under the Moon's influence, is what generates tides.
- Astrophysics — the mass of stars, black holes, and galaxies is often inferred indirectly by observing gravitational effects on nearby objects.
- Precision measurement — extremely sensitive instruments like torsion balances can detect gravitational attraction between lab-scale masses, which is in fact how G itself was first measured.
Just How Weak Gravity Is at Human Scale
| Scenario | Gravitational Force |
|---|---|
| Two 70 kg people standing 1 m apart | 3.270×10&supminus;&sup7; N |
| Earth and Moon (masses 5.972×10²&sup4; kg and 7.342×10²² kg, 3.844×10&sup8; m apart) | 1.980×10²&sup0; N |
The force between two people is roughly a millionth of a newton — far too small to feel — while planet-scale mass produces a force large enough to hold the Moon in orbit.
How to Use This Calculator
- Enter Mass 1 and Mass 2, in kilograms.
- Enter the Distance Between Centers of the two masses, and choose Meters or Kilometers.
- Select Calculate to get the gravitational force in newtons along with the worked formula.
Related Calculations
To compare this with the far stronger electric force between charges, see the Coulomb's Law Calculator, or convert a known force into the resulting acceleration with the Acceleration Calculator.
Principles of Classical Gravitation and Orbital Celestial Mechanics
A gravitational force calculator computes mutual gravitational attraction, planetary surface gravity, gravitational potential energy, orbital velocities, and escape velocities under Newton's Law of Universal Gravitation. In astrophysics, orbital dynamics, and aerospace engineering, gravitation governs satellite orbits, planetary trajectories, and celestial mechanics.
The Fundamental Gravitational Equations
Where G = 6.67430 × 10-11 N·m2/kg2 | m1, m2 = Masses (kg) | r = Center-to-center separation distance (meters)
Surface Gravitational Acceleration: g = ( G · M ) / R2
Gravitational Potential Energy: U = - ( G · M · m ) / r
Circular Orbital Velocity: vorbit = √[ ( G · M ) / r ]
Planetary Escape Velocity: vescape = √[ ( 2 · G · M ) / R ] = √( 2 · g · R )
Planetary Physical Parameters (Solar System Benchmarks)
| Celestial Body | Mass (kg) | Mean Radius (km) | Surface Gravity (g) | Escape Velocity (km/s) |
|---|---|---|---|---|
| Earth | 5.972 × 1024 | 6,371 km | 9.81 m/s2 (1.00 g) | 11.186 km/s |
| Moon | 7.342 × 1022 | 1,737 km | 1.62 m/s2 (0.165 g) | 2.38 km/s |
| Mars | 6.417 × 1023 | 3,390 km | 3.72 m/s2 (0.379 g) | 5.03 km/s |
| Jupiter | 1.898 × 1027 | 69,911 km | 24.79 m/s2 (2.53 g) | 59.5 km/s |
| Sun | 1.989 × 1030 | 696,340 km | 274.0 m/s2 (27.9 g) | 617.5 km/s |
Step-by-Step Worked Calculation Example
Example: Calculating Gravitational Attraction and Orbital Speed of the ISS
Problem: The International Space Station (mass m = 450,000 kg) orbits Earth (Mass M = 5.972 × 1024 kg, Radius R = 6.371 × 106 m) at an orbital altitude of h = 400.0 km (400,000 m). Total radius r = R + h = 6.771 × 106 m. Calculate: (1) Gravitational force acting on the ISS; (2) Gravitational acceleration at orbit; and (3) Required orbital velocity.
Step 1: Calculate Gravitational Force (F = G M m / r2):
r2 = ( 6.771 × 106 )2 = 4.5846 × 1013 m2
G × M × m = ( 6.6743 × 10-11 ) × ( 5.972 × 1024 ) × 450,000 = 1.7937 × 1020 N·m2
F = 1.7937 × 1020 / 4.5846 × 1013 = 3,912,446 N (approx. 3.91 Meganewtons)
Step 2: Calculate Local Orbital Gravitational Acceleration (gorbit = F / m):
gorbit = 3,912,446 N / 450,000 kg = 8.694 m/s2 (approx. 88.6% of Earth surface gravity!)
Step 3: Calculate Circular Orbital Speed (v = √[ G M / r ]):
v = √[ ( 3.9860 × 1014 ) / 6.771 × 106 ] = √( 58,868,705 ) = 7,672.6 m/s (27,621 km/h / 17,163 MPH)
Conclusion: In Low Earth Orbit, gravity is nearly 89% of surface strength; astronauts experience weightlessness because they are in continuous freefall.
Tidal Forces and Differential Gravitational Gradients
Gravitational pull varies with the inverse square of distance (1/r2), creating a Differential Tidal Force stretching astronomical bodies along their orbital axis:
Because tidal force scales with 1/r3, the Moon exerts more than twice the ocean tidal pulling force of the massive Sun on Earth's oceans due to its close proximity.
Lagrange Equilibrium Points (L1 to L5)
In three-body orbital celestial mechanics, Lagrange Points represent positions where the gravitational pull of two large masses (Earth and Sun) precisely equals the centripetal force required for a small object to orbit with them (e.g., the James Webb Space Telescope operates at the Sun-Earth L2 Point 1.5 million km from Earth).
General Relativity and Spacetime Curvature (Beyond Newton)
While Newton's inverse-square law provides exceptional precision for interplanetary spacecraft navigation:
Einstein's General Theory of Relativity (1915) demonstrates that gravity is not a classical pulling force, but the geometrical curvature of 4D spacetime caused by mass-energy density. Near ultra-dense collapsed stars, spacetime curvature creates a Schwarzschild Event Horizon Radius: Rs = 2GM / c2, defining the boundary of black holes.
Gravitational Slingshot (Gravity Assist) Maneuvers
In deep space exploration missions (Voyager 1 & 2, Cassini, New Horizons):
Spacecraft utilize hyperbolic planetary flybys to execute Gravity Assist Maneuvers. By entering a planet's gravitational sphere of influence behind its orbital path, the spacecraft extracts orbital kinetic energy from the moving planet, accelerating by up to 15 to 25 km/s without expending rocket fuel propellant.
Gravitational Lensing in Observational Astronomy
As massive galaxy clusters warp surrounding spacetime, they act as cosmic magnifying glasses (Gravitational Lensing). Light from distant background quasars bends around the massive cluster, producing multiple distorted Einstein Ring images that astrophysics researchers use to map invisible cosmic Dark Matter distributions.
Gravitational Wave Astronomy (LIGO & Virgo)
Catastrophic relativistic collisions between binary neutron stars and orbiting supermassive black holes radiate ripples across spacetime detected as Gravitational Waves by laser interferometer observatories (LIGO).
Cavendish Torsion Balance Measurement of G
Henry Cavendish first measured the universal gravitational constant G in 1798 using a sensitive quartz fiber torsion balance, measuring the mass of the Earth.