Gravitational Force Calculator
Newton's Law of Universal Gravitation states that every pair of masses attracts each other with a force proportional to their masses and inversely proportional to the square of the distance between them. This calculator computes that force in Newtons.
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Formula
F = G × m₁ × m₂ / r²
Gravitational force F equals the gravitational constant G (6.674 × 10⁻¹¹ N·m²/kg²) times mass 1 times mass 2, divided by the square of the distance r between the centers of the two masses. The inverse-square law means doubling the distance reduces force to one-quarter.
How to use the Gravitational Force Calculator
- 1
Enter your mass 1
Value should be in kg.
- 2
Enter your mass 2
Value should be in kg.
- 3
Enter your distance between centers
Value should be in m.
- 4
Read your results instantly
Results update in real time as you type.
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The inverse-square law
Gravity follows an inverse-square law: force drops as the square of distance. Double the distance and the force drops to one-quarter. Triple the distance and it drops to one-ninth. This behavior arises geometrically: gravitational influence spreads over the surface of a sphere (area = 4πr²), so the same total force is diluted over a larger and larger area.
The inverse-square law means gravity never truly reaches zero — it weakens with distance but extends to infinity. Even the Sun's gravitational influence reaches the edge of the solar system, keeping the Oort Cloud of comets loosely bound. The nearest star (Proxima Centauri) is 4 × 10¹⁶ m away; the Sun's gravity there is about 6 × 10⁻¹⁰ N/kg — tiny but not zero.
Gravitational constant G
G = 6.674 × 10⁻¹¹ N·m²/kg² is one of the most precisely measured constants in physics — and one of the hardest to measure accurately. Its small value explains why gravity is the weakest of the four fundamental forces by many orders of magnitude: two 1 kg masses 1 m apart attract with only 6.674 × 10⁻¹¹ N, while the electromagnetic force between two electrons 1 m apart is 2.3 × 10⁻²⁸ N... wait — electromagnetic force between electrons is actually ~2.3 × 10⁻²⁸ N, and gravity between them ~10⁻⁶⁷ N. Gravity between protons: ~10⁻³⁴ N; electrostatic: ~230 N. Electromagnetism is ~10³⁶ times stronger.
Henry Cavendish first measured G experimentally in 1798 using a torsion balance, determining Earth's mass in the process.
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Surface gravity on different planets
Using F = GMm/r² and a = F/m = GM/r², you can compute the surface gravity of any body if you know its mass and radius. Earth: 9.81 m/s². Moon: 1.62 m/s². Mars: 3.72 m/s². Jupiter: 24.8 m/s². The Sun's surface gravity: 274 m/s² — 28 times Earth's.
Neutron stars are extreme: with masses of 1-3 solar masses compressed to ~10 km radius, surface gravity exceeds 10¹² m/s². A marshmallow dropped on a neutron star would release as much energy as an atomic bomb on impact. Black holes take this further — escape velocity exceeds the speed of light.
Tips & Insights
Use Earth's mass for weight calculations
Earth mass = 5.972 × 10²⁴ kg. Earth radius = 6.371 × 10⁶ m. Plugging these in gives a = GM/r² = 9.81 m/s² — Earth's surface gravity.
Distance is center-to-center
The r in the formula is the distance between the centers of mass of the two objects, not their surface-to-surface distance.
Gravity at altitude
At 400 km altitude (ISS orbit): r = 6,371,000 + 400,000 = 6,771,000 m. g = GM/r² ≈ 8.69 m/s² — 89% of surface gravity. The ISS is NOT in zero gravity!
Worked Examples
Earth-person gravitational force
F = 6.674×10⁻¹¹ × 5.972×10²⁴ × 70 / (6.371×10⁶)² ≈ 686.5 N ≈ 154 lbf (your weight on Earth).
Two people standing 1 m apart
F = 6.674×10⁻¹¹ × 70 × 70 / 1 ≈ 3.27 × 10⁻⁷ N — incredibly tiny; gravity between people is negligible.
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Frequently Asked Questions
What is Newton's Law of Universal Gravitation?
F = Gm₁m₂/r². Every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
What is the gravitational constant G?
G = 6.674 × 10⁻¹¹ N·m²/kg². It's one of the fundamental constants of nature, first measured by Henry Cavendish in 1798.
Why is the ISS not in zero gravity?
At 400 km altitude, gravity is about 89% of surface strength. Astronauts feel weightless because they're in continuous free fall — falling around Earth, not standing on it.
How does gravity follow an inverse-square law?
Gravitational influence spreads over spheres (area = 4πr²). The same total force covers a larger area as r increases, reducing intensity by 1/r².
Why is gravity the weakest fundamental force?
The gravitational constant G is extremely small. Between two protons, electromagnetic repulsion is about 10³⁶ times stronger than gravitational attraction.
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