Physics · Classical Mechanics · Dynamics & Forces
Gravitational Force Calculator
Calculates the gravitational force between two masses using Newton's Law of Universal Gravitation.
Calculator
Formula
F is the gravitational force in Newtons (N). G is the universal gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²). m₁ and m₂ are the masses of the two objects in kilograms (kg). r is the distance between the centers of the two masses in meters (m). The force is always attractive, acting along the line joining the two bodies.
Source: Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. CODATA 2018 value for G: 6.67430 × 10⁻¹¹ N·m²·kg⁻².
How it works
Newton's Law of Universal Gravitation states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This elegant inverse-square law was first published by Isaac Newton in 1687 and remains the cornerstone of classical gravitational theory, applicable across an enormous range of scales — from the attraction between two laboratory masses to the orbital dance of galaxies.
The governing formula is F = G(m₁m₂)/r², where G is the universal gravitational constant (6.6743 × 10⁻¹¹ N·m²/kg²), m₁ and m₂ are the masses in kilograms, and r is the center-to-center separation in meters. The resulting force F is expressed in Newtons and is always attractive — there is no gravitational repulsion in Newtonian gravity. The calculator also returns the gravitational field strength (acceleration due to gravity, in N/kg or equivalently m/s²) that each body generates at the location of the other, which equals F divided by the respective mass and is useful for determining free-fall acceleration.
Practical applications span satellite orbit calculations, tidal force analysis, planetary surface gravity, spacecraft trajectory planning, and general education in classical mechanics. Engineers designing interplanetary missions use this formula continuously to compute injection burn requirements, gravitational assists, and Lagrange point positions. Geophysicists use variations of it to model subsurface density anomalies. Even everyday phenomena — the Moon's tidal pull on Earth's oceans, the weight of an object on Mars — follow directly from this single equation.
Worked example
Consider calculating the gravitational force between the Earth (m₁ = 5.972 × 10²⁴ kg) and the Moon (m₂ = 7.342 × 10²² kg), separated by a mean distance of r = 3.844 × 10⁸ m.
Step 1 — Identify constants and inputs:
G = 6.6743 × 10⁻¹¹ N·m²/kg²
m₁ = 5.972 × 10²⁴ kg
m₂ = 7.342 × 10²² kg
r = 3.844 × 10⁸ m
Step 2 — Compute the numerator (G × m₁ × m₂):
6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 7.342 × 10²² = 6.6743 × 5.972 × 7.342 × 10⁻¹¹⁺²⁴⁺²² = 292.49 × 10³⁵ ≈ 2.9249 × 10³⁷ N·m²
Step 3 — Compute the denominator (r²):
(3.844 × 10⁸)² = 14.776 × 10¹⁶ = 1.4776 × 10¹⁷ m²
Step 4 — Divide to find F:
F = 2.9249 × 10³⁷ ÷ 1.4776 × 10¹⁷ ≈ 1.979 × 10²⁰ N
This result — approximately 1.98 × 10²⁰ Newtons — matches the accepted value for the Earth–Moon gravitational attraction. The gravitational field of the Moon at Earth's center is approximately 3.32 × 10⁻⁵ N/kg, which is the tidal acceleration that drives Earth's oceanic tides.
Limitations & notes
Newton's Law of Universal Gravitation assumes point masses or perfectly spherical, uniform-density bodies — real objects with irregular shapes require integration over their mass distributions. At very high masses or strong gravitational fields, General Relativity (Einstein, 1915) supersedes Newtonian gravity and must be used; GPS satellites, for instance, require relativistic corrections. The law also breaks down at quantum scales where gravitational effects are negligible compared to other fundamental forces. The calculator uses the CODATA 2018 recommended value of G (6.6743 × 10⁻¹¹ N·m²/kg²); G is among the least precisely known fundamental constants, with a relative uncertainty of about 22 parts per million. Users should ensure the distance r is the center-to-center separation, not the surface-to-surface gap, particularly for large bodies like planets.
Frequently asked questions
What is the value of the universal gravitational constant G?
The CODATA 2018 recommended value of G is 6.6743 × 10⁻¹¹ N·m²·kg⁻². It is one of the most difficult fundamental constants to measure precisely, with a relative standard uncertainty of approximately 2.2 × 10⁻⁵. This calculator uses the 2018 CODATA value for maximum accuracy.
Why does the gravitational force use the distance squared (inverse-square law)?
The inverse-square relationship arises because gravity propagates isotropically in three-dimensional space. As you move away from a mass, the gravitational 'flux' spreads over the surface area of an expanding sphere (proportional to r²), so the force density — and hence the force on a test mass — falls as 1/r². This same geometry applies to light intensity and electrostatic force.
Can I use this calculator for objects on Earth's surface?
Yes. To find the weight (gravitational force) of an object on Earth, set m₁ to Earth's mass (5.972 × 10²⁴ kg), m₂ to the object's mass in kg, and r to Earth's mean radius (6.371 × 10⁶ m). The result should closely match the object's weight in Newtons (mass × 9.81 m/s²), confirming Newton's law at everyday scales.
Is Newton's gravitational formula accurate for black holes or neutron stars?
No. For extremely massive or compact objects — black holes, neutron stars, or any scenario where gravitational fields are very strong or velocities approach the speed of light — General Relativity must be used. Newtonian gravity is an excellent approximation in weak-field, low-velocity regimes but fails to predict phenomena like gravitational lensing or gravitational waves correctly.
What units should I use for accurate results?
For consistent SI results, enter masses in kilograms (kg) and distance in meters (m). The output will be in Newtons (N). If you have values in other units — for example, astronomical units (AU) for distance or Earth masses for mass — convert them to SI first: 1 AU ≈ 1.496 × 10¹¹ m, and 1 Earth mass ≈ 5.972 × 10²⁴ kg.
Last updated: 2025-01-15 · Formula verified against primary sources.