Newton's universal law of gravitation
Every pair of masses attracts with equal-magnitude, opposite-direction forces. The magnitude grows with both masses and falls with the square of their center-to-center separation.
This solver can calculate force, either unknown mass, or separation distance. All values are converted to SI units before applying the equation, while the answer remains in the unit selected for the unknown.
How to use the gravitational force calculator
- Choose the unknown: Select force, object 1 mass, object 2 mass, or center-to-center distance.
- Enter known quantities: Provide positive values and select the unit next to each input.
- Use a preset if helpful: The optional Earth, Moon, Mars, and Sun presets fill a mass in kilograms converted to the selected unit.
- Calculate: Review the solved value and the complete SI-unit check below it.
Formula and variables
Gravitational force equals the gravitational constant times the product of both masses divided by the square of their center-to-center distance.
F = Gm₁m₂/r²- F — Gravitational force
- Attractive force on each body (N)
- G — Gravitational constant
- 6.67430 × 10⁻¹¹ (N·m²/kg²)
- m₁, m₂ — Masses
- Mass of each interacting body (kg)
- r — Separation
- Distance between the bodies’ centers of mass (m)
Gravity on a person at Earth’s surface
Approximate Earth and a 70 kg person as spherical or point masses separated by Earth’s mean radius.
- Earth mass
- 5.9722 × 10²⁴ kg
- Person mass
- 70 kg
- Center distance
- 6.371 × 10⁶ m
- F = (6.67430 × 10⁻¹¹)(5.9722 × 10²⁴)(70)/(6.371 × 10⁶)²
Result: The gravitational force is approximately 687 N.
This is close to the person’s familiar surface weight; local elevation, rotation, and Earth’s nonuniform shape cause small differences.
Understanding your results
Reading the result
The force shown is the mutual attraction magnitude: each body experiences that same magnitude in the opposite direction.
- Doubling either mass doubles the force.
- Doubling separation reduces force to one quarter.
- Use center-to-center distance, not surface gap.
- Scientific notation is normal for astronomical masses and small laboratory forces.
Assumptions
- Bodies are point masses or spherically symmetric bodies whose separation is measured center to center.
- Newtonian gravity is adequate for the scale and precision required.
- Masses and separation are positive and constant for the instant being modeled.
Limitations
- Does not model general relativity, strong-field gravity, extended irregular mass distributions, rotation, tides, atmospheric effects, or orbital motion over time.
- Celestial presets are approximate reference masses and are not a substitute for mission-grade ephemeris data.
- The result is force, not automatically surface weight unless geometry and context justify that interpretation.
Common mistakes
- Using the gap between surfaces instead of center-to-center distance.
- Entering radius where the problem gives altitude without adding the body radius.
- Mixing pounds-mass with kilograms or miles with meters without conversion.
- Forgetting that distance is squared.
- Confusing gravitational force with the gravitational constant G or local acceleration g.
Practical use cases
Physics coursework
Solve for force, mass, or distance in universal-gravitation problems.
Scale comparisons
Compare how mass and separation change attraction from laboratory to planetary examples.
Frequently asked questions
Is gravitational force the same on both objects?
Yes. Newton’s third law gives equal force magnitudes in opposite directions, although the smaller mass has the larger acceleration.
Which distance should I enter?
Enter the separation between centers of mass. For a person at altitude h above a spherical planet, use planet radius plus h.
Can this calculate weight?
It can approximate weight when one mass is a planet, the other is the object, and center-to-center distance is appropriate. Local measured weight can differ.
Why is the force so small between ordinary objects?
G is very small in SI units, so detectable attraction between everyday masses requires sensitive equipment.
Sources and review
- Newton's Universal Law of Gravitation — OpenStax University Physics Volume 1. Accessed 2026-07-14.
- CODATA recommended values of the fundamental physical constants — National Institute of Standards and Technology. Accessed 2026-07-14.
Reviewed 2026-07-14.