Gravitational force
Magnitude of the attractive force on either mass.
Physics · Mechanics · Motion & Energy
Gravitational force calculator
Calculate the magnitude of the mutual gravitational force between two masses from their center-to-center separation using Newton’s law of universal gravitation.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Acceleration of m₁
Initial acceleration magnitude a₁ = F/m₁ due to m₂.
Acceleration of m₂
Initial acceleration magnitude a₂ = F/m₂ due to m₁.
Relation and numerical substitution
F = 6.67430 × 10⁻¹¹ × m₁ × m₂ / r²—
Positive masses and nonzero center-to-center distance; point masses, or spherically symmetric bodies treated outside their overlapping volumes, in Newtonian gravity.
01Formulas and symbols
Formulas used
F
F = 6.67430 × 10⁻¹¹ × m₁ × m₂ / r²Newtonian gravitational force magnitude using the 2022 CODATA value of G.a₁
a₁ = F / m₁Acceleration magnitude of the first mass from Newton’s second law.a₂
a₂ = F / m₂Acceleration magnitude of the second mass from Newton’s second law.02Assumptions and limits
Scope of validity
- Positive masses and nonzero center-to-center distance; point masses, or spherically symmetric bodies treated outside their overlapping volumes, in Newtonian gravity.
03Validation example
Reference numerical case
- For m₁ = m₂ = 1000 kg and r = 1 m, using G = 6.67430 × 10⁻¹¹ m³·kg⁻¹·s⁻²:
- F = 6.67430 × 10⁻⁵ N and a₁ = a₂ = 6.67430 × 10⁻⁸ m/s².
04References
Technical references
- University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation — OpenStax
- Fundamental Physical Constants — Newtonian constant of gravitation — NIST / CODATA
2022 CODATA value: G = 6.67430(15) × 10⁻¹¹ m³·kg⁻¹·s⁻².
FAQ
What is the difference between G and g?
G is the universal gravitational constant used in F = Gm₁m₂/r². Lowercase g is a local gravitational acceleration, about 9.81 m/s² near Earth’s surface, and varies with location.
Why must r be measured center to center?
Newton’s point-mass law uses the separation between mass locations. For spherically symmetric bodies outside one another, each body acts gravitationally as if its mass were concentrated at its center, so r is the center-to-center distance.
05How it works
Force grows linearly with each mass and falls with the square of separation. Doubling either mass doubles F; doubling r divides F by four. The two force magnitudes are equal and opposite, while the resulting accelerations differ when the masses differ.
06Limits of the model
The point-mass formula is exact for point masses and applies to non-overlapping spherically symmetric bodies by the shell theorem. Close irregular bodies require integrating their mass distributions; relativistic strong-gravity regimes require general relativity.
07Common mistakes
Use center-to-center distance, not the gap between surfaces. Distinguish the universal constant G from local acceleration g, and use scientific notation such as 5.972e24 for astronomical masses when convenient.