Engineering Mechanics

Gravitational Force Calculator

Solve gravitational force, either mass, or center-to-center distance using Newton's universal law of gravitation.

Formula F = G m1 m2 / r²Reviewed Sep 8, 2026

Newton's universal law of gravitation gives the magnitude of the attractive force between two masses: F = G m1 m2 / r². This calculator lets you solve for gravitational force, either mass, or the center-to-center distance when the other three quantities are known. It uses the universal gravitational constant G and reports force magnitude only, not vector direction.

Calculation Bench

Solve for

01Mass 1

m₁ · Mass of the first body.

02Mass 2

m₂ · Mass of the second body.

03Center-to-Center Distance

r · Distance between the centers of mass of the two bodies.

This tool reports force magnitude only. Use center-to-center distance, and enter only positive masses and positive separation.

Solution

Enter the required values to calculate gravitational force.

F = G m1 m2 / r²

Formula Sheet

F = \dfrac{G m_1 m_2}{r^2}

m_1 = \dfrac{F r^2}{G m_2}

m_2 = \dfrac{F r^2}{G m_1}

r = \sqrt{\dfrac{G m_1 m_2}{F}}

FGravitational Force
GUniversal Gravitational Constant
m₁Mass 1
m₂Mass 2
rCenter-to-Center Distance

Variables & Units

Symbol	Variable	Description	Common Units
F	Gravitational Force	Magnitude of the attractive gravitational force between the two masses.	N, kN, lbf
G	Universal Gravitational Constant	Constant that sets the strength of gravitational attraction in Newton's law of gravitation.	m³/kg/s², N·m²/kg²
m₁	Mass 1	Mass of the first body.	g, kg, lb, ton
m₂	Mass 2	Mass of the second body.	g, kg, lb, ton
r	Center-to-Center Distance	Distance between the centers of mass of the two bodies.	cm, m, ft

How to Use This Calculator

01Choose which variable to solve for - Gravitational Force, Mass 1, Mass 2, or Distance.
02Enter the other three known values with their units. Distance here means the separation between the centers of mass of the two bodies, not the gap between their surfaces unless the objects are very small compared with that separation.
03Select Calculate to see the answer in base SI units, along with the formula substitution and a secondary conversion where useful.
04Use only positive magnitudes in this calculator. The force of gravity in this model is always attractive, so the tool reports magnitude rather than a signed direction. For near-Earth weight problems, compare the result with F = mg as a sanity check.

How the Formula Works

Newton's universal law of gravitation states that the magnitude of the force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between their centers: F = G m1 m2 / r². Doubling either mass doubles the force, while doubling the distance reduces the force to one-fourth because of the inverse-square dependence.

This calculator uses the universal gravitational constant G = 6.6743e-11 N·m²/kg² (equivalently m³·kg⁻¹·s⁻²). The formula applies exactly to point masses and also to spherically symmetric bodies when r is taken from center to center, which is why it works well for many planet-object and body-body estimates.

Worked Example 01

Force between Earth and a 75 kg person at the surface

Known

Mass 1 (Earth): 5.972e24 kg
Mass 2 (person): 75 kg
Distance (center to center): 6.371e6 m

Formula

F = G m1 m2 / r²

Substitution

F = (6.6743e-11 × 5.972e24 × 75) / (6.371e6)²

Result

F ≈ 736 N

Using Earth's mass and radius gives the familiar near-surface weight force on a 75 kg person.

Worked Example 02

Solving for distance from a known gravitational force

Known

Mass 1 (m1): 1000 kg
Mass 2 (m2): 1000 kg
Gravitational Force (F): 1e-5 N

Formula

r = √(G m1 m2 / F)

Substitution

r = √((6.6743e-11 × 1000 × 1000) / 1e-5)

Result

r ≈ 2.58 m

Because gravity between modest laboratory masses is very weak, even a micro-newton-scale force corresponds to only a few meters of separation.

Worked Example 03

Solving for a missing mass

Known

Gravitational Force (F): 1.33486e-5 N
Mass 2 (m2): 500 kg
Distance (r): 5 m

Formula

m1 = F r² / (G m2)

Substitution

m1 = (1.33486e-5 × 5²) / (6.6743e-11 × 500)

Result

m1 = 10000 kg

Rearranging Newton's law isolates the missing mass directly once the force, other mass, and distance are known.

Applications

01Estimating attraction between two bodies from Newton's law of gravitation
02Checking center-to-center distance effects in mechanics and astronomy problems
03Relating near-surface weight problems back to universal gravitation
04Checking why ordinary objects have extremely small mutual gravitational attraction compared with everyday contact forces

Assumptions

01The bodies are treated as point masses or as spherically symmetric masses with the distance measured center to center.
02The result reported is the magnitude of the gravitational force; the force direction is attractive along the line joining the masses.
03Masses and distance are positive, and relativistic or tidal effects are neglected.

Where This Model Stops

01Does not account for non-spherical mass distributions, nearby third-body effects, or full orbital dynamics.
02Does not use the local approximation F = mg directly; if you want near-surface weight or acceleration relationships, use a dedicated force or gravity-based tool when appropriate.
03Does not calculate orbital speed, escape velocity, tides, or gravitational potential energy automatically.

References

[1]
13.1 Newton's Law of Universal Gravitation
OpenStax University Physics Volume 1
Presents the vector form of Newton's gravitational law and explains the center-to-center interpretation for spherically symmetric bodies.
[2]
How Do You Measure the Strength of Gravity?
NIST
Summarizes the universal-gravitation relationship and the current best estimate of the gravitational constant G.

Frequently Asked Questions

Is this the same as using F = mg?

Not exactly. F = mg is a near-surface approximation for weight in a local gravitational field. This calculator uses the more general universal-gravitation relation F = Gm1m2/r² and measures distance from center to center.

Why is the distance center to center instead of surface to surface?

Because Newton's law uses the separation between the bodies' centers of mass. For large spherical bodies such as planets, that center-to-center distance is the correct quantity in the formula.

Why are the forces between small everyday objects so tiny?

Because the gravitational constant G is extremely small. Unless at least one body is enormous, such as Earth, the resulting gravitational force between ordinary objects is usually far smaller than forces you notice in daily life.

Can I use surface-to-surface distance?

Usually no. For planets, moons, balls, and other roughly spherical bodies, use center-to-center distance. Surface gap only approximates center distance when the object sizes are tiny compared with the separation.

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