Gravitational Force Calculator
Force equals G times mass 1 times mass 2 divided by distance squared
How It Works
Newton's Law of Universal Gravitation says every pair of objects pulls on each other with a force F = G × m₁ × m₂ / r², where m₁ and m₂ are their masses, r is the center-to-center distance, and G = 6.6743 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant. This calculator solves the equation for any one of three unknowns — force, distance, or one of the masses — given the other three quantities. Inputs accept kg/g/lb for mass, m/km/ft/mi for distance, and N/lbf/dyne for force; the calculator converts to SI internally before computing.
Example Problem
Calculate the gravitational force between Earth (m₁ = 5.972 × 10²⁴ kg) and a 70 kg person standing on its surface (r = 6.371 × 10⁶ m).
- Write Newton's gravitational force formula: F = G × m₁ × m₂ / r².
- Substitute the values: F = (6.6743 × 10⁻¹¹) × (5.972 × 10²⁴) × 70 / (6.371 × 10⁶)².
- Compute the numerator: G × m₁ × m₂ = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 70 ≈ 2.789 × 10¹⁶ N·m².
- Compute the denominator: r² = (6.371 × 10⁶)² ≈ 4.058 × 10¹³ m².
- Divide: F ≈ 2.789 × 10¹⁶ / 4.058 × 10¹³ ≈ 687 N — the textbook weight of a 70 kg adult at Earth's surface.
Key Concepts
Gravitational force scales linearly with each mass and falls off as the inverse square of the distance, so doubling the distance cuts the force to one-quarter. The gravitational constant G is one of the most precisely measured universal constants and appears unchanged in every gravitational equation derived from Newton's law (Kepler's third law, surface gravity, escape velocity). The force always acts along the line connecting the two centers of mass — never along the surfaces — so r in the formula is the center-to-center distance, not surface-to-surface separation.
Applications
- Spacecraft trajectory planning: computing the gravitational pull from Earth, the Moon, and the Sun simultaneously when designing a mission.
- Astronomy: estimating planetary masses from the gravitational forces measured between celestial bodies and their moons.
- Tidal force analysis: separating the gravitational force on Earth's near and far sides to predict ocean tides.
- Geophysics: detecting subsurface density variations from tiny anomalies in measured surface gravitational force.
- Physics education: demonstrating the inverse-square law and the universality of gravitational interaction.
Common Mistakes
- Using surface-to-surface distance instead of center-to-center distance for r — the formula always needs the distance between centers of mass.
- Confusing the universal constant G (6.6743 × 10⁻¹¹ N·m²/kg², constant everywhere) with the surface gravity g (≈ 9.81 m/s², varies with location).
- Forgetting that r is squared in the denominator — doubling the separation reduces the force by a factor of four, not two.
- Mixing units of mass and distance (e.g., grams with kilometers) without converting — the SI form of the equation requires kg and m for the constant G to apply directly.
- Treating gravitational force as if only the larger mass matters — both masses appear in the formula and contribute equally to the attraction.
Frequently Asked Questions
How do you calculate gravitational force?
Multiply the two masses together and the gravitational constant G (6.6743 × 10⁻¹¹ N·m²/kg²), then divide by the square of the distance between their centers: F = G × m₁ × m₂ / r². Make sure the masses are in kilograms and the distance is in meters before applying the formula.
What is the formula for gravitational force?
F = G × m₁ × m₂ / r², where G is the gravitational constant (6.6743 × 10⁻¹¹ N·m²/kg²), m₁ and m₂ are the two masses in kilograms, and r is the distance between their centers in meters. The result F is in newtons.
What is the gravitational constant G?
G = 6.6743 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant — a fixed value that quantifies how strongly any two masses attract each other. Unlike Earth's surface gravity g (≈ 9.81 m/s²), G is identical anywhere in the universe and appears in every gravitational equation.
Why is gravitational force so weak between everyday objects?
Because G is extremely small (≈ 6.7 × 10⁻¹¹). Two 1 kg masses one meter apart pull on each other with only ~6.7 × 10⁻¹¹ N — about a hundred-billionth of a newton. Gravitational force only becomes appreciable when at least one of the masses is planetary in scale.
How does gravitational force change with distance?
It falls off as the inverse square of the distance. Doubling the separation cuts the force to one-quarter; tripling it cuts the force to one-ninth. This is why orbits weaken rapidly with altitude and why the gravitational influence of distant stars on Earth is negligible.
Does gravitational force depend on the masses of both objects?
Yes — both masses appear in the formula and contribute equally. Earth pulls on you with the same force you pull on Earth (Newton's third law). The reason you accelerate noticeably and Earth doesn't is the enormous difference in mass, not in force.
References:
Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
Gravitational constant G: NIST CODATA. https://physics.nist.gov/cgi-bin/cuu/Value?bg
Two-Mass Diagram
Newton's law of universal gravitation says every pair of masses attracts every other pair along the line connecting their centers, with force that scales as the product of the masses and falls off with the square of the separation distance.
m₁, m₂ — the two masses (kg) · r — center-to-center distance (m) · F — gravitational force on each mass; equal in magnitude, opposite in direction (Newton's third law) · G = 6.6743 × 10⁻¹¹ N·m²/kg² — the universal gravitational constant.
Worked Examples
Astronomy
How strong is the gravitational pull between Earth and the Moon?
Earth (m₁ = 5.972 × 10²⁴ kg) and the Moon (m₂ = 7.342 × 10²² kg) sit at a mean center-to-center distance of r = 3.844 × 10⁸ m. The mutual gravity sets ocean tides, locks the Moon's rotation, and keeps it in its orbit.
- Knowns: m₁ = 5.972 × 10²⁴ kg, m₂ = 7.342 × 10²² kg, r = 3.844 × 10⁸ m
- G = 6.6743 × 10⁻¹¹ N·m²/kg²
- F = G × m₁ × m₂ / r²
- F = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 7.342 × 10²² / (3.844 × 10⁸)²
- F = 2.926 × 10³⁷ / 1.478 × 10¹⁷
F ≈ 1.98 × 10²⁰ N
The same force acts on both bodies (Newton's third law). Combined with the Moon's orbital speed of ≈ 1,022 m/s, this is exactly what's needed to hold the Moon in its near-circular path — a hands-on verification of Kepler's laws.
Classroom Physics
How tiny is the attraction in Cavendish's torsion-balance experiment?
In 1798 Henry Cavendish measured G by suspending a 0.73 kg lead ball at distance 0.23 m from a 158 kg fixed lead sphere on a torsion fiber. The deflection let him recover G — the first measurement of the gravitational force between laboratory objects.
- Knowns: m₁ = 158 kg, m₂ = 0.73 kg, r = 0.23 m
- G = 6.6743 × 10⁻¹¹ N·m²/kg²
- F = G × m₁ × m₂ / r²
- F = 6.6743 × 10⁻¹¹ × 158 × 0.73 / (0.23)²
- F = 7.695 × 10⁻⁹ / 0.0529
F ≈ 1.45 × 10⁻⁷ N (145 nN)
Roughly the weight of a single grain of fine sand. Cavendish's apparatus had to detect deflections of just a few millimeters on a torsion fiber tuned to such tiny restoring forces — one of the most remarkable measurements in physics history.
Solar System Dynamics
What is the Sun's gravitational grip on Earth?
The Sun (m₁ = 1.989 × 10³⁰ kg) holds Earth (m₂ = 5.972 × 10²⁴ kg) in orbit at r = 1.496 × 10¹¹ m (1 astronomical unit). This is the centripetal force that makes Earth complete one orbit per year.
- Knowns: m₁ = 1.989 × 10³⁰ kg, m₂ = 5.972 × 10²⁴ kg, r = 1.496 × 10¹¹ m
- G = 6.6743 × 10⁻¹¹ N·m²/kg²
- F = G × m₁ × m₂ / r²
- F = 6.6743 × 10⁻¹¹ × 1.989 × 10³⁰ × 5.972 × 10²⁴ / (1.496 × 10¹¹)²
- F = 7.925 × 10⁴⁴ / 2.238 × 10²²
F ≈ 3.54 × 10²² N
About 178× the Earth-Moon force. Earth's tangential orbital speed of 29.78 km/s converts this gravitational pull into the centripetal acceleration v²/r ≈ 5.93 × 10⁻³ m/s² — the curvature of our annual orbit.
Gravitational Force Formula
Newton's law of universal gravitation says every pair of point masses attracts each other along the line connecting their centers, with a force proportional to the product of the masses and inversely proportional to the square of the separation.
F = G × m₁ × m₂ / r²
Where:
- F — gravitational force on each mass; equal in magnitude, opposite in direction (Newton's third law), measured in newtons (N)
- G — Newton's universal gravitational constant, 6.6743 × 10⁻¹¹ N·m²/kg²
- m₁, m₂ — the two attracting masses (kg)
- r — center-to-center distance between the two masses (m)
The equation treats both bodies as point masses (or as spherically symmetric mass distributions, which behave the same way from outside their surfaces). The inverse-square distance dependence means doubling r reduces the force to a quarter and halving r multiplies it by four. F vanishes only as r → ∞ — gravity has unlimited range.
Masses of Solar-System Bodies
Masses of the Sun, planets, and Moon for plugging into F = G × m₁ × m₂ / r². Click a body to load its mass as Object 1 (m₁); your other mass and distance are preserved. For the full sortable table with radius, surface gravity, and escape velocity, see the planetary data reference.
| Body | Mass (kg) | Type | Source |
|---|---|---|---|
| Sun | 1.99×10³⁰ | Star | NASA Planetary Fact Sheet |
| Mercury | 3.3×10²³ | Rocky planet | NASA Planetary Fact Sheet |
| Venus | 4.87×10²⁴ | Rocky planet | NASA Planetary Fact Sheet |
| Earth | 5.97×10²⁴ | Rocky planet | NASA Planetary Fact Sheet |
| Moon | 7.34×10²² | Moon | NASA Planetary Fact Sheet |
| Mars | 6.42×10²³ | Rocky planet | NASA Planetary Fact Sheet |
| Jupiter | 1.9×10²⁷ | Gas giant | NASA Planetary Fact Sheet |
| Saturn | 5.68×10²⁶ | Gas giant | NASA Planetary Fact Sheet |
Source: NASA Planetary Fact Sheet. Masses are the published values for each body; the calculator treats each as a point mass at the center-to-center distance you enter.
Related Calculators
- Gravitational Acceleration Calculator — surface gravity from planet mass and radius using g = GM/r²
- Escape Velocity Calculator — minimum launch speed to leave a planet via v = √(2GM/R)
- Kepler's Third Law Calculator — orbital period from radius and central body mass
- Gravity Equations Hub — all four gravitational formulas in one comprehensive tool
- Weight Equation Calculator — compute weight from mass and gravitational acceleration
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