# How strong is the gravitational force?

Computes the gravitational force between two masses a distance apart with Newton’s law F = G m₁ m₂ ÷ r², in newtons and pounds-force, with the acceleration each mass feels.

- Page: https://www.acalculator.org/physics/gravitational-force-calculator
- JSON spec: https://www.acalculator.org/physics/gravitational-force-calculator.json
- Version: 466bf71be7aa

## Default answer

Example with the default inputs (First mass 5,972,000,000,000,000,000,000,000 kg, Second mass 73,460,000,000,000,000,000,000 kg, Distance between centers 384,400 km): The gravitational force between the two masses is 1.98157 × 10²⁰ N (4.45475 × 10¹⁹ lbf).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| m1 | First mass | The mass of the first object: Earth is 5.972 × 10²⁴ kg. |
| m2 | Second mass | The mass of the second object: the Moon is 7.346 × 10²² kg. |
| r | Distance between centers | The distance from the center of one mass to the center of the other. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| newtons | Force (N) | The force of gravity between the two masses, in newtons. |
| lbf | Force (lbf) | The same force in pounds-force (1 lbf = 4.4482216152605 N). |
| a1 | Acceleration of the first mass (m/s²) | F ÷ m₁: how fast the pull speeds up the first mass. |
| a2 | Acceleration of the second mass (m/s²) | F ÷ m₂: how fast the pull speeds up the second mass. |

## Method

F = G × m₁ × m₂ ÷ r², G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻².

## Assumptions

- The masses are points or spheres (whose mass is spread evenly in shells), and r is the distance between their centers.
- G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2022). Exact unit sizes: 1 lb = 0.45359237 kg; 1 short ton = 907.18474 kg; 1 mi = 1,609.344 m; 1 ft = 0.3048 m.

## Worked examples

1. m1 = 5,972,000,000,000,000,000,000,000, m2 = 73,460,000,000,000,000,000,000, r = 384,400,000 gives newtons = 198,157,123,241,918,700,000. Source: OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg.
2. m1 = 1, m2 = 1, r = 1 gives newtons = 0, a1 = 0. Source: OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg.
3. m1 = 5,972,000,000,000,000,000,000,000, m2 = 70, r = 6,371,000 gives newtons = 687.39814, a2 = 9.819973. Source: OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg.

## FAQ

### What is the formula for gravitational force?

Newton’s law of universal gravitation: F = G × m₁ × m₂ ÷ r², where m₁ and m₂ are the masses in kilograms, r is the distance between their centers in meters, and G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² is the gravitational constant.

### What is the gravitational force between the Earth and the Moon?

With m₁ = 5.972 × 10²⁴ kg, m₂ = 7.346 × 10²² kg and r = 384,400 km, F = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 7.346 × 10²² ÷ (3.844 × 10⁸)² ≈ 1.98 × 10²⁰ N.

### Why is gravity between everyday objects so weak?

G is tiny. Two 1 kg masses 1 m apart pull on each other with only 6.67 × 10⁻¹¹ N. Gravity only becomes large when one mass is as big as a planet.

### What happens to the force if the distance doubles?

It drops to a quarter, because the distance is squared. Three times the distance gives one ninth of the force.

### Does the heavier object pull harder?

No. Both masses feel the same size of force, in opposite directions (Newton’s third law). The lighter one speeds up more: its acceleration is F ÷ m, which the page shows for each mass.

### How does this give g on Earth?

Put Earth’s mass and radius in: a 70 kg person at 6,371 km from Earth’s center feels 687 N, and 687 ÷ 70 = 9.82 m/s². The standard value 9.80665 m/s² is a little lower, partly because Earth spins.

## Sources

- OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²). https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation (retrieved 2026-10-03)
- NIST, CODATA 2022 recommended value: Newtonian constant of gravitation, G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻². https://physics.nist.gov/cgi-bin/cuu/Value?bg (retrieved 2026-10-03)
- NASA, Earth Fact Sheet (mass 5.9722 × 10²⁴ kg, volumetric mean radius 6,371 km) and Moon Fact Sheet (mass 0.07346 × 10²⁴ kg, mean distance 384,400 km). https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html (retrieved 2026-10-03)