Gravitational Force Calculator

Last updated: 2026-05-18

Gravitational force

Gravitational force is the mutual attraction that acts between any two objects that have mass. It keeps the Moon in orbit around Earth, Earth in orbit around the Sun, and objects resting on the ground. This calculator applies Newton's law of universal gravitation to find the attractive force between any two masses separated by a given distance, along with the acceleration that force produces on each body.

Newton's Law of Universal Gravitation

The formula is:

$F = G \frac{m_1 m_2}{r^2}$

where:

• F is the gravitational force in newtons (N)
• G = 6.6743 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant
• m₁ and m₂ are the masses of the two objects in kilograms
• r is the distance between the centres of mass in metres

The resulting acceleration on each body follows Newton's second law: a = F/m, so the smaller body accelerates more noticeably while the larger one barely moves.

Worked Example: Earth–Moon System

• Mass of Earth (m₁): 5.972 × 10²⁴ kg
• Mass of Moon (m₂): 7.342 × 10²² kg
• Distance between centres (r): 3.844 × 10⁸ m

$F = 6.6743 \times 10^{-11} \times \frac{5.972 \times 10^{24} \times 7.342 \times 10^{22}}{(3.844 \times 10^8)^2} \approx 1.98 \times 10^{20} \text{ N}$

That is 198 quintillion newtons — the force that both maintains the Moon's orbit and drives ocean tides.

The Inverse-Square Law

The force falls off as 1/r², not 1/r. This means:

This is why satellites in low Earth orbit (≈400 km) experience about 89% of Earth's surface gravity, while a spacecraft at the Moon's distance feels only about 0.028% of it.

Earth's Surface Gravity (g ≈ 9.82 m/s²)

The familiar gravitational acceleration at Earth's surface is simply Newton's law with r = Earth's radius:

$g = \frac{GM_\oplus}{R_\oplus^2} = \frac{6.6743 \times 10^{-11} \times 5.972 \times 10^{24}}{(6.371 \times 10^6)^2} \approx 9.82 \ \text{m/s}^2$

The calculator's default values reproduce this exactly. Increasing the distance shows how gravity weakens with altitude: at the International Space Station (400 km up), g ≈ 8.7 m/s² — astronauts are not weightless because gravity is absent, but because the station and everything inside it are falling together.

The Gravitational Constant G

G = 6.6743 × 10⁻¹¹ N·m²/kg² is one of the most fundamental — and least precisely known — constants in physics. Henry Cavendish first measured it in 1798 using a torsion balance: two small lead balls attracted to two large ones, twisting a thin wire by a measurable angle. Modern measurements have reduced the uncertainty to about 22 parts per million, but G remains harder to measure precisely than almost any other constant because gravity is so weak that even tiny vibrations in the laboratory floor introduce noise.

Limitations

This calculator treats objects as point masses (or uniform spheres, for which the result is identical). It does not account for:

• Non-spherical mass distributions — Earth is slightly oblate; the real gravitational field varies with latitude.
• Relativistic effects — at very strong fields or very high velocities, general relativity gives more accurate predictions than Newton's law.
• Tidal forces — the differential pull across an extended object (which deforms the Moon and causes ocean tides) is not computed here.

For everyday astronomical distances and non-relativistic speeds, Newton's law is accurate to better than one part in a billion.