Gravitational Force Calculator (Newton)

Understanding the gravitational force calculator — concept and principle

Every pair of masses attracts each other. Newton's law of universal gravitation gives the size of that pull from the two masses and the distance between their centres. This calculator returns the force and the gravitational acceleration produced by mass 1 at that distance. The defaults describe the Earth and the Moon.

Both bodies feel the same 1.98 × 10²⁰ N, but the light Moon is moved far more.

How it is calculated

Force: F = G m₁ m₂ ÷ r², in newtons (N).
Gravitational acceleration due to mass 1: g = G m₁ ÷ r², in m/s².
m₁ and m₂ are masses in kilograms; r is the centre-to-centre distance in metres.
G = 6.6743 × 10⁻¹¹ N·m²/kg², the CODATA 2018 recommended value of the gravitational constant.

The principle

Newton proposed that the same force making an apple fall also keeps the Moon in orbit, and that it weakens with the square of distance. The inverse-square law has a simple geometric picture: the influence spreading from a mass is shared over the surface of a sphere, whose area grows as 4πr². Doubling the distance spreads it over four times the area, so the force falls to a quarter.

For spherical bodies with symmetric layers, the force outside them is the same as if all the mass were concentrated at the centre, which is why r is measured between centres. The force is mutual: the Moon pulls the Earth exactly as hard as the Earth pulls the Moon (Newton's third law), but the Earth accelerates far less because of its larger mass.

Worked example

Inputs: m₁ = 5.972 × 10²⁴ kg (Earth), m₂ = 7.348 × 10²² kg (Moon), r = 3.844 × 10⁸ m.
r² = (3.844 × 10⁸)² = 1.47763 × 10¹⁷ m².
G m₁ = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ = 3.98589 × 10¹⁴ m³/s².
g = 3.98589 × 10¹⁴ ÷ 1.47763 × 10¹⁷ ≈ 0.0026974833 m/s².
F = g × m₂ = 0.0026974833 × 7.348 × 10²² ≈ 1.982111 × 10²⁰ N.

Each point is the calculator re-run with only “Distance (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Distance (m)	Force (N)
9.61e+07	3.171377e+21
1.92e+08	7.928443e+20
2.88e+08	3.523752e+20
3.84e+08	1.982111e+20
4.8e+08	1.268551e+20
5.77e+08	8.809381e+19
7.69e+08	4.955277e+19

How to read the result

The acceleration at the Moon's distance is about 1/3,600 of the 9.8 m/s² at Earth's surface, because the Moon is roughly 60 Earth radii away and 60² = 3,600. Newton checked exactly this: the Moon's centripetal acceleration from its 27.3-day orbit is about 0.00272 m/s², very close to the value here. For everyday objects the force is tiny; two 1 kg masses 1 m apart attract with only 6.67 × 10⁻¹¹ N.

Common mistakes

Using the distance between surfaces instead of between centres.
Entering kilometres instead of metres, which makes the force a million times too large.
Forgetting to square the distance.

Keep in mind: The formula is classical Newtonian gravity for point masses or spherically symmetric bodies. It does not include relativistic effects, tides, irregular shapes or the influence of third bodies, which matter for precise spacecraft navigation.

Frequently asked questions

Why is G so hard to measure?

Gravity between laboratory masses is extremely weak, so torsion-balance experiments, starting with Cavendish in 1798, must detect tiny forces. G is still known less precisely than most other fundamental constants.

How do I find surface gravity of a planet?

Use the planet's mass as m₁ and its radius as r; the acceleration output is the surface gravity, ignoring rotation.

Sources: CODATA 2018 recommended values of the fundamental physical constants (NIST) · Isaac Newton, Philosophiæ Naturalis Principia Mathematica (1687), Book III

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

Your numbers