What this calculator does
Newton's insight was that the force pulling an apple down and the force holding the Moon in orbit are the same force, following one rule: every mass attracts every other mass, in proportion to both masses and inversely with the square of the distance between them.
The constant that sets the strength of this, big G, is remarkably small, which is why gravity is only noticeable when at least one of the masses is astronomical. Two people standing next to each other do attract one another, but by an utterly negligible amount.
The formula
Multiply the gravitational constant by both masses and divide by the square of the distance between their centres. Gravitational potential energy uses the same product divided by distance rather than distance squared, and is negative by convention.
| Term | Meaning |
|---|---|
| Gravitational constant (G) | 6.6743 × 10⁻¹¹ m³/(kg·s²). One of the least precisely known physical constants, because gravity is so weak. |
| Inverse square law | Force falls with the square of separation: twice as far apart means a quarter the force. |
| Gravitational potential energy | Energy of the configuration, taken as zero at infinite separation and therefore negative at any finite distance. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass 1 (kg) | The first mass, in kilograms. |
| Mass 2 (kg) | The second mass, in kilograms. Earth is 5.972 × 10²⁴. |
| Distance between centres (m) | The distance between the centres of the two masses, in metres. For an object on Earth's surface this is Earth's radius, about 6,371,000 m. |
When to use it
Checking surface gravity
Putting in a planet's mass and radius returns the acceleration at its surface, which is where 9.81 m/s² comes from for Earth.
Finding the force between astronomical bodies
The same equation covers everything from a satellite to a galaxy, with only the numbers changing.
Understanding why gravity feels weak
Running the calculation for two ordinary objects shows just how tiny the force is without an astronomical mass involved.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does gravitational force fall with distance?
A 70 kilogram person at a range of distances from Earth's centre.
| Distance from centre | Gravitational force | Acceleration of mass 1 |
|---|---|---|
| 6,371 km | 687.398 N | 9.820 m/s² |
| 9,000 km | 344.460 N | 4.921 m/s² |
| 12,742 km | 171.850 N | 2.455 m/s² |
| 42,164 km | 15.694 N | 0.2242 m/s² |
Questions
Why is potential energy negative?
Because the zero point is defined at infinite separation. Bringing two masses together releases energy, so any finite separation has less energy than that reference, which makes the value negative. Only differences in potential energy matter physically.
Does gravity have infinite range?
In this formulation, yes. The force falls off with distance squared but never reaches zero, so every mass in the universe technically pulls on every other. In practice it becomes negligible very quickly.
Why is big G so hard to measure?
Because gravity is extraordinarily weak and cannot be shielded, so laboratory measurements have to detect a minute attraction between test masses while excluding every other influence. It remains among the least precisely known constants in physics.
Is Newton's law still correct?
It is an excellent approximation for almost everything, but general relativity supersedes it. The difference shows up in strong fields and at high precision, which is why GPS satellites must account for relativistic effects to stay accurate.
For the speed needed to escape that pull, see the escape velocity calculator. For orbital periods under the same force, see the Kepler's third law calculator.