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How to calculate gravitational force
Gravitational force between any two objects is given by Newton’s law of universal gravitation:
\[F = \frac{Gm_1 m_2}{r^2}\]Where F is the force in newtons, G is the gravitational constant, m₁ and m₂ are the masses of the two objects in kilograms, and r is the distance between their centers in meters.
The gravitational constant G has been measured experimentally and equals:
\[G = 6.674 \times 10^{-11} \, \text{N m}^2 \text{ kg}^{-2}\]This number is extraordinarily small. Gravity is by far the weakest of the four fundamental forces. Two one-kilogram masses sitting one meter apart attract each other with a force of only 6.674 × 10⁻¹¹ newtons, which is roughly the weight of a single bacterium. Gravity only becomes significant at astronomical scales because planets and stars have enormous masses.
Worked example: force between Earth and the Moon
The Moon orbits Earth due to the gravitational attraction between them. Here are the values needed:
- Mass of Earth: m₁ = 5.972 × 10²⁴ kg
- Mass of the Moon: m₂ = 7.342 × 10²² kg
- Average Earth-Moon distance: r = 3.844 × 10⁸ m
Plug into the formula:
\[F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(7.342 \times 10^{22})}{(3.844 \times 10^8)^2}\]Work through the numerator first:
\[6.674 \times 10^{-11} \times 5.972 \times 10^{24} = 3.985 \times 10^{14}\] \[3.985 \times 10^{14} \times 7.342 \times 10^{22} = 2.925 \times 10^{37}\]Now the denominator:
\[(3.844 \times 10^8)^2 = 1.478 \times 10^{17}\]Divide:
\[F = \frac{2.925 \times 10^{37}}{1.478 \times 10^{17}} \approx 1.98 \times 10^{20} \, \text{N}\]The gravitational force between Earth and the Moon is approximately 1.98 × 10²⁰ newtons, or about 198 quintillion newtons. This force is what keeps the Moon in its orbit rather than flying off into space.
Connecting to surface weight: F = mg
On the surface of a planet, the same formula simplifies to the familiar weight equation. If you are standing on Earth’s surface, m₂ is your mass, m₁ is Earth’s mass, and r is Earth’s radius (6.371 × 10⁶ m).
The ratio Gm₁/r² reduces to a constant called the surface gravitational acceleration, g:
\[g = \frac{Gm_1}{r^2} = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{(6.371 \times 10^6)^2}\] \[g = \frac{3.985 \times 10^{14}}{4.059 \times 10^{13}} \approx 9.81 \, \text{m/s}^2\]So the force on a 70 kg person standing on Earth’s surface is:
\[F = mg = 70 \times 9.81 = \textbf{686.7 N}\]This is their weight in newtons. The pound-force equivalent is roughly 154 lbs. The F = mg formula is not a separate equation; it is simply the full gravitational formula with Earth’s mass and radius pre-computed into the constant g.
Gravitational acceleration on other planets
Every planet has a different surface gravity because each has a different mass and radius. The formula g = GM/R² gives the surface gravitational acceleration for any body.
| Body | Surface g (m/s²) | Relative to Earth |
|---|---|---|
| Mercury | 3.70 | 0.38g |
| Venus | 8.87 | 0.90g |
| Earth | 9.81 | 1.00g |
| Moon | 1.62 | 0.17g |
| Mars | 3.72 | 0.38g |
| Jupiter | 24.79 | 2.53g |
| Saturn | 10.44 | 1.06g |
| Uranus | 8.87 | 0.90g |
| Neptune | 11.15 | 1.14g |
A 70 kg person would weigh approximately 686 N on Earth, 114 N on the Moon, and 1,735 N on Jupiter. The mass does not change; only the weight (the gravitational force) changes with location.
Mars has the same surface gravity as Mercury despite being much larger, because Mars has a lower average density. Jupiter’s gravitational pull at its cloud tops is 2.53 times Earth’s, which is why humans visiting Jupiter (if that were possible) would find it very difficult to move.
The inverse square law
The r² in the denominator means gravitational force decreases with the square of distance. If you double the distance between two objects, the force drops to one-fourth. If you triple the distance, the force drops to one-ninth.
| Distance multiplier | Force relative to original |
|---|---|
| 1x (original) | F |
| 2x | F/4 |
| 3x | F/9 |
| 4x | F/16 |
| 10x | F/100 |
This is why gravity weakens so rapidly with distance. At twice Earth’s radius above the surface (about 12,742 km up), you are three Earth radii from Earth’s center, so the gravitational pull is only 1/9 of what it is on the surface. Astronauts in low Earth orbit are not truly in zero gravity: they experience about 89% of surface gravity. They appear weightless because they are in freefall around Earth at the same rate as their spacecraft.
Why gravity is weak compared to other forces
Gravity feels dominant in everyday life because it is always attractive (never repulsive), acts at unlimited range, and operates on everything with mass. But at the particle scale, it is negligible compared to electromagnetism.
The gravitational force between two protons is approximately 10⁻³⁶ times weaker than the electromagnetic repulsion between them. The ratio is so extreme that the electrical repulsion between two protons overwhelms their mutual gravitational attraction by a factor of about 10³⁶. It is only because large objects (planets, stars) are electrically neutral (positive and negative charges cancel out) that gravity becomes the dominant force at astronomical scales.
The strong nuclear force and weak nuclear force operate only at subatomic distances and do not factor into gravitational calculations at any macroscopic scale.
Escape velocity
A related application of the gravitational formula is escape velocity: the minimum speed needed for an object to escape a planet’s gravity without further propulsion.
\[v_e = \sqrt{\frac{2GM}{r}}\]For Earth, this works out to approximately 11,200 m/s (about 11.2 km/s or 25,000 mph). Anything launched slower than this will eventually return to Earth due to gravity, assuming it does not achieve a stable orbit first.
The Moon’s lower mass gives it an escape velocity of only about 2.4 km/s, which is why the Moon has no atmosphere: gas molecules moving at typical thermal speeds can exceed this threshold and escape permanently over geological timescales.
Calculating it quickly
The formula F = Gm₁m₂/r² involves multiplying very large and very small numbers, which makes arithmetic errors likely. The force calculator handles Newton’s second law (F = ma), and can complement gravitational calculations when working through multi-step physics problems.
The worked examples above show the arithmetic at each step. Once you understand the structure of the formula, the pattern becomes predictable: larger masses produce more force, and greater distance produces less force, decreasing as the square of the separation between the two objects.