Newton's Law of Gravitation, Explained
After reading this you can compute the gravitational force between any two masses, find the field one mass creates, estimate potential energy, and know when the simple inverse-square formula stops being trustworthy.
What the law says and one example to anchor it
Every pair of masses pulls on each other. The force grows with each mass and shrinks with the square of the distance between their centres. That single rule, written by Newton in 1687, governs falling apples, the orbit of the Moon, and the timing of tides.
Here is the anchor: the Earth has a mass of about 5.972 \times 10^{24} kg, the Moon about 7.342 \times 10^{22} kg, and their centres sit roughly 3.844 \times 10^{8} m apart. Plug those in and the force comes out near 1.98 \times 10^{20} N. That is the pull holding the Moon in its orbit. You will reproduce that exact number below.
The same formula applied to two 1 kg masses one metre apart gives a force of only 6.674 \times 10^{-11} N. That is about the weight of a bacterium. Gravity is the weakest of the four fundamental forces by a wide margin. It dominates the cosmos only because it always attracts and never cancels, so large masses accumulate huge total pulls.
When to use this formula and when not to
The inverse-square law is exact for two point masses, and it is exact for two spherically symmetric bodies when you measure from their centres. That second fact is Newton's shell theorem, and it is why you can treat the Earth as if all its mass sat at its core.
Use it freely for planets, moons, stars and satellites separated by clear space. Be careful in three situations. First, if the bodies are not spherical or not far apart compared to their size, the "centre to centre" distance loses meaning and you need to integrate over the mass distribution. Second, if you are inside a body, only the mass closer to the centre than you pulls you inward. Third, near very compact objects or at very high speeds, Newtonian gravity fails and you need general relativity.
The distance is measured between centres of mass, not between surfaces. For a satellite 400 km above the Earth's surface, use r = 6{,}371 + 400 = 6{,}771 km, not 400 km. Getting this wrong changes the force by a factor of hundreds.
The formula and the intuition behind each piece
Here F is the magnitude of the attractive force in newtons, m_1 and m_2 are the two masses in kilograms, r is the distance between their centres in metres, and G = 6.674 \times 10^{-11} N·m²/kg² is the gravitational constant. The force points along the line joining the two masses, and each mass feels an equal and opposite pull.
The product m_1 m_2 means doubling either mass doubles the force. The r^2 in the denominator means moving twice as far apart cuts the force to a quarter, not a half. Triple the distance and the force drops to one ninth. This inverse-square shape appears because the influence of a point source spreads over a sphere whose surface area grows as r^2.
Two related quantities come straight out of the same setup. The gravitational field of mass m_1 is the force per kilogram it would exert on a test mass:
with units of m/s², the acceleration a small object would feel. The gravitational potential energy of the pair, taking zero at infinite separation, is:
The energy is negative because the two masses are bound: you must add energy to pull them apart. Note the single r in the denominator here, not r^2. Force falls off faster with distance than energy does.
Reproducing the Earth and Moon demo
Use the default values: m_1 = 5.972 \times 10^{24} kg, m_2 = 7.342 \times 10^{22} kg, r = 3.844 \times 10^{8} m.
- Multiply the masses: 5.972 \times 10^{24} \times 7.342 \times 10^{22} = 4.385 \times 10^{47} kg².
- Square the distance: (3.844 \times 10^{8})^2 = 1.478 \times 10^{17} m².
- Divide: 4.385 \times 10^{47} \div 1.478 \times 10^{17} = 2.968 \times 10^{30} kg²/m².
- Multiply by G: 6.674 \times 10^{-11} \times 2.968 \times 10^{30} = 1.981 \times 10^{20} N.
The force is about 1.98 \times 10^{20} N. For a check, the field of the Earth at the Moon's distance is g = 6.674 \times 10^{-11} \times 5.972 \times 10^{24} / 1.478 \times 10^{17} = 2.70 \times 10^{-3} m/s². Multiply that field by the Moon's mass and you recover the same force: 2.70 \times 10^{-3} \times 7.342 \times 10^{22} = 1.98 \times 10^{20} N. The potential energy is U = -1.98 \times 10^{20} \times 3.844 \times 10^{8} = -7.61 \times 10^{28} J.
Reading and interpreting your results
The force number alone rarely means much until you compare it. The Moon's 1.98 \times 10^{20} N sounds enormous, but the Moon is so massive that this force only bends its path gently, giving it a monthly orbit rather than an immediate plunge.
The field value g is often more intuitive because it is an acceleration. At the Earth's surface, r = 6.371 \times 10^{6} m gives g = 9.82 m/s², the familiar surface gravity. The table below shows how field and force fall off as you move away from the Earth.
| Distance from centre (m) | Multiple of Earth radius | Field g (m/s²) | Force on 1 kg (N) |
|---|---|---|---|
| 6.371e6 | 1.0 | 9.820 | 9.820 |
| 1.274e7 | 2.0 | 2.455 | 2.455 |
| 2.548e7 | 4.0 | 0.6138 | 0.6138 |
| 3.844e8 | 60.3 | 0.002700 | 0.002700 |
Common mistakes
The errors that produce wrong answers cluster in a few places.
- Surface distance instead of centre distance
- Always add the body's radius when you start from an altitude. This is the single most frequent slip.
- Forgetting to square the distance
- The denominator is r^2. Writing r alone inflates the force by a factor equal to r itself, which for planetary scales is astronomically large.
- Mixing units
- Because G is defined in SI, mass must be in kilograms and distance in metres. Kilometres or grams give results off by factors of thousands.
- Confusing weight with mass
- A "70 kg" reading on a scale is a mass. Its weight on Earth is 70 \times 9.82 = 687 N. Use mass in this formula, not weight.
Sanity-check any planetary force by computing the field first. If g comes out near 9.8 m/s² at a planet's surface, your setup is sound. Wildly different numbers usually signal a unit or radius error.
Related tools
Once you have the force and field, several nearby calculations follow naturally. To turn a surface field into what you would weigh standing there, use the Weight on Other Planets Calculator. To find how fast something must move to orbit or escape a mass, the Escape & Orbital Velocity Calculator uses the same G and masses. When your separations grow into astronomical units or light-years, convert them with the Astronomical Distance Converter before plugging metres into the force formula. And to plan observations of the bodies you are modelling, the Telescope Calculator handles magnification and resolution.
Frequently asked questions
Why is the gravitational force between two people so tiny?
Two 70 kg people standing 1 m apart attract each other with F = 6.674 \times 10^{-11} \times 70 \times 70 / 1 = 3.27 \times 10^{-7} N. That is far smaller than friction or a breath of air. Human masses are simply too small for gravity to be felt between them.
Does the formula give the force on both masses or just one?
It gives the magnitude of the force that each mass exerts on the other. By Newton's third law those forces are equal and opposite, so the Earth pulls the Moon with the same 1.98 \times 10^{20} N that the Moon pulls the Earth.
Why is the potential energy negative?
Zero energy is defined at infinite separation, where the masses no longer interact. Bringing them together releases energy, so any finite separation has energy below zero. To fully separate them again you must supply that energy back.
How accurate is Newton's law compared to general relativity?
For the Solar System it is accurate to better than one part in ten million in most cases. The famous exception is Mercury's orbit, which precesses about 43 arcseconds per century more than Newton predicts. General relativity accounts for that difference exactly.
Can I use the calculator for a small object near a planet?
Yes. Set one mass to the planet, the other to your object, and the distance to the planet's radius plus your altitude. The resulting field g is the local surface gravity, and the force is the object's weight there.
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