Gravitational Force Calculator
Pick what you want to find — the force between two masses, either mass, the distance, or the strength of gravity at some height above a planet — and enter the rest. You get the answer in the units you choose, the accelerations it causes, and the equation with your numbers substituted in. Free, no sign-up, and it runs entirely in your browser.
Measured centre to centre. For a person standing on Earth, use Earth's radius.
Results
1.982 × 10²⁰ N
Gravitational force
3.318 × 10⁻⁵ m/s²
Acceleration of m₁
0.002698 m/s²
Acceleration of m₂
The working
F = G m₁ m₂ / r²
F = 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ × 7.346 × 10²² / (3.844 × 10⁸)²
F = 1.982 × 10²⁰ N
Assumes point masses or uniform spheres, whose pull acts as if all their mass sat at the centre (Newton's shell theorem). Rotation, tides and the Moon's or Sun's pull on a surface point are ignored.
How to Use the Gravitational Force Calculator
- 1. Choose the calculation. Force between two masses applies Newton's law to any pair of bodies. Gravitational field strength gives g at a height above a planet, moon or star.
- 2. Pick what to solve for. The box for the unknown turns into the answer; fill in the others. Every box has its own unit menu — kilograms to Earth masses, metres to astronomical units — and scientific notation such as 5.97e24 works. Start from an example if you want a working set of numbers.
- 3. Check the working. The equation is shown with your values substituted in SI units, so you can match it line by line with your own calculation.
- 4. Set the precision. Results are rounded to 3–6 significant figures; the arithmetic always runs at full precision.
Newton's Law of Universal Gravitation
Force
F = G m₁ m₂ / r²
Field strength
g = G M / (R + h)²
Every mass attracts every other mass with a force proportional to the product of the masses and inversely proportional to the square of the distance between their centres. G is the gravitational constant, 6.6743 × 10−11 N·m²/kg² (CODATA 2022, with an uncertainty of about 0.0023%), which is the least precisely known of the fundamental constants (see the NIST CODATA tables). Because it is so small, gravity only becomes noticeable when one of the masses is planet-sized.
The two forces in the formula are a pair. Body 1 pulls on body 2 exactly as hard as body 2 pulls on body 1, in the opposite direction (Newton's third law), but the accelerations differ because acceleration is force divided by mass. That is why the calculator reports a₁ and a₂ separately. The law and its worked problems are in OpenStax University Physics, section 13.1.
Worked Examples
- Earth and Moon (the default example). With M⊕ = 5.9722 × 1024 kg, m☾ = 7.346 × 1022 kg and a mean separation of 384,400 km, F = 1.982 × 1020 N. Earth accelerates toward the Moon at only 3.318 × 10−5 m/s²; the Moon accelerates toward Earth at 0.002698 m/s².
- A 70 kg person on Earth. Using Earth's mean radius, 6,371 km, as the distance, F = 687.4 N and the person's acceleration is 9.820 m/s². That is just weight, W = mg, and it is why g ≈ 9.8 m/s².
- Two 70 kg people one metre apart. F = 3.270 × 10−7 N — about a third of a micronewton, far too small to feel or to beat friction.
- The Sun and Earth. With the Sun's 1.9884 × 1030 kg and a distance of 149,598,000 km, F = 3.542 × 1022 N, the force that keeps Earth in its orbit.
Gravity at Altitude
Gravity weakens with the square of the distance from the centre of the planet, not from its surface. Climb 408 km above Earth, about the height of the International Space Station's orbit, and the distance goes from 6,371 km to 6,779 km. The field strength drops to g = 8.674 m/s², which is 88.33% of its surface value. At 100 km, the edge of space by the Kármán convention, it is still 9.519 m/s² (96.93%). Even at the 35,786 km altitude of a geostationary orbit, g is 0.2243 m/s², about 2.3% of the surface value, which is not zero.
Astronauts are not weightless because gravity is gone
At 408 km gravity is still almost 90% of what you feel on the ground. Astronauts float because they and the station are in continuous free fall around Earth, accelerating together. Escaping gravity entirely would take an infinite distance: it only gets weaker.
Assumptions and Limits
- Point masses or uniform spheres. Newton's shell theorem shows a spherically symmetric body attracts as if all its mass were at its centre, so planets and stars can use the same formula. For irregular or non-spherical objects the result is an approximation.
- Distance is centre to centre. Entering the gap between surfaces is the most common mistake in homework answers; add both radii.
- Newtonian gravity. Fine for everything from falling apples to planetary orbits. Extremely strong fields or very precise orbits (Mercury's perihelion, GPS clocks) need general relativity.
- Body data. The presets use masses and volumetric mean radii from NASA's Planetary Fact Sheet. Gas giants have no solid surface, so their radius is taken at the 1-bar pressure level, and actual surface g varies with latitude and rotation.
Frequently Asked Questions
- What is the formula for gravitational force?
- F = G m₁ m₂ / r², where G = 6.6743 × 10−11 N·m²/kg², m₁ and m₂ are the two masses in kilograms, and r is the distance between their centres in metres. The answer is in newtons. Doubling either mass doubles the force; doubling the distance cuts it to a quarter.
- What is the value of G, and why is it so hard to measure?
- The 2022 CODATA recommended value is 6.67430 × 10−11 m³ kg⁻¹ s⁻², uncertain by about 22 parts per million. Gravity is by far the weakest force, so laboratory measurements must detect tiny forces between small masses against vibration and thermal noise, and different experiments still disagree slightly.
- How do I find g on another planet?
- Use the field-strength mode with the planet preset and an altitude of zero: g = GM/R². For Mars it gives about 3.73 m/s², and for the Moon about 1.62 m/s². The mass and radius come from NASA's Planetary Fact Sheet.
- Why is my answer so small?
- Gravity between everyday objects really is tiny. Two 70 kg people a metre apart attract each other with only about 3 × 10−7 N. Check your units too: a distance typed in kilometres but read as metres changes the force by a factor of a million.
- What is the difference between gravitational force and weight?
- Weight is the gravitational force a large body, usually Earth, exerts on an object, W = mg, where g = GM/R² at that spot. The force calculator and the weight formula agree: a 70 kg person on Earth's surface feels 687.4 N.
- Does gravity ever reach zero?
- Not at any finite distance. The force falls as 1/r², becoming very small but never exactly zero. Between two bodies there are points where their pulls cancel, but each pull is still there.