Gravitational Force Calculator
Find the gravitational force between two masses, the acceleration each one feels, or the surface gravity, weight and escape speed of a planet.
🌍 What is the Gravitational Force?
Gravitational force is the attraction between any two objects that have mass. Newton's law of universal gravitation says that every particle pulls on every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. The pull is always attractive, it acts along the line joining the two bodies, and it never switches off, although it becomes extremely weak at large distances. This calculator applies the law F = G m₁ m₂ / r² with the CODATA 2018 value of the gravitational constant, G = 6.67430 × 10−11 N m² kg−2.
The calculator has two modes. The first finds the force between two masses at a chosen separation, then shows the acceleration each body experiences, the gravitational field strength of body 1 at that distance, and what happens to the force when the separation doubles. The second mode starts from a planet, moon or star: give its mass and radius and you get the surface gravity g = G M / R², the weight of any test mass standing on it, the escape speed and the mean density. Students use it to check homework on orbits and weight, teachers use it to show the inverse square law, and writers and hobbyists use it to compare how heavy a person would feel on the Moon or Mars.
Two points cause most mistakes. First, r is the distance between the centres of the two bodies. For a person on the ground, that is the radius of the Earth, not the height above the surface. Second, the two bodies feel exactly equal and opposite forces, even when one is enormously heavier. The Earth pulls a 70 kg person with 687 N, and the person pulls the Earth with the same 687 N. The difference is the acceleration, which is force divided by mass: about 9.82 m/s² for the person and about 10−22 m/s² for the Earth.
The law is exact for point masses and for uniform spheres, and it is a very good approximation for planets and stars treated from outside. It is a Newtonian result, so it is not the right tool for black holes or for precision work such as the orbit of Mercury, where general relativity is needed. For the speed needed to leave a planet in full detail, see the escape velocity calculator, and for circular orbits use the orbital period and velocity calculator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — The Earth and the Moon
m₁ = 5.972 × 1024 kg, m₂ = 7.342 × 1022 kg, r = 3.844 × 108 m
Example 2 — Two 1 kg Masses 1 m Apart
m₁ = 1 kg, m₂ = 1 kg, r = 1 m
Example 3 — A 70 kg Person and the Earth
m₁ = 5.972 × 1024 kg (Earth), m₂ = 70 kg, r = 6.371 × 106 m (Earth radius)
Example 4 — Surface Gravity of the Moon
M = 7.342 × 1022 kg, R = 1737.4 km, test mass 70 kg
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for gravitational force?
Newton's law of universal gravitation gives F = G m1 m2 / r^2, where G = 6.67430e-11 N m^2 kg^-2 (CODATA 2018), m1 and m2 are the masses in kilograms and r is the distance between their centres in metres. The force is attractive and acts along the line joining the two bodies.
What is the gravitational force between the Earth and the Moon?
About 1.98e20 N. Using Earth mass 5.972e24 kg, Moon mass 7.342e22 kg and a mean separation of 3.844e8 m, F = 6.6743e-11 x 5.972e24 x 7.342e22 / (3.844e8)^2 = 1.98e20 N. The Moon accelerates toward the Earth at 2.7e-3 m/s^2.
What is the gravitational force between two 1 kg masses 1 m apart?
Only 6.674e-11 N, numerically equal to G. That is far too small to notice, which is why everyday objects do not visibly attract each other and why the constant G was so hard to measure in the laboratory.
How do you calculate surface gravity?
Use g = G M / R^2, where M is the mass of the body and R its radius. For Earth, g = 6.6743e-11 x 5.972e24 / (6.371e6)^2 = 9.82 m/s^2. The weight of any object is then its mass times g.
Does the force depend on the size of the objects?
Not directly. The force depends only on the masses and the distance between their centres. Size matters only because it limits how close the centres can get. A uniform sphere acts as if all its mass sat at its centre.
What happens to gravity when the distance doubles?
The force falls to one quarter. Gravity follows an inverse square law, so F is proportional to 1/r^2. Ten times the distance gives one hundredth of the force.
Why do both bodies feel the same force but accelerate differently?
Newton's third law makes the pair of forces equal and opposite. Acceleration is force divided by mass, so the less massive body accelerates more. A 70 kg person and the Earth pull each other with 687 N, yet the person accelerates at 9.82 m/s^2 and the Earth at about 1e-22 m/s^2.
What is gravitational field strength?
It is the force per unit mass that a body creates at a point, g = G M / r^2, measured in N/kg or m/s^2. At Earth's surface it is about 9.82 m/s^2, and at the Moon's distance it falls to about 2.7e-3 m/s^2.
How is escape speed related to surface gravity?
Escape speed is v = sqrt(2 G M / R), which equals sqrt(2 g R). For Earth it is 11.19 km/s and for the Moon 2.375 km/s. It is the minimum launch speed to leave the body without further propulsion, ignoring air drag.
Is the value of G exact?
No. G = 6.67430e-11 N m^2 kg^-2 with a standard uncertainty of 0.00015e-11 (CODATA 2018). It is the least precisely known of the major constants because gravity is so weak and laboratory results disagree slightly.