Short answer
- The force of gravity is the attractive force between any two objects that have mass. In the Newtonian definition used in introductory physics, its size is given by F equals G times m1 times m2 divided by r squared. Here F is the gravitational force, G is the universal gravitational constant, m1 and m2 are the two masses, and r is the distance between their centers of mass. The force always pulls the objects toward each other, it grows when either mass grows, and it falls quickly when distance grows. Near Earth, the same idea is often written in the shorter form W equals m times g, where W is weight, m is mass, and g is the local acceleration due to gravity. Weight is a force measured in newtons. Mass is a quantity of matter measured in kilograms, and it does not change when an object moves to a new location.
What the Force of Gravity Means
In the broad sense, gravity is the name for the attraction between masses. In the narrower, quantitative sense, the force of gravity is the size of that attraction for a particular pair of objects at a particular distance. A definition is not complete if it only says that gravity pulls things down. It must also say what pulls on what, in which direction, and how the size of the pull changes.
NASA describes gravity as a force of attraction that exists between any two masses, any two bodies, and any two particles, and it stresses that gravity is not only the attraction between objects and Earth. A book on a table, a person standing on a floor, and the Moon in orbit are all examples of the same general pattern, even though the masses and distances are very different. The force is attractive only in this Newtonian picture. It does not push masses apart. Each object pulls the other toward itself, along the line joining their centers. That pair of pulls is equal in size and opposite in direction. Earth pulls a person down, and the person pulls Earth up by the same amount of force, even though Earth does not noticeably move because its mass is so much larger.
NASA states that the effect of gravity extends from each object out into space in all directions, and for an infinite distance in the mathematical model. In practice, the force becomes extremely small at very large distances, but the formula does not set a cutoff distance beyond which the force suddenly becomes zero. Gravitation is the general phenomenon: masses attract each other. Gravity is often used for the same phenomenon, especially for attraction near a planet. The force of gravity is the quantitative version: the actual force for a given pair of masses at a given distance, measured in newtons.
The Definition in One Sentence, Then in Full
A useful one sentence definition is this: the force of gravity is the attractive force between two masses, whose magnitude is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers.
Directly proportional to the product of the masses means that if one mass doubles while the other mass and the distance stay the same, the force doubles. If both masses double, the force becomes four times as large. Inversely proportional to the square of the distance means that if the distance doubles, the force does not fall to one half. It falls to one fourth, because two squared equals four and the force is divided by that number. If the distance is cut in half, the force becomes four times as large. NASA gives the same doubling example for the Moon: if the Moon were twice its current distance from Earth, the gravitational tug between the two would be just one fourth of what it is now.
The full form is F equals G times m1 times m2 divided by r squared, written F = G * m1 * m2 / r^2. The caret in r^2 means r squared, or r multiplied by itself. Direction must be added in words: the force on each mass points toward the other mass. This law is called Newton’s law of universal gravitation. The word universal means that the same law is intended to apply to falling objects near Earth and to bodies far away in the solar system. OpenStax describes the formula as valid everywhere in the universe for masses and distances that vary from the tiny to the immense, and as a force that acts at a distance without physical contact. An object does not need to touch Earth to feel Earth’s gravitational force.
Reading the Formula Symbol by Symbol
F stands for the gravitational force between the two masses. Its unit in the International System of Units, usually called SI, is the newton, written N. One newton is the force needed to accelerate a one kilogram mass at one meter per second squared.
G stands for the gravitational constant, sometimes called big G to distinguish it from little g. G is a proportionality factor that converts the product of the masses divided by the squared distance into a force. OpenStax describes G as a universal constant, meaning that it is thought to be the same everywhere in the universe. The CODATA 2022 recommended value of G is 6.67430 times 10 to the power of negative 11 cubic meters per kilogram per second squared, which can also be written as N m^2 / kg^2. OpenStax rounds the value to 6.673 times 10 to the power of negative 11 N m^2 / kg^2 in one section and to 6.67 times 10 to the power of negative 11 in another, which is normal rounding for teaching examples. For most general reader calculations, 6.67 times 10 to the power of negative 11 is enough, but a reader who needs the best current recommended value should use 6.67430 times 10 to the power of negative 11 and note that it is the CODATA 2022 recommendation. G is extremely small, and that smallness is the main reason gravity between everyday objects is negligible in ordinary life, even though the same law governs planets. When a mass of about 10 to the power of 24 kilograms is put into the formula, the tiny value of G is balanced by a huge mass, and the result is a familiar force of hundreds of newtons for a person on Earth.
m1 and m2 stand for the two masses, both measured in kilograms in SI calculations. The order does not matter, because multiplication is commutative. What matters is that both masses are included. Leaving one mass out, or using weight in place of mass in this formula, will give a wrong answer. r stands for the distance between the centers of mass of the two objects, measured in meters. It is not the distance between their surfaces, and it is not the height above the ground, unless those numbers happen to match the center to center distance. For an object on Earth’s surface, r is very nearly Earth’s radius, because the object is tiny compared with Earth and sits on its surface. OpenStax uses 6.38 times 10 to the power of 6 meters for Earth’s radius in its worked example. Getting r right is often more important than getting extra decimal places for G, because r is squared, so a small error in r becomes a larger error in the force.
Worked Examples: A Person on Earth and Two Small Masses
Using the inputs from OpenStax’s worked example, take a person with a mass of 60.0 kilograms. Take Earth’s mass as 5.97 times 10 to the power of 24 kilograms, and Earth’s radius as 6.38 times 10 to the power of 6 meters. Put those numbers into F equals G times m times M divided by r squared, using G as 6.673 times 10 to the power of negative 11 N m^2 / kg^2. The recomputed result is about 587 newtons, although OpenStax itself prints 584 newtons for the same inputs. The same person can be checked with the shorter near Earth formula, W equals m times g, using g as 9.8 meters per second squared. That gives 60 kilograms times 9.8 meters per second squared, which equals 588 newtons. The two results differ by only 1 newton, because of rounding in G, Earth’s mass, Earth’s radius, and g. Rounded inputs give rounded outputs, and a reader should not claim more precision than the inputs support.
This example also shows why the full formula and the short formula agree near Earth. The short formula is not a separate law. It is the full formula with Earth’s mass and radius already combined into a single number, g. OpenStax shows this directly: start with m times g equals G times m times M divided by r squared, cancel m from both sides, and the result is g equals G times M divided by r squared. The mass of the small object cancels because it appears on both sides. That cancellation is the reason why g near Earth does not depend on the mass of the falling object in this model.
Now take two masses of 1 kilogram each, with their centers 1 meter apart. Multiplying by G gives about 6.67 times 10 to the power of negative 11 newtons. That force is far too small to feel, and far too small to move a 1 kilogram object in any noticeable way against friction and other forces in a normal room. At 0.5 meters, half the original distance, the force becomes about 2.67 times 10 to the power of negative 10 newtons, four times larger but still extremely small. At 2 meters, twice the original distance, the force falls to one fourth of the original value, about 1.67 times 10 to the power of negative 11 newtons. This answers a common question: if gravity acts between all masses, why do we not feel pulled toward a book or another person? G is so small that ordinary masses produce forces far below human sensation.
Big G and Little g: Two Different Quantities
Big G is the universal gravitational constant. It is the same number in the formula wherever the formula is applied, at least in the Newtonian model. Its value is about 6.67430 times 10 to the power of negative 11, and its units involve cubic meters, kilograms, and seconds. It does not depend on which planet is involved, and it does not change when a person walks from a valley to a mountain top.
Little g is the local acceleration due to gravity. It is the acceleration that a freely falling object would gain near a particular body, if other forces such as air resistance could be ignored. On Earth, g is about 9.80 meters per second squared in OpenStax teaching examples. It varies from place to place because it depends on the mass and radius of the body and on the distance from its center. The relationship between them near a spherical body is g equals G times M divided by r squared, where M is the mass of the large body and r is the distance from its center. If M is Earth’s mass and r is Earth’s radius, the result is Earth’s surface g. If M is the Moon’s mass and r is the Moon’s radius, the result is the Moon’s surface g, which is much smaller.
Standard gravity is a related but different idea. Standard gravity, written g sub 0 or g sub n, is a defined nominal value of 9.80665 meters per second squared, used as a fixed reference value, not as the local g at any particular place. Local g varies slightly with latitude, altitude, and nearby mass.
Units, and How to Check Them
Units are part of the definition, not an optional extra. In SI, force F is in newtons, N; gravitational constant G is in newton square meters per kilogram squared, N m^2 / kg^2, which is the same as cubic meters per kilogram per second squared; masses m1 and m2 are in kilograms, kg; distance r is in meters, m; and acceleration due to gravity g is in meters per second squared, m/s^2, which is the same as newtons per kilogram, N/kg.
A quick unit check can catch many errors. Multiply G by the two masses and divide by distance squared: N m^2 / kg^2 times kg times kg divided by m^2. The kilograms cancel, the square meters cancel, and newtons remain. For W equals m times g, kilograms times meters per second squared gives kilogram meters per second squared, which is the definition of a newton. OpenStax states this directly: 1 N equals 1 kg m/s^2.
In the United States, pounds are also used for weight in everyday speech. OpenStax notes that 1 newton equals 0.225 pounds of force, so a 225 pound person weighs about 1000 newtons. The pound in that sentence is a unit of force, not a unit of mass. A scale that reads in kilograms is really measuring a force and then converting it to a mass reading using Earth’s g. That conversion is correct only where the local g matches the calibration.
Weight Versus Mass
Mass is a measure of how much matter an object contains, and in Newtonian physics it does not vary with location. OpenStax states the point plainly: the mass of an object is the same on Earth, in orbit, or on the surface of the Moon. Mass also measures resistance to acceleration. A more massive object needs a larger net force to gain the same acceleration as a less massive object.
Weight is the gravitational force acting on an object. OpenStax defines weight as the pull of Earth on an object in the Earth case, and more generally as w equals m times g, where w is weight, m is mass, and g is the local acceleration due to gravity. Because g changes with location, weight changes with location even when mass stays the same.
Take a mass of 70 kilograms. On Earth, using g as 9.80 meters per second squared, the weight is 70 times 9.80, which equals 686 newtons. On the Moon, using g as 1.625 meters per second squared from OpenStax, the weight is 70 times 1.625, which equals 113.75 newtons. The mass is 70 kilograms in both places. The weight falls to about one sixth of its Earth value on the Moon because the Moon’s surface gravity is about one sixth of Earth’s. A bathroom scale calibrated for Earth would therefore give a misleading mass reading on the Moon, because it would divide the smaller Moon weight by Earth’s g.
OpenStax gives another location example on Earth itself. A person at a low elevation, such as New Orleans, weighs slightly more than the same person at a higher elevation, such as Denver, even with the same mass, because weight depends on distance from Earth’s center. The everyday phrase “losing weight” usually means losing mass: when body mass falls, weight also falls because W equals m times g and g has not changed.
What Changes the Force, and What Does Not
Increasing either mass increases the force in proportion to that increase, if the other mass and the distance stay the same. Doubling one mass doubles the force. Doubling both masses quadruples the force. Increasing distance decreases the force faster than a simple proportional decrease because distance is squared. A person who climbs a tall building moves only a tiny fraction of Earth’s radius farther from Earth’s center, so the change in force is tiny. A satellite in a high orbit moves a substantial fraction of Earth’s radius or more, so the change is large and must be included. The formula does not change between those cases. Only the value of r changes.
Changing the material of an object does not change the force in this formula, except insofar as the material affects the mass. A 1 kilogram block of iron and a 1 kilogram block of wood have the same mass, and at the same center to center distance they experience the same gravitational force in this model. Changing color, temperature, or electric charge does not change the gravitational force in this formula either. Motion by itself does not change the formula, except insofar as motion changes position and therefore distance.
Direction, pairs, and the inverse square pattern complete the definition. Direction: the force on each object points toward the other object, along the line joining their centers. Pairs: gravity in this model is always a pair interaction, and the formula needs two masses. In a system with many masses, the total force on one object is the sum of the forces from each other mass, with direction taken into account. Inverse square: doubling distance quarters the force, tripling distance reduces the force to one ninth, and halving distance quadruples the force.
How to Calculate, Common Mistakes, and Limits
A reliable calculation follows the same steps every time. Step 1: identify the two masses in kilograms, converting grams first if needed, since 1000 grams equals 1 kilogram. Step 2: identify r as the center to center distance in meters, converting kilometers first if needed, since 1 kilometer equals 1000 meters. If an object sits on a planet’s surface, use the planet’s radius as r, unless the height above the surface is large enough to matter, in which case add the height to the radius. Step 3: square r. Forgetting this step is one of the most common mistakes. Step 4: multiply G by m1 and by m2, then divide by r squared, and check the final units. The answer should be in newtons. Step 5: state the direction and the meaning of the answer.
Use W equals m times g when the object is near a planet’s surface and the local g is known, and when only the weight is needed. Use the full formula when the distance is not close to the planet’s radius, when neither mass is a planet, or when g itself must be calculated from mass and radius. If the two methods give very different answers near Earth, then at least one input is wrong. Other common mistakes are using weight where mass belongs, using surface to surface distance for r, mixing G and g, reporting mass in newtons or weight in kilograms without explanation, treating gravity as a push, and assuming that gravity stops at some distance. The formula has no cutoff. A final common error is assuming that heavier objects fall faster because they feel a larger force. With air resistance ignored, a heavier object feels a larger force but also has more mass to accelerate, so g is independent of the falling object’s mass, as OpenStax stresses.
The Newtonian formula is an excellent model for most everyday, planetary, and introductory problems, and the United States Department of Energy notes that it is still taught in school. At the same time, modern physics describes gravity in a different way in Einstein’s theory of general relativity, as curvature of space-time caused by mass and energy, rather than as a force in the Newtonian sense. NASA’s Forces page gives the same picture: mass makes a dent in space-time, and that geometry affects the motions of galaxies, stars, planets, and even light. So F equals G times m1 times m2 divided by r squared defines the force of gravity in Newton’s model, which is the model used for the calculations in this article. The simple formula also applies most cleanly to point masses and to spherically symmetric bodies. Real bodies are not perfect spheres, and local mass variations cause small variations in g. Finally, G is a measured constant with a recommended value, not an exact number fixed by definition.
Frequently Asked Questions
What is the force of gravity in simple terms?
The force of gravity is the pull between any two objects that have mass. It always attracts, it gets stronger when the masses get larger, and it gets weaker quickly when the distance between them gets larger. Near Earth, it is the force we call weight when it acts on an object.
What is the formula for the force of gravity?
The formula is F equals G times m1 times m2 divided by r squared. F is the gravitational force in newtons, G is the gravitational constant, m1 and m2 are the masses in kilograms, and r is the distance between their centers in meters. Near a planet’s surface, weight can also be calculated with the shorter formula W equals m times g.
What is the value of G?
The CODATA 2022 recommended value of G is 6.67430 times 10 to the power of negative 11 cubic meters per kilogram per second squared, which is the same as 6.67430 times 10 to the power of negative 11 N m^2 / kg^2. In teaching examples it is often rounded to 6.67 times 10 to the power of negative 11.
What is the difference between G and g?
Big G is the universal gravitational constant in the formula. It is the same in the Newtonian model wherever the formula is applied. Little g is the local acceleration due to gravity at a particular place, such as about 9.80 meters per second squared on Earth and about 1.625 meters per second squared on the Moon in OpenStax examples. Little g changes with location, while big G does not change in the model.
Is weight the same as mass?
No. Mass is the quantity of matter in an object, measured in kilograms, and it does not change with location in Newtonian physics. Weight is the gravitational force on that object, measured in newtons, and it changes when g changes. A 70 kilogram mass weighs about 686 newtons on Earth and about 114 newtons on the Moon using the values in this article, but its mass stays 70 kilograms.
Why do heavier objects not fall faster than lighter objects in the model?
A heavier object feels a larger gravitational force, but it also has more mass to accelerate. In the Newtonian model with air resistance ignored, those effects cancel, so the acceleration g is independent of the falling object’s mass. That is why the formula for g near a planet, g equals G times M divided by r squared, contains the planet’s mass M but not the falling object’s mass.
Does gravity act between small objects?
Yes, in the model gravity acts between all masses, including small ones. The force between two 1 kilogram masses 1 meter apart is only about 6.67 times 10 to the power of negative 11 newtons, which is far too small to feel. That is why we notice gravity mainly for planets and other very large masses.
Does gravity ever stop working at a certain distance?
No cutoff distance appears in the Newtonian formula. The force gets smaller as distance grows, following the inverse square pattern, and it becomes extremely small at very large distances, but the formula does not set a distance beyond which it becomes exactly zero.
What units are used for the force of gravity?
Force is measured in newtons, mass in kilograms, and distance in meters in SI calculations. G has units of N m^2 / kg^2, and g has units of meters per second squared, which is the same as newtons per kilogram. Checking units is a quick way to catch errors in a calculation.
How is the force of gravity different from the mechanism of gravity?
This article defines the force and gives its size. It does not explain the deeper mechanism by which gravity works, and it does not retell the history of its discovery. Those topics are covered separately, and readers who want them should look for articles on how gravity works and on when gravity was discovered, rather than expecting a full mechanism or history inside a definition.
Sources & references
- NASA, StarChild: What is gravity? http://starchild.gsfc.nasa.gov/docs/StarChild/questions/question30.html
- NASA Science: Forces. https://science.nasa.gov/universe/overview/forces/
- NASA Glenn Research Center: Weight Equation, Weight and Gravitational Force. https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/weight-gravitational-force/
- OpenStax, Physics: 7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity. https://openstax.org/books/physics/pages/7-2-newtons-law-of-universal-gravitation-and-einsteins-theory-of-general-relativity
- OpenStax, College Physics 2e: 6.5 Newton's Universal Law of Gravitation. https://openstax.org/books/college-physics-2e/pages/6-5-newtons-universal-law-of-gravitation
- OpenStax, University Physics Volume 1: 5.4 Mass and Weight. https://openstax.org/books/university-physics-volume-1/pages/5-4-mass-and-weight
- OpenStax, College Physics: 4.3 Newton's Second Law of Motion: Concept of a System. https://openstax.org/books/college-physics/pages/4-3-newtons-second-law-of-motion-concept-of-a-system
- United States Department of Energy, DOE Explains Gravity. https://www.energy.gov/science/doe-explainsgravity
- NIST Reference on Constants, Units, and Uncertainty, 2022 CODATA Value: Newtonian constant of gravitation, as reported in search results citing NIST (value 6.67430 times 10 to the power of negative 11 m^3 kg^-1 s^-2). No NIST URL was supplied in the search result listing, so no direct NIST link is asserted here.
