Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, Newton's Law of Gravitation | Physics

Learning objectives

Teacher Support

The learning objectives in this section will help your students master the following standards:

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:
- (D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects;

(5) Science concepts. The student knows the nature of forces in the physical world. The student is expected to:
- (A) research and describe the historical development of the concepts of gravitational, electromagnetic, weak nuclear, and strong nuclear forces;
- (B) describe and calculate how the magnitude of the gravitational force between two objects depends on their masses and the distance between their centers.

Section Key Terms

Einstein’s theory of general relativity

gravitational constant

Newton’s universal law of gravitation

Concepts Related to Newton’s Law of Universal Gravitation

Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See Figure_07_02_apple. But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner, Galileo Galilei, had contended that falling bodies and planetary motions had the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph. It had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose an explanation of the mechanism that caused them to follow these paths and not others.

View original figure (external; image not reproduced in this text edition)

Original figure description: A painting shows a man sitting under a tree, from which an apple is falling.

The popular legend that Newton suddenly discovered the law of universal gravitation when an apple fell from a tree and hit him on the head has an element of truth in it. A more probable account is that he was walking through an orchard and wondered why all the apples fell in the same direction with the same acceleration. Great importance is attached to it because Newton’s universal law of gravitation and his laws of motion answered very old questions about nature and gave tremendous support to the notion of underlying simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature.

The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Expressed in modern language, Newton’s universal law of gravitation states that every object in the universe attracts every other object with a force that is directed along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This attraction is illustrated by Figure_07_02_gravity.

View original figure (external; image not reproduced in this text edition)

Original figure description: A figure shows a small circle and a large circle. A black line goes from one circle to the other. A red line with an arrow extends outward from each circle. The red lines have arrows on one end, pointing away from the circles, towards the midpoint of the space in between the circles.

Gravitational attraction is along a line joining the centers of mass (CM) of the two bodies. The magnitude of the force on each body is the same, consistent with Newton’s third law (action-reaction).

For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law of gravitation is

F = G \frac{m M}{r^{2}}

where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G is a universal constant, meaning that it is thought to be the same everywhere in the universe. It has been measured experimentally to be $G = 6.673 \times 10^{- 11} N \cdot m^{2} {/kg}^{2}$ .

If a person has a mass of 60.0 kg, what would be the force of gravitational attraction on him at Earth’s surface? G is given above, Earth’s mass M is 5.97 × 10²⁴ kg, and the radius r of Earth is 6.38 × 10⁶ m. Putting these values into Newton’s universal law of gravitation gives

F = G \frac{m M}{r^{2}} = (6.673 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{(60.0 kg) (5.97 \times 10^{24} kg)}{{(6.38 \times 10^{6} m)}^{2}}) = 584 N

We can check this result with the relationship: $F = m g = (60 kg) (9.8 {m/s}^{2}) = 588 N$

You may remember that g, the acceleration due to gravity, is another important constant related to gravity. By substituting g for a in the equation for Newton’s second law of motion we get $F = m g$ . Combining this with the equation for universal gravitation gives

m g = G \frac{m M}{r^{2}}

Cancelling the mass m on both sides of the equation and filling in the values for the gravitational constant and mass and radius of the Earth, gives the value of g, which may look familiar.

g = G \frac{M}{r^{2}} = (6.67 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{5.98 \times 10^{24} kg}{{(6.38 \times 10^{6} m)}^{2}}) = 9.80 {m/s}^{2}

This is a good point to recall the difference between mass and weight. Mass is the amount of matter in an object; weight is the force of attraction between the mass within two objects. Weight can change because g is different on every moon and planet. An object’s mass m does not change but its weight mg can.

Take-Home Experiment: Falling Objects

In this activity you will study the effects of mass and air resistance on the acceleration of falling objects. Make predictions (hypotheses) about the outcome of this experiment. Write them down to compare later with results.

Four sheets of $8 - 1 / 2 \times 11$ -inch paper

Procedure

Take four identical pieces of paper.
- Crumple one up into a small ball.
- Leave one uncrumpled.
- Take the other two and crumple them up together, so that they make a ball of exactly twice the mass of the other crumpled ball.
- Now compare which ball of paper lands first when dropped simultaneously from the same height.
  1. Compare crumpled one-paper ball with crumpled two-paper ball.
  2. Compare crumpled one-paper ball with uncrumpled paper.

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It is possible to derive Kepler’s third law from Newton’s law of universal gravitation. Applying Newton’s second law of motion to angular motion gives an expression for centripetal force, which can be equated to the expression for force in the universal gravitation equation. This expression can be manipulated to produce the equation for Kepler’s third law. We saw earlier that the expression r³/T² is a constant for satellites orbiting the same massive object. The derivation of Kepler’s third law from Newton’s law of universal gravitation and Newton’s second law of motion yields that constant:

\frac{r^{3}}{T^{2}} = \frac{G M}{4 π^{2}}

where M is the mass of the central body about which the satellites orbit (for example, the sun in our solar system). The usefulness of this equation will be seen later.

The universal gravitational constant G is determined experimentally. This determination was first done accurately in 1798 by English scientist Henry Cavendish (1731–1810), more than 100 years after Newton published his universal law of gravitation. The measurement of G is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most) by using an apparatus like that in Figure_07_02_sphere. Remarkably, his value for G differs by less than 1% from the modern value.

View original figure (external; image not reproduced in this text edition)

Original figure description: Apparatus consisting of two large spheres, one on each end of a horizontal rod centered on a pivot. Two smaller spheres are at the extremity of a smaller horizontal rod also centered on same pivot point. Light source shines laser beam that hits mirror and reflects onto scale to show pivot angle of spheres.

Cavendish used an apparatus like this to measure the gravitational attraction between two suspended spheres (m) and two spheres on a stand (M) by observing the amount of torsion (twisting) created in the fiber. The distance between the masses can be varied to check the dependence of the force on distance. Modern experiments of this type continue to explore gravity.

Einstein’s Theory of General Relativity

Einstein’s theory of general relativity explained some interesting properties of gravity not covered by Newton’s theory. Einstein based his theory on the postulate that acceleration and gravity have the same effect and cannot be distinguished from each other. He concluded that light must fall in both a gravitational field and in an accelerating reference frame. Figure_07_02_light shows this effect (greatly exaggerated) in an accelerating elevator. In Figure_07_02_light (a), the elevator accelerates upward in zero gravity. In Figure_07_02_light (b), the room is not accelerating but is subject to gravity. The effect on light is the same: it “falls” downward in both situations. The person in the elevator cannot tell whether the elevator is accelerating in zero gravity or is stationary and subject to gravity. Thus, gravity affects the path of light, even though we think of gravity as acting between masses, while photons are massless.

View original figure (external; image not reproduced in this text edition)

Original figure description: Left panel shows a person in an elevator who shines a light beam on the wall. Beam of light bends down. Elevator is labelled “accelerated up.” Right panel shows same picture, but person is in a room on Earth.

(a) A beam of light emerges from a flashlight in an upward-accelerating elevator. Since the elevator moves up during the time the light takes to reach the wall, the beam strikes lower than it would if the elevator were not accelerated. (b) Gravity must have the same effect on light, since it is not possible to tell whether the elevator is accelerating upward or is stationary and acted upon by gravity.

Einstein’s theory of general relativity got its first verification in 1919 when starlight passing near the sun was observed during a solar eclipse. (See Figure_07_02_sun.) During an eclipse, the sky is darkened and we can briefly see stars. Those on a line of sight nearest the sun should have a shift in their apparent positions. Not only was this shift observed, but it agreed with Einstein’s predictions well within experimental uncertainties. This discovery created a scientific and public sensation. Einstein was now a folk hero as well as a very great scientist. The bending of light by matter is equivalent to a bending of space itself, with light following the curve. This is another radical change in our concept of space and time. It is also another connection that any particle with mass or energy (e.g., massless photons) is affected by gravity.

View original figure (external; image not reproduced in this text edition)

Original figure description: Schematic drawing of sun at left and Earth at right. A person is standing on Earth. Two stars to left of sun. Lines drawn from starts to person’s eyes. Lower star has curved solid line, and upper star has straight dashed line.

This schematic shows how light passing near a massive body like the sun is curved toward it. The light that reaches the Earth then seems to be coming from different locations than the known positions of the originating stars. Not only was this effect observed, but the amount of bending was precisely what Einstein predicted in his general theory of relativity.

To summarize the two views of gravity, Newton envisioned gravity as a tug of war along the line connecting any two objects in the universe. In contrast, Einstein envisioned gravity as a bending of space-time by mass.

NASA gravity probe B

NASA’s Gravity Probe B (GP-B) mission has confirmed two key predictions derived from Albert Einstein’s general theory of relativity. The probe, shown in Figure_07_02_NASA was launched in 2004. It carried four ultra-precise gyroscopes designed to measure two effects hypothesized by Einstein’s theory:

The geodetic effect, which is the warping of space and time by the gravitational field of a massive body (in this case, Earth)
The frame-dragging effect, which is the amount by which a spinning object pulls space and time with it as it rotates

View original figure (external; image not reproduced in this text edition)

Original figure description: Photo shows satellite in space.

Artist concept of Gravity Probe B spacecraft in orbit around the Earth. (credit: NASA/MSFC)

Both effects were measured with unprecedented precision. This was done by pointing the gyroscopes at a single star while orbiting Earth in a polar orbit. As predicted by relativity theory, the gyroscopes experienced very small, but measureable, changes in the direction of their spin caused by the pull of Earth’s gravity.

The principle investigator suggested imagining Earth spinning in honey. As Earth rotates it drags space and time with it as it would a surrounding sea of honey.

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Calculations Based on Newton’s Law of Universal Gravitation

The mass of an object is constant, but its weight varies with the strength of the gravitational field. This means the value of g varies from place to place in the universe. The relationship between force, mass, and acceleration from the second law of motion can be written in terms of g.

F = m a = m g

In this case, the force is the weight of the object, which is caused by the gravitational attraction of the planet or moon on which the object is located. We can use this expression to compare weights of an object on different moons and planets.

Mass and Weight Clarification

This video shows the mathematical basis of the relationship between mass and weight. The distinction between mass and weight are clearly explained. The mathematical relationship between mass and weight are shown mathematically in terms of the equation for Newton’s law of universal gravitation and in terms of his second law of motion.

Mass and weight clarification | Centripetal force and gravitation | Physics | Khan Academy

Would you have the same mass on the moon as you do on Earth? Would you have the same weight?

You would weigh more on the moon than on Earth because gravity on the moon is stronger than gravity on Earth.
You would weigh less on the moon than on Earth because gravity on the moon is weaker than gravity on Earth.
You would weigh less on the moon than on Earth because gravity on the moon is stronger than gravity on Earth.
You would weigh more on the moon than on Earth because gravity on the moon is weaker than gravity on Earth.

Two equations involving the gravitational constant, G, are often useful. The first is Newton’s equation, $F = G \frac{m M}{r^{2}}$ . Several of the values in this equation are either constants or easily obtainable. F is often the weight of an object on the surface of a large object with mass M, which is usually known. The mass of the smaller object, m, is often known, and G is a universal constant with the same value anywhere in the universe. This equation can be used to solve problems involving an object on or orbiting Earth or other massive celestial object. Sometimes it is helpful to equate the right-hand side of the equation to mg and cancel the m on both sides.

The equation $\frac{r^{3}}{T^{2}} = \frac{G M}{4 π^{2}}$ is also useful for problems involving objects in orbit. Note that there is no need to know the mass of the object. Often, we know the radius r or the period T and want to find the other. If these are both known, we can use the equation to calculate the mass of a planet or star.

Earth’s g at the Moon

Find the acceleration due to Earth’s gravity at the distance of the moon.

\begin{matrix} G = 6 .67 \times 10^{–11} {N · m}^{2} {/kg}^{2} \\ Earth-moon distance = 3.84 \times 10^{8} m \\ Earth’s mass = 5.98 \times 10^{24} kg \end{matrix}

Express the force of gravity in terms of g.

F = W = m a = m g

Combine with the equation for universal gravitation.

m g = m G \frac{M}{r^{2}}

Solution

Cancel m and substitute.

g = G \frac{M}{r^{2}} = (6.67 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{5.98 \times 10^{24} kg}{{(3.84 \times 10^{8} m)}^{2}}) = 2.70 \times 10^{- 3} {m/s}^{2}

Discussion

The value of g for the moon is 1.62 m/s². Comparing this value to the answer, we see that Earth’s gravitational influence on an object on the moon’s surface would be insignificant.

Practice Problems

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Check Your Understanding

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Newtons law of universal gravitation explains the paths of what?

A charged particle
A ball rolling on a plane surface
A planet moving around the sun
A stone tied to a string and whirled at constant speed in a horizontal circle

Section Summary

Newton’s law of universal gravitation provides a mathematical basis for gravitational force and Kepler’s laws of planetary motion.
Einstein’s theory of general relativity shows that gravitational fields change the path of light and warp space and time.
An object’s mass is constant, but its weight changes when acceleration due to gravity, g, changes.

Key Equations

Newton’s second law of motion	$F = m a = m g$
Newton’s universal law of gravitation	$F = G \frac{m M}{r^{2}}$
acceleration due to gravity	$g = G \frac{M}{r^{2}}$
constant for satellites orbiting the same massive object	$\frac{r^{3}}{T^{2}} = \frac{G M}{4 π^{2}}$

Concept Items

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Saturn’s moon Titan has an orbital period of 15.9 days. If Saturn has a mass of 5.68×10²⁶ kg, what is the average distance from Titan to the center of Saturn?

1.22×10⁹ m
4.26×10⁷ m
5.25×10⁴ km
4.26×10¹⁰ km

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The Cavendish experiment marked a milestone in the study of gravity.
Part A. What important value did the experiment determine?
Part B. Why was this so difficult in terms of the masses used in the apparatus and the strength of the gravitational force?

Part A. The experiment measured the acceleration due to gravity, g. Part B. Gravity is a very weak force but despite this limitation, Cavendish was able to measure the attraction between very massive objects.
Part A. The experiment measured the gravitational constant, G. Part B. Gravity is a very weak force but, despite this limitation, Cavendish was able to measure the attraction between very massive objects.
Part A. The experiment measured the acceleration due to gravity, g. Part B. Gravity is a very weak force but despite this limitation, Cavendish was able to measure the attraction between less massive objects.
Part A. The experiment measured the gravitational constant, G. Part B. Gravity is a very weak force but despite this limitation, Cavendish was able to measure the attraction between less massive objects.

Critical Thinking

Rhea, with a radius of 7.63×10⁵ m, is the second-largest moon of the planet Saturn. If the mass of Rhea is 2.31×10²¹ kg, what is the acceleration due to gravity on the surface of this moon?

2.65×10⁻¹ m/s
2.02×10⁵ m/s
2.65×10⁻¹ m/s²
2.02×10⁵ m/s²

Earth has a mass of 5.971×10²⁴ kg and a radius of 6.371×10⁶ m. Use the data to check the value of the gravitational constant.

$6.66 \times 10^{- 11} \frac{N \cdot m}{k g^{2}},$ it matches the value of the gravitational constant G.
$1.05 \times 10^{- 17} \frac{N \cdot m}{k g^{2}},$ it matches the value of the gravitational constant G.
$6.66 \times 10^{- 11} \frac{N \cdot m^{2}}{k g^{2}},$ it matches the value of the gravitational constant G.
$1.05 \times 10^{- 17} \frac{N \cdot m^{2}}{k g^{2}},$ it matches the value of the gravitational constant G.

The orbit of the planet Mercury has a period of 88.0 days and an average radius of 5.791×10¹⁰ m. What is the mass of the sun?

3.43×10¹⁹ kg
1.99×10³⁰ kg
2.56×10²⁹ kg
1.48×10⁴⁰ kg

Performance Task

Design an experiment to test whether magnetic force is inversely proportional to the square of distance. Gravitational, magnetic, and electrical fields all act at a distance, but do they all follow the inverse square law? One difference in the forces related to these fields is that gravity is only attractive, but the other two can repel as well. In general, the inverse square law says that force F equals a constant C divided by the distance between objects, d, squared: $F = C / d^{2}$ .

Incorporate these materials into your design:

Two strong, permanent bar magnets
A spring scale that can measure small forces
A short ruler calibrated in millimeters

Use the magnets to study the relationship between attractive force and distance.

What will be the independent variable?
What will be the dependent variable?
How will you measure each of these variables?
If you plot the independent variable versus the dependent variable and the inverse square law is upheld, will the plot be a straight line? Explain.
Which plot would be a straight line if the inverse square law were upheld?

Test Prep Multiple Choice

What did the Cavendish experiment measure?

The mass of Earth
The gravitational constant
Acceleration due to gravity
The eccentricity of Earths orbit

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Test Prep Short Answer

Newtons third law of motion says that, for every action force, there is a reaction force equal in magnitude but that acts in the opposite direction. Apply this law to gravitational forces acting between the Washington Monument and Earth.

The monument is attracted to Earth with a force equal to its weight, and Earth is attracted to the monument with a force equal to Earths weight. The situation can be represented with two force vectors of unequal magnitude and pointing in the same direction.
The monument is attracted to Earth with a force equal to its weight, and Earth is attracted to the monument with a force equal to Earths weight. The situation can be represented with two force vectors of unequal magnitude but pointing in opposite directions.
The monument is attracted to Earth with a force equal to its weight, and Earth is attracted to the monument with an equal force. The situation can be represented with two force vectors of equal magnitude and pointing in the same direction.
The monument is attracted to Earth with a force equal to its weight, and Earth is attracted to the monument with an equal force. The situation can be represented with two force vectors of equal magnitude but pointing in opposite directions.

True or false—Gravitational force is the attraction of the mass of one object to the mass of another. Light, either as a particle or a wave, has no rest mass. Despite this fact gravity bends a beam of light.

True
False

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Test Prep Extended Response

It took scientists a long time to arrive at the understanding of gravity as explained by Galileo and Newton. They were hindered by two ideas that seemed like common sense but were serious misconceptions. First was the fact that heavier things fall faster than light things. Second, it was believed impossible that forces could act at a distance. Explain why these ideas persisted and why they prevented advances.

Heavier things fall faster than light things if they have less surface area and greater mass density. In the Renaissance and before, forces that acted at a distance were considered impossible, so people were skeptical about scientific theories that invoked such forces.
Heavier things fall faster than light things because they have greater surface area and less mass density. In the Renaissance and before, forces that act at a distance were considered impossible, so people were skeptical about scientific theories that invoked such forces.
Heavier things fall faster than light things because they have less surface area and greater mass density. In the Renaissance and before, forces that act at a distance were considered impossible, so people were quick to accept scientific theories that invoked such forces.
Heavier things fall faster than light things because they have larger surface area and less mass density. In the Renaissance and before, forces that act at a distance were considered impossible because of people’s faith in scientific theories.

The masses of Earth and the moon are 5.97×10²⁴ kg and 7.35×10²² kg, respectively. The distance from Earth to the moon is 3.80×10⁵ km. At what point between the Earth and the moon are the opposing gravitational forces equal? (Use subscripts e and m to represent Earth and moon.)

3.42×10⁵ km from the center of Earth
3.80×10⁵ km from the center of Earth
3.42×10⁶ km from the center of Earth
3.10×10⁷ km from the center of Earth

Glossary

Einstein’s theory of general relativity: the theory that gravitational force results from the bending of spacetime by an object’s mass
gravitational constant: the proportionality constant in Newton’s law of universal gravitation
Newton’s universal law of gravitation: states that gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Attribution. Physics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 56e71824e7 of openstax/osbooks-physics (2026-10-13). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.