# Law of Universal Gravitation

by Ron Kurtus (revised 17 December 2018)

In 1687, Isaac Newton combined his observations with theories from other scientists about gravity and gravitation into a scientific law: the Law of Universal Gravitation. This law states that the mass of an object is attracted toward the mass of other objects by a force called gravitation.

The force of attraction between two masses is defined by the Universal Gravitation Equation. Although the law and its equation were effective in predicting many phenomena, several discrepancies later arose in astronomical measurements. It wasn't until 1915 that Einstein's Theory of General Relativity provided a solution to these discrepancies.

Questions you may have include:

• How was the Law of Universal Gravitation established?
• What is the Universal Gravitation Equation?
• What are some problems with Universal Gravitation?

This lesson will answer those questions. Useful tool: Units Conversion

## Establishing the Law of Universal Gravitation

Scientists and philosophers from ancient times have made observations about gravity on Earth. In 628, Indian astronomer Brahmagupta recognized gravity as a force of attraction. In the 1600s, Galileo Galilei, Robert Hooke and Johannes Kepler formulated laws of gravity near the Earth.

### Newton plays off Kepler's Laws

In studying the orbits of planets around the Sun, Kepler determined that gravitational attraction varied with separation, while formulating his Laws of Planetary Motion.

In 1687, Isaac Newton's observations on planetary motion and empirical measurements, allowed him to establish the Law of Universal Gravitation, which was explained in the publication of Philosophiæ Naturalis Principia Mathematica (or simply Principia) set of three books that stated his Laws of Motion, the Law of Universal Gravitation and a derivation of Kepler's Laws of Planetary Motion.

The story that Newton was sitting under a tree when he was hit in the head by a falling apple, thus causing him to discover gravity, is a fable. Although he may have been inspired by seeing an apple fall from a tree, his studies and correspondence with other scientists who were studying gravitation probably influenced him more than seeing an apple fall. Nevertheless, it is a good story.

### Statement of Universal Gravitation

The Law of Universal Gravitation states that every object of mass in the Universe attracts every other object of mass with a force which is directly proportional to the product of their masses and inversely proportional to the square of the separation between their centers. This was then formalized into the Universal Gravitation Equation. Masses attract each other

This law was originally stated for point masses. However, it was shown that the gravitation from a large uniform sphere is approximately the same as if all the mass was concentrated at its center.

#### Law versus Theory

Note: Newton's Law of Universal Gravitation is sometimes incorrectly called the Theory of Universal Gravitation. In the sciences, a law generalizes observations and shows no exceptions. On the other hand, a theory tells why a phenomenon happens. The Law of Universal Gravitation only makes predictions of how bodies are attracted and does not explain why it happens.

#### Universal Gravitation versus Universal Law

Note: Also, the Law of Universal Gravitation is sometimes called the Universal Law of Gravitation. A universal law is one that is said to apply to the whole Universe. On the other hand, universal gravitation implies that gravitation is throughout the universe, which is slightly different than a universal law.

### Force at a distance

Newton could not explain the mechanism of how and why gravitation occurred, except that it was some sort of force at a distance. Critics said that this explanation was bringing the occult or mysticism into science, which Newton denied.

## Universal Gravitation Equation

From the Law of Universal Gravitation, Newton formulated the Universal Gravitation Equation, which defines the gravitational force between two objects. The equation is:

F = GMm/R2

where

• F is the force of attraction between two objects in newtons (N)
• G is the Universal Gravitational Constant = 6.67*10−11 N-m2/kg2
• M and m are the masses of the two objects in kilograms (kg)
• R is the separation in meters (m) between the objects, as measured from their centers of mass

### Useful in determining orbits

Application of this equation is especially useful in determining the orbits of celestial objects, such as that of the planets around the Sun. This is relatively simple when considering one object orbiting another.

However, in most cases there are more than two objects involved, which can result in some complicated mathematics.

For example, the motion of the planet Mars around the Sun is not only determined by the masses of Mars and the Sun but also by the influence of the gravitation from Mar's moons, the Earth and other planets. The result is that orbit of Mars includes perturbations or deviations caused by the other planets.

## Problems with Universal Gravitation

In most case, the Law of Universal Gravitation and resulting calculations work well. But there are some observed discrepancies and unexplained phenomena where the law fails.

### Orbit of planet Mercury

Predictions of the orbit of the planet Mercury proved to be slightly inaccurate using the Law of Universal Gravitation. The law could not explain the precession or deviations of Mercury's orbit.

### Deflection of light rays

Electromagnetic waves or light rays are deflected by gravitation, especially when they pass by a large mass, such as the Sun. However, calculations from Newton's law result in only one-half of the deflection that is actually observed by astronomers.

### Speed of gravitation

The Law of Universal Gravitation requires that the gravitational force is transmitted instantaneously over large distances to maintain stability in planetary and stellar orbits. Although light from the Sun may take eight minutes to reach the Earth, gravitational changes would have to occur immediately to maintain orbital stability between the two objects.

This implies that what is called the speed of gravitation (also carelessly called the speed of gravity) is infinite. That seems counterintuitive.

### Solutions found

Solutions to these problems were finally found in Einstein's Theory of General Relativity that he published in 1915. The theory reduces to the Law of Universal Gravitation for simple cases.

## Summary

Since ancient times, scientists and philosophers studied gravity and then later gravitation. Isaac Newton formalized the observations into a scientific law: the Law of Universal Gravitation, which states that states that all objects of mass are attracted toward other objects of mass, due to a force called gravitation.

The force of attraction between two masses is defined by the Universal Gravitation Equation. Although the law and its equation were effective in predicting many phenomena, several discrepancies arose in measurements that were made. Einstein's Theory of General Relativity provided a solution to the Universal Gravitation problems.

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## Resources and references

Ron Kurtus' Credentials

### Websites

Scientific Hypothesis, Theory, Law Definitions: Learn the Language of Science - Anne Marie Helmenstine, Ph.D., About.com

Understanding General Relativity - Rafi Moor, Israel

Einstein's Theory of General Relativity - Andrew Zimmerman Jones, About.com

General Relativity - Wikipedia

The Meaning of Einstein's Equation - John C. Baez, University of California-Riverside and Emory F. Bunn, University of Richmond (Virginia)

The Einstein Field Equations - Larry Smarr, University of Illinois

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