# MechanicsGravity: Newtonian Relationships

by Nathaniel Stites, M.A./M.S.

Did you know that the same force that makes an apple fall to the ground holds vast galaxies together? Gravity affects our activities every day, and yet it is not well understood by scientists. However, whether it is a small marble dropping from someone's hand or the motion of planets around the sun, the behavior of objects under the influence of gravity can be described mathematically.

Isaac Newton's description of gravity was not the first explanation of this phenomenon, nor was it the last. This module explores how Newton built on the work of early astronomers and how his theory was confirmed and built upon by others. Mathematical equations are presented for (1) the Law of Universal Gravitation, (2) the Gravitational Constant, (3) Earth's mass, and (4) the gravitational attraction between two people.

What causes objects to fall toward Earth? Why do the planets orbit the sun? What holds galaxies together? If you traveled to another planet, why would your weight change?

All of these questions relate to one aspect of physics: gravity. For all of its influence on our daily lives, for all of its control over the cosmos, and for all of our ability to describe and model its effects, we do not understand the actual mechanisms of gravitational force. Of the four fundamental forces identified by physicists – strong nuclear, electroweak, electrostatic, and gravitational – the gravitational force is the least understood. Physicists today strive toward a "Grand Unified Theory," wherein all four of these forces are united into one physical model that describes the behavior of everything in the universe. At this point in time, the gravitational force is the troublesome one, the force that resists unification.

In spite of the mystery behind the mechanisms of gravity, physicists have been able to describe the behavior of objects under the influence of gravity quite thoroughly. Isaac Newton, a seventeenth into eighteenth century English scientist and mathematician (among other things), was the first person to propose a mathematical model to describe the gravitational attraction between objects. Albert Einstein built upon this model in the twentieth century and devised a more complete description of gravity in his theory of general relativity. In this module, we will explore Newton's description of gravity and some of the experimental confirmations of his theory that came many years after he proposed his original idea.

## The Apple

Whether or not Isaac Newton actually sat under an apple tree while pondering the nature of gravity, the fact that objects fall toward the surface of Earth was well understood long before Newton's time. Everyone has experience with gravity and its effects near the surface of Earth, and our intuitive view of the world includes an understanding that what goes up must come down.

Galileo Galilei (1564–1642) demonstrated that all objects fall to the surface of Earth with the same acceleration, and that this acceleration was independent of the mass of the falling object (see the Concept Simulation *Leaning Tower of Pisa Experiment* below). Isaac Newton was no doubt familiar with this concept, and he would eventually formulate a broad and far-reaching theory of gravitation. Newton's theory would encompass not only the behavior of an apple near the surface of Earth, but also the motions of much larger bodies quite far away from Earth.

## The Planets

Early conceptions of the universe were "geocentric" – they placed Earth at the center of the universe and had the planets and stars move around Earth. This Ptolemaic Model of the universe dominated scientific thought for many centuries, until the work of such careful astronomers as Tycho Brahe, Nicolaus Copernicus, Galileo Galilei, and Johannes Kepler supplanted this view of the cosmos. The "Copernican Revolution" placed the sun at the center of the solar system and the planets, including Earth, in orbit around the sun. This major shift in perception laid the foundation for Isaac Newton to begin thinking about gravitation as it related to the motions of the planets.

## An early unification theory

Just as physicists today are searching for ways to unify the fundamental forces, Isaac Newton also sought to unify two seemingly disparate phenomena: the motion of objects falling toward Earth and the motion of the planets orbiting the sun. Isaac Newton's breakthrough was not that apples fall to Earth because of gravity; it was that the planets are constantly falling toward the sun for exactly the same reason: gravity!

Newton built upon the work of early astronomers, in particular Johannes Kepler, who in 1596 and 1619 published his laws of planetary motion. One of Kepler's central observations was that the planets move in elliptical orbits around the sun. Newton expanded Kepler's description of planetary motion into a theory of gravitation.

## Comprehension Checkpoint

### What was Newton's most important contribution to our understanding of gravity?

## Newton's Law of Universal Gravitation

The essential feature of Newton's Law of Universal Gravitation is that the force of gravity between two objects is inversely proportional to the square of the distance between them. Such a connection is known as an "inverse square" relationship. Newton derived this relationship from Kepler's assertion that the planets follow elliptical orbits. To understand this, consider the light radiating from the surface of the sun. The light has some intensity at the surface of the sun. As the light travels away from the sun, its intensity diminishes. The intensity of the light at any distance away from the sun equals the strength of the source divided by the surface area of a sphere surrounding the sun at that radius.

As the distance away from the sun (r) doubles, the area of the sphere surrounding the sun quadruples. Thus, the intensity of the sun's light depends inversely on the square of the distance away from the sun. Newton envisioned the gravitational force as radiating equally in all directions from a central body, just as sunlight in the previous example. Newton recognized that his gravitational model must take the form of an inverse square relationship. Such a model predicts that the orbits of objects around a central body will be conic sections, and years of astronomical observations have borne this out. Although this idea is most commonly attributed to Isaac Newton, the English mathematician Robert Hooke claimed that he originated the idea of the inverse square relationship. Nonetheless, Newton eventually published his theory of gravitation and became famous as a result.

The relationship that Newton came up with looks like this:

where F is the force of gravity (in units now referred to as newtons), m_{1} and m_{2} are the masses of the two objects in kilograms (for example, the sun and Earth), r is the distance separating the centers of mass of the objects and G is the "gravitational constant." The equation shows that the force of gravity is directly proportional to the product of the two masses, but inversely proportional to the square of the distance between the centers of those two masses. To understand the formula, keep in mind that the force of gravity *decreases*, as distance *increases* (an inverse relationship). The distance (r) is squared due to the relationship between the increasing distance and the growth of the area over which the force is exerted (just as rays of light spread out as they get farther from the sun). Finally, since both masses exert some force due to gravity, it is the product of their masses – not just a single mass – that makes a difference.

This relationship has come to be known as Newton's Law of Universal Gravitation. It is "universal" because all objects in the universe are attracted to all other objects in the universe according to this relationship. Two people sitting across a room from each other are actually attracted gravitationally. As we know from everyday experience, human-sized objects don't crash into each other as a result of this force, but it does exist even if it is very small. Although Newton correctly identified this relationship between force, mass, and distance, he was able only to estimate the value of the gravitational constant between these quantities. The world would have to wait more than a century for an experimental measurement of the constant of proportionality: G.

## Comprehension Checkpoint

### The farther away two objects are from each other, the stronger the gravitational pull between them.

## Measuring the mass of Earth: The Cavendish experiment

In 1797 and 1798 Henry Cavendish set out to confirm Newton's theory and to determine the constant of proportionality in Newton's Law of Universal Gravitation. His ingenious experiment, based on the work of John Michell, was successful on both fronts. To accomplish this, Cavendish created a "torsion balance," which consisted of two masses at either end of a bar that was suspended from the ceiling by a thin wire (see Figure 3).

Attached to the wire was a mirror, off of which a beam of light was reflected. Cavendish brought a third mass close to one of the masses on the torsion balance. As the third mass attracted one of the ends of the torsion balance, the entire apparatus, including the mirror, rotated slightly and the beam of light was deflected. Through careful measurement of the angular deflection of the beam of light, Cavendish was able to determine the extent to which the known mass was attracted to the introduced mass. Not only did Cavendish confirm Newton's theory, but also he determined the value of the gravitational constant to an accuracy of about 1 percent.

Cavendish cleverly referred to his research as "Measuring the Mass of Earth." Since he had determined the value of G, he could do some simple calculations to determine the mass of Earth. By Newton's Second Law, the force between an object and Earth equals the product of the acceleration (g) and the mass of the object (m):

Galileo had determined the acceleration of all objects near the surface of Earth in the early 1600s as g = 9.8 m/s^{2}.

Therefore, setting this equation equal to Newton's Law of Universal Gravitation described above, Cavendish found:

where m is the mass of the object, m_{E} is the mass of Earth, and r_{E} is the radius of Earth. Solving for the mass of Earth yields the following result:

Cavendish had determined the mass of Earth with great accuracy. We can also use this relationship to calculate the force of attraction between two people across a room. To do this, we simply need to use Newton's Law of Universal Gravitation with Cavendish's gravitational constant. Assume the two people have masses of 75 and 100 kilograms, respectively, and that they are 5 meters apart. The force of gravitation between them is:

Although it is small, there is still a force!

## Conclusion

Newton's Law of Universal Gravitation grew in importance as scientists realized its utility in predicting the orbits of the planets and other bodies in space. In 1705, Sir Edmund Halley, after studying comets in great detail, predicted correctly that the famous comet of 1682 would return 76 years later, in December of 1758. Halley had used Newton's Law to predict the behavior of the comet orbiting the sun. With the advent of Cavendish's accurate value for the gravitational constant, scientists were able to use Newton's law for even more purposes. In 1845, John Couch Adams and Urbain Le Verrier predicted the existence of a new, yet unseen, planet based on small discrepancies between predictions for and observations of the position of Uranus. In 1846, the German astronomer Johann Galle confirmed their predictions and officially discovered the new planet, Neptune.

While Newton's Law of Universal Gravitation remains very useful today, Albert Einstein demonstrated in 1915 that the law was only approximately correct, and that it fails to work when gravitation becomes extremely strong. Nonetheless, Newton's gravitational constant plays an important role in Einstein's alternative to Newton's Law, the Theory of General Relativity. The value of G has been the subject of great debate even in recent years, and scientists are still struggling to determine a very accurate value for this most elusive of fundamental physical constants.