# The Cavendish Experiment and the Quest to Determine the Gravitational Constant

# The Cavendish Experiment and the Quest to Determine the Gravitational Constant

*Overview*

The determination of a precise value for the gravitational constant (G) has proved a frustrating, but fruitful, exercise for scientists since the constant was first described by English physicist Sir Isaac Newton (1642-1727) in his influential 1687 work, *Philosophiae Naturalis Principia Mathematica* ("Mathematical Principles of Natural Philosophy"). In many ways as enigmatic as mathematicians' search for a proof to Fermat's last theorem (proved only in the last decade of the twentieth century), the determination of an exact value of the gravitational constant has eluded physicists for more than 300 years. The quest for "G" provides a continuing challenge to the experimental ingenuity of physicists and often spurs new generations of physicists to recapture the inventiveness and delicacy of measurement first embodied in the elegant experiments conducted by English physicist Henry Cavendish (1731-1810).

*Background*

In *Principia* Newton put forth a grand synthesis of theory regarding the physical universe. According to Newtonian theory, the universe was bound together by the mutual gravitational attraction of its constituent particles. With regard to gravity, Newton formulated that the gravitational attraction between two bodies was directly proportional to the masses, and inversely proportional to the square of the distance between the masses. Accordingly, if one doubled a mass one would double the gravitational attraction; if one doubled the distance between masses one would reduce the gravitational attraction to one-fourth of its former value. What was missing from Newton's formulation, however, was a value for a gravitational constant that would accurately translate these fundamental qualitative relationships into experimentally verifiable numbers.

The gravitational constant essentially measures the strength of gravity. Newton's law of gravitation is mathematically expressed as F = (G)(m_{1})m_{2})/r^{2} where F is the gravitational force between two bodies, m_{1} and m_{2}) are the two masses, and r is the radius or distance between the two masses. The gravitational constant has been experimentally calculated by modern methods to approximately 6.6726 × 10^{-11} m^{3} kg^{-1} s^{-2}. Prior to the determination of the Cavendish value for the gravitational constant (usually referred to as the "Cavendish constant"), however, scientists could only derive a value for the product of the gravitational constant and the mass of the Earth—there was no practical way to separate them. Essentially, although it was possible to calculate the product of the gravitational constant and the mass (GM) of the Earth, there seemed no easy way to determine the value of G and M separately.

The units of the gravitational constant (m^{3} kg^{-1} s^{-2}) dictate that, by itself, the gravitational constant is not a tangible physical entity. Accordingly, the determination of G requires experimental ingenuity. Prior to Cavendish, English
astronomer Nevil Maskelyne (1732-1811) attempted to determine the gravitational constant using a plumb bob in an experiment inspired by Newton's *Principia*, an estimated mass of Mount Shiehallion. Maskelyne's experiments failed, however, to yield a practical value for the gravitational constant.

Although Cavendish was wealthy, he devoted himself to scientific works and study. Considered eccentric by his colleagues, he worked quietly and apart from the academic mainstream. Cavendish attended Cambridge University but did not complete his studies. He was so reclusive that he kept many of his scientific discoveries to himself. Much of Cavendish's work with electricity, for example, remained unpublished for almost another century, when his notes were compiled and published by Scottish physicist James Clerk Maxwell (1831-1879).

In 1798 Cavendish performed an ingenious experiment that led to the determination of the gravitational constant. Cavendish used a carefully constructed experiment that utilized a torsion balance to measure the very small gravitational attraction between two masses suspended by a thin fiber support. (Cavendish actually measured the restoring torque of the fiber support.) Cavendish's experimental methodology and device design was not novel. A similar apparatus had been designed by French physicist Charles Coulomb (1736-1806) and others for electrical measurements and calibrations. Cavendish's use of the torsion balance to measure the gravitational constant of the Earth, however, was a triumph of empirical skill.

Cavendish balanced his apparatus by placing balls of identical mass at both ends of a crossbar suspended by a thin wire. By using lead balls of known mass, Cavendish was able to account for both the masses in the Newtonian calculation and thereby allow a determination of the gravitational constant. The Cavendish experiment worked because not much force was required to twist the wire suspending the balance. In addition, Cavendish brought relatively large masses close to the smaller weights—actually on symmetrically opposite sides of the weights—so as to double the actual force and make the small effects more readily observable. Over time, due to the mutual gravitational attraction of the weights, the smaller balls moved toward the larger masses. The smaller balls moved because of their smaller mass and inertia (resistance to movement). Cavendish was able to measure the force of the gravitational attraction as a function of the time it took to produce and given the twist in the suspending wire. The value of the gravitational constant determined by this method was not precise by modern standards but was an exceptional value for the eighteenth century given the small forces being measured. Because all objects exert a gravitational "pull," precision in Cavendish-type experiments is often hampered by a number of factors, including underlying geology or factors as subtle as movements of furniture or objects near the experiment.

Because of his age (Cavendish was nearly 70 years old) the tedious and intricate experiment was taxing on Cavendish, and the determination of the gravitational constant proved to be his last great experiment. Decades after his death, a portion of the substantial Cavendish estate was used to fund the endowment of the great Cavendish Laboratory at Cambridge University, where so much of the fundamental work on the detailed structure of the atom and of the small scale forces underlying modern physics was to take place.

*Impact*

The effort to determine a precise and accurate value for the gravitational constant was a natural consequence to the fact that, by the end of the eighteenth century, Newtonian physics dominated Western scientific and intellectual thought. Cavendish's experiment essentially allowed scientists
to "weigh" Earth (more properly to determine its mass and density). Once the value of the gravitational constant was determined, the mass of Earth could be calculated from the experimentally determined gravitational acceleration of 9.8 m/s^{2}.

An accurate value for the gravitational constant was essential to determine the mass of the Sun, Moon, planets, and other astronomical bodies. As a result of the Cavendish determination, revised applications of Kepler's third law allowed for subsequent refined estimates of the mass of the Sun and planets.

Eighteenth-century astronomical estimations of the radius and of the distances between the planets were accurate enough to make reasonably good estimates of the size and, hence, the volume of the planets. One unexpected outcome of the eighteenth-century quest for G was the ability to more accurately measure the density of Earth. Maskelyne's data established a greater than expected density of Earth—and inferred that Earth must have a core much denser than its crust. Cavendish's values for G indicated an even greater density for Earth (approximately 5500 kg / m^{3}) than Maskelyne had proposed.

The gravitational constant is a fundamental quantity of the universe. It was the first great universal constant of physics (the others subsequently being the speed of light and Planck's constant), and modern physicists still argue its importance and relationship to cosmology. Regardless, almost all the major theoretical frameworks dictate that the value for the gravitational constant is in some regard related to the large-scale structure of the cosmos. Ironically, despite centuries of research, the gravitational constant is—by a substantial margin—the least understood, most difficult to determine, and least precisely known fundamental constant value.

Although profoundly influential and powerful on the cosmic scale, the force of gravity is weak in terms of human dimensions. Accordingly, the masses must be very large before gravitational effects can be easily measured. Even using modern methods, different laboratories often report significantly different values for G.

The Cavendish experiment was, therefore, a milestone in the advancement of scientific empiricism. In fact, accuracy of the Cavendish determination remained unimproved for almost another century until Charles Vernon Boys (1855-1944) used the Cavendish balance to make a more accurate determination of the gravitational constant. More importantly, the Cavendish experiment proved that scientists could construct experiments that were able to measure very small forces. Cavendish's work spurred analysis of the fundamental force of electromagnetism (a fundamental force far stronger than gravity) and gave confidence to the scientific community that Newton's laws were not only valid but testable on exceedingly small scales.

The ability to measure the mass and movements of the planets more critically became increasingly important throughout the eighteenth century in philosophical and theological circles. The precise delineation of planetary movements under the influence of gravity was an issue often raised by all sides in the increasingly heated debate about whether there was any role for God in a mechanical, clockwork cosmos.

In modern physics, the speed of light, Planck's constant, and the gravitational constant are among the most important of fundamental constants. According to relativity theory, G is related to the amount of space-time curvature caused by a given mass. Modern concepts of gravity and of the ramifications of the value of the gravitational constant are subject to seemingly constant revision as scientists aim to extend the linkage between the gravitational constant and other fundamental constants.

**K. LEE LERNER**

*Further Reading*

Bronowski, J. *The Ascent of Man*. Boston: Little, Brown, 1973.

Hoyle, F. *Astronomy*. Crescent Books, 1962.

Jungnickel, C. and R. McCormmach. "Cavendish."* Memoirs of the American Philosophical Society Series. *American Philosophical Society, 1996.

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