Newton’s law of universal gravitation, first having appeared in the Philosophiæ Naturalis Principia Mathematica in July 1687, states that every particle in the universe attracts another with a force proportional to the product of their masses, and inversely so to the distance between their center’s squared. The proportionality constant, which connects both parts of the equation, and will eventually also connect the geometry of space-time and the stress-energy tensor in Einstein’s field equations, is an empirical physical constant which takes on the accepted value of 6.67259 X 10-11 [ m3 kg-1 s-2].
However, it only came to be measured to high accuracy, around a century after its first mention, by a scientist named Henry Cavendish and inspired by a renowned geologist, John Michell, although the initial motivations for the Cavendish experiment were not to calculate G but rather to determine Earth’s density.
The setup and conduct of the Cavendish experiment
Henry Cavendish used a torsion balance (developed by Charles Coulomb), a long rigid rod suspended in its center by a thin wire, to successfully model the first low scale model of gravitational interactions in a laboratory. To each end of that 2 feet-long rod was hanging small lead spheres close to which were brought much larger ones in an attempt to simulate gravitational force between both spheres and measure it by observing and quantifying the twist in the wire.
In order to accurately measure that angular displacement of the torsion pendulum, Cavendish’s set-up included a small mirror on which a beam of light was deflected. As such, any change in angle between the incident and reflected ray would indicate a small twist in the wire and thus a change in the distance r between the larger and smaller sphere and indicate the specific parameters of sphere positioning when the balance attained equilibrium.
Before even bringing the larger sphere into the vicinity of the torsion balance, the pendulum would naturally oscillate between a certain maximal and minimal amplitude over a defined period, which Cavendish both recorded. For, the more the rod rotated, the more the twisted wire was strained and would exert an opposite force to counter the change induced in the system and reattain a state of equilibrium.
This phenomenon enabled him to determine the torque exerted by the wire while covering a certain angle or, in other words, the torsion coefficient. This constant was important to take into account when deriving the force induced by the larger spheres as the balance was always subject to exterior noise.
Derivation of the universal gravitational constant
Cavendish then proceeded in bringing the large spheres to point where their influence on the natural oscillation of the torsion balance could be felt and measured. The movement of the balance would eventually be reversed for the small spheres would be attracted by the larger ones, and the balance would start moving towards them. The balance would eventually come to rest when the force of gravitational attraction induced by the large sphere and the opposing torsion force from the wire canceled out.
Having weighed the masses, having at hand Newton’s formulation of the law of universal gravitation as well as having formulated expression of the torque in the function of the determined the torsion coefficient for the chosen wire and the measured the angle of displacement, Cavendish was able to calculate Earth’s density and the value of G which he set at 6.75 X 10-11 [ N m2 kg-2].
Challenges and difficulties
In comparison to the perceptible and visible gravitational effect of the earth’s mass on smaller objects, the gravitational interaction between smaller, and more maneuverable objects, such as the two spheres becomes almost negligible and extremely difficult to observe and quantify. As such, an important challenge faced during the set-up and unfolding of the Cavendish experiment is that of canceling out the effects of Earth’s gravity on components of the experiment.
In addition, and as previously mentioned, airflow as well as large displacements of mass might account for a significant increase in noise and exacerbate the natural oscillation of the torsion balance. Thus, in order to mitigate any haphazard interferences with his experiment, Cavendish placed his set-up inside a wooden box, itself inside a closed shed, and observed the unfolding of his experiment from the outside. This in turn prevented excessive air currents and temperature changes.
To conclude, not only did the Cavendish experiment lead to the determination of the universal constant of gravitation. It also had the interesting repercussion of providing for first-hand evidence of the composition of Earth’s core, proving that it was made of metal or more specifically, an iron dense core.