Cavendish experiment

# Cavendish experiment

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The Cavendish experiment, performed in 1797–98 by British scientist Henry Cavendish
Henry Cavendish
Henry Cavendish FRS was a British scientist noted for his discovery of hydrogen or what he called "inflammable air". He described the density of inflammable air, which formed water on combustion, in a 1766 paper "On Factitious Airs". Antoine Lavoisier later reproduced Cavendish's experiment and...

was the first experiment to measure the force of gravity
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

between masses in the laboratory, and the first to yield accurate values for the gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

. Because of the unit conventions then in use, the gravitational constant does not appear explicitly in Cavendish's work. Instead, the result was originally expressed as the specific gravity
Specific gravity
Specific gravity is the ratio of the density of a substance to the density of a reference substance. Apparent specific gravity is the ratio of the weight of a volume of the substance to the weight of an equal volume of the reference substance. The reference substance is nearly always water for...

of the Earth, or equivalently the mass of the Earth; and were the first accurate values for these geophysical constants. The experiment was devised sometime before 1783 by geologist John Michell
John Michell
John Michell was an English natural philosopher and geologist whose work spanned a wide range of subjects from astronomy to geology, optics, and gravitation. He was both a theorist and an experimenter....

, who constructed a torsion balance apparatus for it. However, Michell died in 1793 without completing the work, and after his death the apparatus passed to Francis John Hyde Wollaston and then to Henry Cavendish, who rebuilt the apparatus but kept close to Michell's original plan. Cavendish then carried out a series of measurements with the equipment, and reported his results in the Philosophical Transactions of the Royal Society
Philosophical Transactions of the Royal Society
The Philosophical Transactions of the Royal Society is a scientific journal published by the Royal Society of London. It was established in 1665, making it the first journal in the world exclusively devoted to science, and it has remained in continuous publication ever since, making it the world's...

in 1798.

## The experiment

The apparatus constructed by Cavendish was a torsion balance made of a six-foot (1.8 m) wooden rod suspended from a wire, with a 2 inches (5 cm) diameter 1.61 pound (0.7302837157 kg) lead
Lead is a main-group element in the carbon group with the symbol Pb and atomic number 82. Lead is a soft, malleable poor metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed...

sphere attached to each end. Two 12 inches (30.5 cm) 348 pounds (157.9 kg) lead balls were located near the smaller balls, about 9 inches (22.9 cm) away, and held in place with a separate suspension system. The experiment measured the faint gravitational attraction between the small balls and the larger ones.

The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod, and knowing the twisting force (torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the earth to be calculated, using Newton's law of gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

.

Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water (due to a simple arithmetic error, found in 1821 by F. Baily, the erroneous value 5.48 ± 0.038 appears in his paper).

To find the wire's torsion coefficient, the torque exerted by the wire for a given angle of twist, Cavendish timed the natural oscillation period of the balance rod as it rotated slowly clockwise and counterclockwise against the twisting of the wire. The period was about 20 minutes. The torsion coefficient could be calculated from this and the mass and dimensions of the balance. Actually, the rod was never at rest; Cavendish had to measure the deflection angle of the rod while it was oscillating.

Cavendish's equipment was remarkably sensitive for its time. The force involved in twisting the torsion balance was very small, 1.74 x 10–7 N, about 1/50,000,000 of the weight of the small balls or roughly the weight of a large grain of sand. To prevent air currents and temperature changes from interfering with the measurements, Cavendish placed the entire apparatus in a wooden box about 2 foot (0.6096 m) thick, 10 feet (3 m) tall, and 10 feet (3 m) wide, all in a closed shed on his estate. Through two holes in the walls of the shed, Cavendish used telescopes to observe the movement of the torsion balance's horizontal rod. The motion of the rod was only about 0.16 inch (0.4064 cm). Cavendish was able to measure this small deflection to an accuracy of better than one hundredth of an inch using vernier scale
Vernier scale
A vernier scale is an additional scale which allows a distance or angle measurement to be read more precisely than directly reading a uniformly-divided straight or circular measurement scale...

s on the ends of the rod.

Cavendish's experiment was repeated by Reich (1838), Baily (1843), Cornu & Baille (1878), and many others. Its accuracy was not exceeded for 97 years, until C. V. Boys' 1895 experiment. In time, Michell's torsion balance became the dominant technique for measuring the gravitational constant (G)
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

, and most contemporary measurements still use variations of it. This is why Cavendish's experiment became the Cavendish experiment.

## Did Cavendish determine G?

The formulation of Newtonian gravity
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

in terms of a gravitational constant did not become standard until long after Cavendish's time. Indeed, one of the first references to G is in 1873, 75 years after Cavendish's work. Cavendish expressed his result in terms of the density of the Earth, and he referred to his experiment in correspondence as 'weighing the world'. Later authors reformulated his results in modern terms. thus:
After converting to SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, Cavendish's value for the Earth's density, 5.448 g cm−3, gives
,

which differs by only 1% from the currently accepted value: 6.67428 × 10−11 m3 kg−1 s−2.

For this reason, historians of science have argued that Cavendish did not measure the gravitational constant.

Physicists, however, often use units where the gravitational constant takes a different form. The Gaussian gravitational constant
Gaussian gravitational constant
The Gaussian gravitational constant is an astronomical constant first proposed by German polymath Carl Friedrich Gauss in his 1809 work Theoria motus corporum coelestium in sectionibus conicis solem ambientum , although he had already used the concept to great success in predicting the...

used in space dynamics is a defined constant, and the Cavendish experiment can be considered as a measurement of the astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

.
In Cavendish's time, physicists used the same units for mass and weight, in effect taking as a standard acceleration. Then, since was known, played the role of an inverse gravitational constant. The density of the Earth was hence a much sought-after quantity at the time, and there had been earlier attempts to measure it, such as the Schiehallion experiment
Schiehallion experiment
The Schiehallion experiment was an 18th-century experiment to determine the mean density of the Earth. Funded by a grant from the Royal Society, it was conducted in the summer of 1774 around the Scottish mountain of Schiehallion, Perthshire. The experiment involved measuring the tiny deflection of...

in 1774.

For these reasons, physicists generally do credit Cavendish with the first measurement of the gravitational constant.

## Derivation of G and the Earth's mass

For the definitions of terms, see the drawing below and the table at the end of this section.

The following is not the method Cavendish used, but shows how modern physicists would use his results. From Hooke's law, the torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

on the torsion wire is proportional to the deflection angle of the balance. The torque is where is the torsion coefficient of the wire. However, the torque can also be written as a product of the attractive forces between the balls and the distance to the suspension wire. Since there are two pairs of balls, each experiencing force F at a distance L / 2 from the axis of the balance, the torque is LF. Equating the two formulas for torque gives the following:

For F, Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's law of universal gravitation is used to express the attractive force between the large and small balls:

Substituting F into the first equation above gives

To find the torsion coefficient () of the wire, Cavendish measured the natural resonant
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

oscillation period T of the torsion balance:

Assuming the mass of the torsion beam itself is negligible, the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

of the balance is just due to the small balls:
,

and so:

Solving this for , substituting into (1), and rearranging for G, the result is:

Once G has been found, the attraction of an object at the Earth's surface to the Earth itself can be used to calculate the Earth's mass and density:

 Definition of terms Deflection of torsion balance beam from its rest position Gravitational force between masses M and m Gravitational constant Mass of small lead ball Mass of large lead ball Distance between centers of large and small balls when balance is deflected Length of torsion balance beam between centers of small balls Torsion coefficient of suspending wire Moment of inertia of torsion balance beam Period of oscillation of torsion balance Acceleration of gravity at the surface of the Earth Mass of the Earth Radius of the Earth Density of the Earth