Cavendish experiment

The Cavendish experiment, done in 1797 – 1798 by Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory, [ [http://books.google.com/books?id=ZrloHemOmUEC&pg=PA355 Boys 1894] p.355] and the first to yield accurate values for the gravitational constant and the mass of the Earth. [ [http://books.google.com/books?id=DgTALFa3sa4C&pg=PA385 Encyclopaedia Britannica 1910] p.385 'The aim [of experiments like Cavendish's] may be regarded either as the determination of the mass of the Earth,...conveniently expressed...as its "mean density", or as the determination of the "gravitation constant", G'. Cavendish's experiment is generally described today as a measurement of G (Clotfelter 1987 p.210).] [Many sources state erroneously that this was the first measurement of G (or the Earth's density), such as Citation|last=Feynman|first=Richard P.|title=Lectures on Physics, Vol.1|date=1963|publisher=Addison-Wesley|pages=p.6–7|url=http://books.google.com/books?id=k6MQrphL-NIC&pg=PA28|id=ISBN 0201021161|format=dead link|date=June 2008 – ^{[http://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AFeynman+intitle%3ALectures+on+Physics%2C+Vol.1&as_publication=&as_ylo=&as_yhi=&btnG=Search Scholar search]} . There were previous measurements, chiefly Bouguer (1740) and Maskelyne (1774), but they were very inaccurate ( [http://books.google.com/books?id=dg0RAAAAIAAJ Poynting 1894] )( [http://books.google.com/books?id=DgTALFa3sa4C&pg=PA385 Encyclopedia Britannica 1910] ).] However, these were derived by others from Cavendish's result, which was a value for the Earth's density. [Clotfelter 1987, p.210] The experiment was devised sometime before 1783 [ [http://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA336&sig=--1AlZ9rl_0AEL7h73LZvtK01S4 McCormmach & Jungnickel 1996] , p.336: A 1783 letter from Cavendish to Michell contains '...the earliest mention of weighing the world'. Not clear whether 'earliest mention' refers to Cavendish or Michell.] by John Michell, [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA59 Cavendish 1798] , p.59 Cavendish gives full credit to Michell for devising the experiment] 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" in 1798. [Cavendish, H. 'Experiments to determine the Density of the Earth', "Philosophical Transactions of the Royal Society of London", (part II) 88 p.469-526 (21 June 1798), reprinted in [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA59 Cavendish 1798] ]

The experiment

The apparatus constructed by Cavendish was a torsion balance made of a six-foot wooden rod suspended from a wire, with a convert|2|in|mm|sing=on diameter 1.61 pound lead sphere attached to each end. Two 12 inch 348 pound lead balls were located near the smaller balls, about 9 inches away, and held in place with a separate suspension system. [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA59 Cavendish 1798] , p.59] 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) 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.

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). [ [http://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA45 Poynting 1894] , p.45]

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 7 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. [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA64 Cavendish 1798] , p.64]

Cavendish's equipment was remarkably sensitive for its time. [ [http://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA45 Poynting 1894] , p.45] The force involved in twisting the torsion balance was very small, 1.47 x 10^–7 N, [ [http://books.google.com/books?id=ZrloHemOmUEC&pg=PA357 Boys 1894] p.357] about 1/50,000,000 of the weight of the small balls [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA60 Cavendish 1798] p. 60] or roughly the weight of a large grain of sand. [A 2 mm sand grain weighs ~13 mg. cite web|last=Theodoris|first=Marina|year=2003|url=http://hypertextbook.com/facts/2003/MarinaTheodoris.shtml|title=Mass of a Grain of Sand|work=The Physics Factbook] To prevent air currents and temperature changes from interfering with the measurements, Cavendish placed the entire apparatus in a wooden box about convert|2|ft|m thick, convert|10|ft|m tall, and convert|10|ft|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. [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA99 Cavendish 1798] , p. 99, Result table, (scale graduations = 1/20 inch) The total deflection shown in most trials was twice this since he compared the deflection with large balls on opposite sides of the balance beam.] Cavendish was able to measure this small deflection to an accuracy of better than one hundredth of an inch using vernier scales on the ends of the rod. [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA63 Cavendish 1798] , p.63]

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"), and most contemporary measurements still use variations of it. This is why Cavendish's experiment became "the" Cavendish experiment. [ [http://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA341&sig=--1AlZ9rl_0AEL7h73LZvtK01S4 McCormmach & Jungnickel 1996] , p.341]

Did Cavendish determine "G"?

It is not unusual to find books that state erroneously that Cavendish's purpose was determining the gravitational constant ("G"), [Citation|last=Halliday|first=David|last2=Resnick|first2=Robert|title=Fundamentals of Physics|date=1993|publisher=John Wiley & Sons|id=ISBN 0471147311|pages=p.418|url=http://books.google.com/books?id=-AjnmJHPiKMC&pg=PA418 'The apparatus used in 1798 by Henry Cavendish to measure the gravitational constant'] [Citation|last=Feynman|first=Richard P.|title=Lectures on Physics, Vol.1|date=1963|publisher=Addison-Wesley|pages=p.6–7|url=http://books.google.com/books?id=k6MQrphL-NIC&pg=PA28|id=ISBN 0201021161|format=dead link|date=June 2008 – ^{[http://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AFeynman+intitle%3ALectures+on+Physics%2C+Vol.1&as_publication=&as_ylo=&as_yhi=&btnG=Search Scholar search]} 'Cavendish claimed he was weighing the Earth, but what he was measuring was the coefficient G...'] [Citation|last=Feynman|first=Richard P.|title=The Character of Physical Law|date=1967|publisher=MIT Press|pages=p.28|url=http://books.google.com/books?id=k6MQrphL-NIC&pg=PA28|id=ISBN 0262560038|format=dead link|date=June 2008 – ^{[http://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AFeynman+intitle%3AThe+Character+of+Physical+Law&as_publication=&as_ylo=&as_yhi=&btnG=Search Scholar search]} 'Cavendish was able to measure the force, the two masses, and the distance, and thus determine the gravitational constant G'] [Citation|title=Cavendish Experiment, Harvard Lecture Demonstrations, Harvard Univ|url=http://www.fas.harvard.edu/~scdiroff/lds/NewtonianMechanics/CavendishExperiment/CavendishExperiment.html|accessdate=2007-08-26. ' [the torsion balance was] ...modified by Cavendish to measure G.'] [Citation|last=Shectman|first=Jonathan|title=Groundbreaking Experiments, Inventions, and Discoveries of the 18th Century|date=2003|publisher=Greenwood|pages=p.xlvii|url=http://books.google.com/books?id=SsbChdIiflsC&pg=PAxlvii|id=ISBN 0313320152 'Cavendish calculates the gravitational constant, which in turn gives him the mass of the earth...'] and this mistake has been pointed out by several authors. [Clotfelter 1987] [ [http://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA336&sig=--1AlZ9rl_0AEL7h73LZvtK01S4 McCormmach & Jungnickel 1996] , p.337] [ [http://www.public.iastate.edu/~lhodges/Michell.htm Hodges 1999] ] [Lally 1999] In actuality, Cavendish's only goal was to measure the density of the Earth; he called it 'weighing the world'. The method Cavendish used to calculate the Earth's density consists in measuring the force on a small ball caused by a large ball of known mass, and comparing it with the force on the small ball caused by the Earth, so the Earth can be calculated to be "N" times more massive than the large ball without the need to obtain a numeric value for "G". [ [http://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA336&sig=--1AlZ9rl_0AEL7h73LZvtK01S4 McCormmach & Jungnickel 1996] , p.337] The gravitational constant does not appear in Cavendish's paper, and there is no indication that he regarded it as a goal of his experiment. One of the first references to "G" is in 1873, 75 years after Cavendish's work. [Cornu, A. and Baille, J. B. (1873), Mutual determination of the constant of attraction and the mean density of the earth, "C. R. Acad. Sci.", Paris Vol. 76, 954-958.]

In Cavendish's time, "G" did not have the importance among scientists that it has today; it was simply a proportionality constant in Newton's law. [ [http://books.google.com/books?id=ZrloHemOmUEC&pg=PA353 Boys 1894] , p.330 In this lecture before the Royal Society, Boys introduces G and argues for its acceptance] The purpose of measuring the force of gravity was instead to determine the Earth's density. This was a much-desired quantity in 18th-century astronomy, since once the Earth's density was known, the densities of the Moon, Sun, and the other planets could be found from it. [ [http://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA4 Poynting 1894] , p.4]

A further complication is that up through the mid-nineteenth century, scientists did not use a specific unit of measurement for force. [ [http://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA336&sig=--1AlZ9rl_0AEL7h73LZvtK01S4 McCormmach & Jungnickel 1996] , p.337] This unnecessarily tied "G" to the mass of the Earth, as opposed to "G" being recognized as a universal constant. However, even though Cavendish did not report a value for "G", the results of his experiment allowed it to be determined. During the late 1800s, as scientists began to recognize "G" as a fundamental constant of nature, they calculated it from Cavendish's accurate results, thus: [ [http://books.google.com/books?id=O58mAAAAMAAJ&pg=PA1 MacKenzie 1900] , p.vi]

:$G\; =\; gfrac\{R\_\{earth\}^2\}\{M\_\{earth\; =\; frac\{3g\}\{4pi\; R\_\{earth\}\; ho\_\{earth,$

After converting to SI units, Cavendish's value for the Earth's density, 5.45 g cm⁻³, gives "G" = 6.74 × 10⁻¹¹ m³ kg⁻¹ sec⁻², which is within 1% of the currently accepted value.

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. [ [http://www.fas.harvard.edu/~scdiroff/lds/NewtonianMechanics/CavendishExperiment/CavendishExperiment.html Cavendish Experiment, Harvard Lecture Demonstrations, Harvard Univ.] ] [ [http://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA41 Poynting 1894] , p.41] [Clotfelter 1987 p.212 explains Cavendish's original method of calculation] From Hooke's law, the torque on the torsion wire is proportional to the deflection "$heta$" of the balance. The torque is "$kappa\; heta$" where "$kappa$" is the torsion coefficient. However, the torque can also be written as a product of the attractive forces and the distance to the 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:

:$kappa\; heta\; =\; LF\; ,$

For "F", Newton's law of universal gravitation is used to express the attractive force between the large and small balls:

:$F\; =\; frac\{G\; m\; M\}\{r^2\},$

Substituting F into the first equation above gives

:$kappa\; heta\; =\; Lfrac\{GmM\}\{r^2\}\; qquadqquadqquad(1),$

To find the torsion coefficient ($kappa,$) of the wire, Cavendish measured the natural resonant oscillation period "T" of the torsion balance:

:$T\; =\; 2pisqrt\{I/kappa\}$

Assuming the mass of the torsion beam itself is negligible, the moment of inertia of the balance is just due to the small balls:

:$I\; =\; m(L/2)^2\; +\; m(L/2)^2\; =\; 2m(L/2)^2\; =\; mL^2/2,$,

and so:

:$T\; =\; 2pisqrt\{frac\{mL^2\}\{2kappa,$

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

:$G\; =\; frac\{2\; pi^2\; L\; r^2\}\{M\; T^2\}\; heta,$

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:

:$mg\; =\; frac\{GmM\_\{earth\{R\_\{earth\}^2\},$

:$M\_\{earth\}\; =\; frac\{gR\_\{earth\}^2\}\{G\},$

:$ho\_\{earth\}\; =\; frac\{M\_\{earth\{4\; pi\; R\_\{earth\}^3/3\}\; =\; frac\{3g\}\{4\; pi\; R\_\{earth\}\; G\},$

References

*
* Online copy of Cavendish's 1798 paper, and other early measurements of gravitational constant.
* Establishes that Cavendish didn't determine G.
*
*cite encyclopedia | title=Gravitation Constant and Mean Density of the Earth | encyclopedia=Encyclopaedia Britannica, 11th Ed. | volume=12 | publisher=The Encyclopaedia Britannica Co. | date=1910 | page=385-389 | url=http://books.google.com/books?id=DgTALFa3sa4C&pg=PA385
* Discusses Michell's contributions, and whether Cavendish determined G.
*
* cite book
last = McCormmach
first = Russell
coauthors = Jungnickel, Christa
title = Cavendish
location = Philadelphia, Pennsylvania
publisher = American Philosophical Society
date = 1996
pages =
url = http://books.google.com/books?id=EUoLAAAAIAAJ
isbn = 0-87169-220-1
* Review of gravity measurements since 1740.

Notes

External links

* [http://www.sas.org/tcs/weeklyIssues_2005/2005-07-01/feature1/index.html Sideways Gravity in the Basement, "The Citizen Scientist", July 1, 2005] , retrieved Aug. 9, 2007. Homebrew Cavendish experiment, showing calculation of results and precautions necessary to eliminate wind and electrostatic errors.
* [http://www.physicscentral.com/action/2001/bigg-measure.html Measuring Big G, Physics Central] , retrieved Aug. 9, 2007. Recent experiment at Univ. of Washington to measure the gravitational constant using variation of Cavendish method.
* [http://www.npl.washington.edu/eotwash/experiments/bigG/bigG.html The Controversy over Newton's Gravitational Constant, Eot-Wash Group, Univ. of Washington] , retrieved Aug. 9, 2007. Discusses current state of measurements of G.
* [http://www.scienceandsociety.co.uk/results.asp?
] , retrieved Aug. 28, 2007, at Science Museum, London.
* [http://www.juliantrubin.com/bigten/cavendishg.html Weighing the Earth] - background and experiment