Editing Cavendish experiment

{{Short description|Experiment measuring the force of gravity (1797–1798)}}
[[File:Cavendish_experiment_schematic.png|upright=1.5|thumb|Cavendish's diagram of his torsion pendulum, seen from above. The pendulum consists of two small spherical lead weights ''(h, h)'' hanging from a 6-foot horizontal wooden beam supported in the center by a fine torsion wire. The beam is protected from air currents inside a wooden box ''(A, A, A, A)''. The two large weights ''(W, W)'' attached to a separate suspension attract the small weights, causing the beam to rotate slightly. The rotation is read off of vernier scales ''(S)'' at either end of the rod. The large weights can be rotated to the other side of the torsion beam ''(w, w)'', causing the beam to rotate in the opposite direction.]]

The '''Cavendish experiment''', performed in 1797–1798 by English scientist [[Henry Cavendish]], was the first experiment to measure the force of gravity between [[mass]]es in the laboratory<ref>[https://books.google.com/books?id=ZrloHemOmUEC&pg=PA355 Boys 1894] p. 355</ref> and the first to yield accurate values for the [[gravitational constant]].{{sfn|Poynting|1911|p=385}}<ref>'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).</ref><ref>Many sources incorrectly state that this was the first measurement of '''''G''''' (or Earth's density); for instance: {{Cite book|last = Feynman|first = Richard P.|publisher = California Institute of Technology|isbn = 9780465025626|location = Pasadena, California|url = https://feynmanlectures.caltech.edu/I_07.html#Ch1-S1|series = The Feynman lectures on physics|volume = I|title = mainly mechanics, radiation and heat|publication-date = 2013|year = 1963|chapter-url = https://feynmanlectures.caltech.edu/I_07.html|chapter = 7. The Theory of Gravitation|at = 7–6 Cavendish’s experiment|accessdate = December 9, 2013}}
There were previous measurements, chiefly by Bouguer (1740) and Maskelyne (1774), but they were very inaccurate ([http://ebooks.library.ualberta.ca/local/meandensityofear00poynuoft Poynting 1894]){{harv|Poynting1911|p=386}}.
</ref> 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 [[relative density]] of [[Earth]],<ref>Clotfelter 1987, p. 210</ref> or equivalently the [[Earth mass|mass of Earth]]. His experiment gave the first accurate values for these [[geophysics|geophysical]] constants.

The experiment was devised sometime before 1783 by geologist [[John Michell]],<ref>[https://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA336 Jungnickel & McCormmach 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.</ref><ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA59 Cavendish 1798], p. 59 Cavendish gives full credit to Michell for devising the experiment</ref> who constructed a [[torsion balance]] apparatus for it. However, Michell died in 1793 without completing the work. After his death the apparatus passed to [[Francis Wollaston (philosopher)|Francis John Hyde Wollaston]] and then to Cavendish, who rebuilt it, 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.<ref name="Cavendish">{{cite journal
  | last = Cavendish
  | first = Henry
  | title = XXI Experiments to determine the Density of the Earth
  | journal = Philosophical Transactions of the Royal Society
  | volume = 88
  | issue = 
  | pages = 469-526
  | publisher = The Royal Society of London
  | location = London
  | date = 21 June 1798
  | language = 
  | url = https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1798.0022
  | jstor = 
  | issn = 0261-0523
  | doi = 10.1098/rstl.1798.0022
  | id = 
  | mr = 
  | zbl = 
  | jfm = 
  | access-date = 14 January 2025}}</ref>

==The experiment==
The apparatus consisted of a [[torsion balance]] made of a {{convert|6|ft|spell=in|adj=on}} wooden rod horizontally suspended from a wire, with two {{convert|2|in|0|adj=mid|-diameter}}, {{convert|1.61|lb|adj=on}} [[lead]] spheres, one attached to each end. Two massive {{convert|12|in|adj=on}}, {{convert|348|lb|adj=on}} lead balls, suspended separately, could be positioned away from or to either side of the smaller balls, {{convert|8.85|in}} away.<ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA59 Cavendish 1798], p. 59</ref> The experiment measured the faint gravitational attraction between the small and large balls, which deflected the torsion balance rod by about 0.16" (or only 0.03" with a stiffer suspending wire).
[[File:Cavendish Experiment.png|thumb|left|250px|Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated by a pulley from outside. Figure 1 of Cavendish's paper]]
[[File:CavendishSchematic111.jpg|thumb|left|250px|Detail showing torsion balance arm (''m''), large ball (''W''), small ball (''x''), and isolating box (''ABCDE'').]]

The two large balls could be positioned either away from or to either side of the torsion balance rod. Their mutual attraction to the small balls caused the arm to rotate, twisting the suspension wire. The arm rotated until 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 [[relative density]] of the Earth to be calculated, using [[Newton's law of universal gravitation|Newton's law of gravitation]].

Cavendish found that the Earth's density was {{val|5.448|0.033}} times that of water (although due to a simple [[arithmetic]] error, found in 1821 by [[Francis Baily]], the erroneous value {{val|5.480|0.038}} appears in his paper).<ref name="Poynting 1894">[https://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA45 Poynting 1894], p. 45</ref><ref>{{Cite EB1911 |wstitle=Cavendish, Henry |volume=5 |pages=580–581}}</ref> The current accepted value is 5.514 g/cm<sup>3</sup>.

To find the wire's [[Torsion spring#Torsion coefficient|torsion coefficient]], the torque exerted by the wire for a given angle of twist, Cavendish timed the natural [[Torsion spring#Torsional harmonic oscillators|oscillation period]] of the balance rod as it rotated slowly clockwise and counterclockwise against the twisting of the wire. For the first 3 experiments the period was about 15 minutes and for the next 14 experiments the period was half of that, about 7.5 minutes. The period changed because after the third experiment Cavendish put in a stiffer wire. 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.<ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA64 Cavendish 1798], p. 64</ref>

Cavendish's equipment was remarkably sensitive for its time.<ref name="Poynting 1894"/> The force involved in twisting the torsion balance was very small, {{val|1.74e-7|u=N}},<ref>[https://books.google.com/books?id=ZrloHemOmUEC&pg=PA357 Boys 1894] p. 357</ref> (the weight of only 0.0177 milligrams) or about {{frac|50,000,000}} of the weight of the small balls.<ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA60 Cavendish 1798] p. 60</ref> To prevent air currents and temperature changes from interfering with the measurements, Cavendish placed the entire apparatus in a mahogany box about 1.98 meters wide, 1.27 meters tall, and 14&nbsp;cm thick,[http://cavendish-deneyi.com/pdf/Cavendish-c%CC%A7izim-03.pdf] 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 key observable was of course the deflection of the torsion balance rod, which Cavendish measured to be about 0.16" (or only 0.03" for the stiffer wire used mostly).<ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA99 Cavendish 1798], p. 99, Result table, (scale graduations = {{frac|20}}&nbsp;in ≈ 1.3&nbsp;mm) 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.</ref> Cavendish was able to measure this small deflection to an accuracy of better than {{convert|0.01|in}} using [[vernier scale]]s on the ends of the rod.<ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA63 Cavendish 1798], p. 63</ref>
The accuracy of Cavendish's result was not exceeded until [[C. V. Boys]]' experiment in 1895. 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.<ref>[https://books.google.com/books?id=EUoLAAAAIAAJ&pg=PA341 Jungnickel & McCormmach 1996], p. 341</ref>

Cavendish's result provided additional evidence for a [[outer core|planetary core]] made of metal, an idea first proposed by [[Charles Hutton]] based on his analysis of the 1774 [[Schiehallion experiment]].<ref name="Danson_p153-154">{{Cite book|last=Danson|first=Edwin |title=Weighing the World |publisher=Oxford University Press|date=2006|pages=153–154|isbn=978-0-19-518169-2|url=https://books.google.com/books?id=UNH_Y7ERFeoC&pg=PA153}}</ref> Cavendish's result of 5.4 g·cm<sup>&minus;3</sup>, 23% bigger than Hutton's, is close to 80% of the density of liquid [[iron]], and 80% higher than the density of the Earth's outer [[Crust (geology)|crust]], suggesting the existence of a dense iron core.<ref>see e.g. Hrvoje Tkalčić, ''The Earth's Inner Core'', Cambridge University Press (2017), [https://books.google.com/books?id=wa7DDQAAQBAJ&pg=PA2 p. 2].</ref>

==Reformulation of Cavendish's result to ''G'' ==
The formulation of [[Newton's law of universal gravitation|Newtonian gravity]] 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.<ref>{{cite journal |last1=Cornu |first1=A. |last2=Baille |first2=J. B. |date=1873 |url=http://gallica.bnf.fr/ark:/12148/bpt6k3033b/f954.image |title=Détermination nouvelle de la constante de l'attraction et de la densité moyenne de la Terre |language=fr |trans-title=New Determination of the Constant of Attraction and the Average Density of Earth |journal=C. R. Acad. Sci. |location=Paris |volume=76 |pages=954–958 }}</ref>

Cavendish expressed his result in terms of the density of the Earth. He referred to his experiment in correspondence as 'weighing the world'. Later authors reformulated his results in modern terms.<ref>[https://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</ref><ref>[https://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA4 Poynting 1894], p. 4</ref><ref>[https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA1 MacKenzie 1900], p. vi</ref>
:<math>G = g\frac{R_\text{earth}^2}{M_\text{earth}} = \frac{3g}{4\pi R_\text{earth}\rho_\text{earth}}\,</math>
After converting to [[SI]] units, Cavendish's value for the Earth's density, 5.448&nbsp;g&nbsp;cm<sup>−3</sup>, gives
:''G'' = {{val|6.74e-11|u=m<sup>3</sup>&nbsp;kg<sup>–1</sup>&nbsp;s<sup>−2</sup>}},<ref>{{cite journal|first=Adam |last=Mann |title=The curious case of the gravitational constant |date=September 6, 2016 |journal=Proceedings of the National Academy of Sciences |volume=113 |issue=36 |pages=9949–9952 |doi=10.1073/pnas.1612597113 |doi-access=free |pmid=27601579 |pmc=5018785 }}</ref>
which differs by only 1% from the 2014 [[CODATA]] value of {{val|fmt=none|6.67408e-11|u=m<sup>3</sup>&nbsp;kg<sup>−1</sup>&nbsp;s<sup>−2</sup>}}.<ref>{{cite journal |first=Jennifer Lauren |last=Lee |title=Big G Redux: Solving the Mystery of a Perplexing Result |date=November 16, 2016 |journal=NIST |url=https://www.nist.gov/news-events/news/2016/11/big-g-redux-solving-mystery-perplexing-result}}</ref>
Today, physicists often use units where the gravitational constant takes a different form. The [[Gaussian gravitational constant]] used in space dynamics is a defined constant and the Cavendish experiment can be considered as a measurement of this constant.
In Cavendish's time, physicists used the same units for mass and weight, in effect taking ''g'' as a standard acceleration. Then, since ''R''{{Sub|earth}} was known, ''ρ''{{sub|earth}} 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]] in 1774.

==Derivation of ''G'' and the Earth's mass==
{{hatnote|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 describes how modern physicists would calculate the results from his experiment.<ref name=HarvLect>{{cite web|title=Cavendish Experiment, Harvard Lecture Demonstrations, Harvard Univ|url=http://sciencedemonstrations.fas.harvard.edu/icb/icb.do?keyword=k16940&pageid=icb.page80669&pageContentId=icb.pagecontent277503&state=maximize&view=view.do&viewParam_name=indepth.html#a_icb_pagecontent277503|accessdate=2013-12-30}}. '[the torsion balance was]...modified by Cavendish to measure ''G''.'</ref><ref>[https://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA41 Poynting 1894], p. 41</ref><ref>Clotfelter 1987 p. 212 explains Cavendish's original method of calculation.</ref> From [[Hooke's law]], the [[torque]] on the torsion wire is proportional to the deflection angle <math>\theta</math> of the balance. The torque is <math>\kappa\theta</math> where <math>\kappa</math> is the [[torsion coefficient]] of the wire. However, a torque in the opposite direction is also generated by the gravitational pull of the masses. It can be written as a product of the attractive force of a large ball on a small ball and the distance L/2 to the suspension wire. Since there are two balls, each experiencing force ''F'' at a distance {{sfrac|''L''|2}} from the axis of the balance, the torque due to gravitational force is ''LF''. At equilibrium (when the balance has been stabilized at an angle <math>\theta</math>), the total amount of torque must be zero as these two sources of torque balance out. Thus, we can equate their magnitudes given by the formulas above, which gives the following:

:<math>\kappa\theta\ = LF \,</math>

For ''F'', [[Isaac Newton|Newton]]'s [[law of universal gravitation]] is used to express the attractive force between a large and small ball:

[[File:Cavendish Torsion Balance Diagram.svg|thumb|220px|Simplified diagram of torsion balance]]

:<math>F = \frac{G m M}{r^2}\,</math>

Substituting ''F'' into the first equation above gives

:<math>\kappa\theta\ = L\frac{GmM}{r^2} \qquad\qquad\qquad(1)\,</math>

To find the torsion coefficient (<math>\kappa</math>) of the wire, Cavendish measured the natural [[Resonance|resonant]] [[Torsion spring#Torsional harmonic oscillators|oscillation period]] ''T'' of the torsion balance:

:<math>T = 2\pi\sqrt{\frac{I}{\kappa}}</math>

Assuming the mass of the torsion beam itself is negligible, the [[moment of inertia]] of the balance is just due to the small balls. Treating them as point masses, each at L/2 from the axis, gives:

:<math>I = m\left (\frac{L}{2}\right )^2 + m\left (\frac{L}{2}\right )^2 = 2m\left (\frac{L}{2}\right )^2 = \frac{mL^2}{2}\,</math>,

and so:

:<math>T = 2\pi\sqrt{\frac{mL^2}{2\kappa}}\,</math>

Solving this for <math>\kappa</math>, substituting into (1), and rearranging for ''G'', the result is:

:<math>G = \frac{2 \pi^2 L r^2 \theta}{M T^2} \,</math>.

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 mass|Earth's mass]] and density:

:<math>mg = \frac{GmM_{\rm earth}}{R_{\rm earth}^2}\,</math>

:<math>M_{\rm earth} = \frac{gR_{\rm earth}^2}{G}\,</math>

:<math>\rho_{\rm earth} = \frac{M_{\rm earth}}{\tfrac{4}{3} \pi R_{\rm earth}^3} = \frac{3g}{4 \pi R_{\rm earth} G}\,</math>

===Definitions of terms===

{| class="wikitable"
!Symbol ||Unit ||Definition
|-
|<math>\theta</math>||[[radian]]s||Deflection of torsion balance beam from its rest position
|-
|''F''||[[newton (unit)|N]]||Gravitational force between masses ''M'' and ''m''
|-
|''G''||m{{sup|3}} kg{{sup|−1}} s{{sup|−2}}||Gravitational constant
|-
|''m''||kg||Mass of small lead ball
|-
|''M''||kg||Mass of large lead ball
|-
|''r''||m||Distance between centers of large and small balls when balance is deflected
|-
|''L''||m||Length of torsion balance beam between centers of small balls
|-
|<math>\kappa</math>||N m rad{{sup|−1}}||Torsion coefficient of suspending wire
|-
|{{math|''I''}}||kg m{{sup|2}}||Moment of inertia of torsion balance beam
|-
|''T''||s||Period of oscillation of torsion balance
|-
|''g''||m s{{sup|−2}}||Acceleration of gravity at the surface of the Earth
|-
|''M''{{sub|earth}}||kg||Mass of the Earth
|-
|''R''{{sub|earth}}||m||Radius of the Earth
|-
|<math>\rho</math>{{sub|earth}}||kg m{{sup|−3}}||Density of the Earth
|}

==References==
{{Reflist|30em}}

===Sources===
{{refbegin}}
* {{cite journal | author=Boys, C. Vernon | title=On the Newtonian constant of gravitation | journal=Nature | year=1894 | volume=50 | issue=1292 | pages=330–334 | url=https://books.google.com/books?id=ZrloHemOmUEC&pg=PA353 | doi=10.1038/050330a0 | bibcode=1894Natur..50..330. | accessdate=2013-12-30 | doi-access=free }}
* {{Cite journal | last=Cavendish | first=Henry | title=Experiments to Determine the Density of the Earth | year=1798 | journal=Philosophical Transactions of the Royal Society| volume=88 |pages=469–526 | doi=10.1098/rstl.1798.0022| doi-access=free }}
* {{cite journal | author = Clotfelter, B. E. | title = The Cavendish experiment as Cavendish knew it | journal = American Journal of Physics | year = 1987 | volume = 55 | issue = 3 | pages = 210&ndash;213 | doi = 10.1119/1.15214|bibcode = 1987AmJPh..55..210C }} Establishes that Cavendish didn't determine G.
* {{cite journal | author = Falconer, Isobel | title = Henry Cavendish: the man and the measurement | journal = Measurement Science and Technology | year = 1999 | volume = 10 | issue = 6 | pages = 470&ndash;477 | doi = 10.1088/0957-0233/10/6/310 |bibcode = 1999MeScT..10..470F | s2cid = 250862938 }}
* {{cite web | last=Hodges | first=Laurent | year=1999 | title=The Michell-Cavendish Experiment, faculty website, Iowa State Univ. | url=http://www.public.iastate.edu/~lhodges/Michell.htm | accessdate=2013-12-30 | archive-url=https://web.archive.org/web/20170906134148/http://www.public.iastate.edu/~lhodges/Michell.htm | archive-date=2017-09-06 | url-status=dead }} Discusses Michell's contributions, and whether Cavendish determined G.
* {{cite book
 | last2 = McCormmach
 | first2 = Russell|author2-link= Russell McCormmach
 | last1 = Jungnickel
 | first1 = Christa|author1-link= Christa Jungnickel
 | title = Cavendish
 | location = Philadelphia, Pennsylvania
 | publisher = [[American Philosophical Society]]
 | year = 1996
 | url = https://archive.org/details/bub_gb_EUoLAAAAIAAJ
 | isbn = 978-0-87169-220-7
 | accessdate = 2013-12-30
 }}
* {{cite journal | author = Lally, Sean P. | title = Henry Cavendish and the Density of the Earth | journal = The Physics Teacher | year = 1999 | volume = 37 | issue = 1 | pages = 34&ndash;37 | bibcode=1999PhTea..37...34L | doi = 10.1119/1.880145}}
* {{cite book | last=Poynting | first=John H. | title=The Mean Density of the Earth: An essay to which the Adams prize was adjudged in 1893 | year=1894 | publisher=C. Griffin & Co. | location=London | url=https://archive.org/details/meandensityeart00poyngoog | accessdate=2013-12-30 }} Review of gravity measurements since 1740.
*{{Cite EB1911 |wstitle=Gravitation |volume=12 |pages=384–389|first=John Henry|last=Poynting|author-link=John Henry Poynting}}{{refend}}

==External links==
{{Commons}}
{{Portal|Physics}}
* [https://feynmanlectures.caltech.edu/I_07.html#Ch7-S6 Cavendish’s experiment in the Feynman Lectures on Physics]
* [https://web.archive.org/web/20080508011932/http://www.sas.org/tcs/weeklyIssues_2005/2005-07-01/feature1/index.html Sideways Gravity in the Basement, ''The Citizen Scientist'', July 1, 2005]. Homebrew Cavendish experiment, showing calculation of results and precautions necessary to eliminate wind and electrostatic errors.
* [http://www.physicscentral.com/explore/action/bigg.cfm "Big 'G'", Physics Central], retrieved Dec. 8, 2013. Experiment at Univ. of Washington to measure the gravitational constant using variation of Cavendish method. <!-- page captured by archive.is on 8 Dec 2013 -->
* {{cite web |url=http://www.npl.washington.edu/eotwash/bigG |title=The Controversy over Newton's Gravitational Constant |author=Eöt-Wash Group, Univ. of Washington |accessdate=December 8, 2013 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304031910/http://www.npl.washington.edu/eotwash/bigG}}. Discusses current state of measurements of '''''G'''''.
* [http://www.scienceandsociety.co.uk/results.asp?image=10314095 Model of Cavendish's torsion balance], retrieved Aug. 28, 2007, at Science Museum, London.

{{DEFAULTSORT:Cavendish Experiment}}
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