Editing Pendulum (section)

== Gravity measurement ==
The presence of the [[Gravitational acceleration|acceleration of gravity]] ''g'' in the periodicity equation (1) for a pendulum means that the local gravitational acceleration of the Earth can be calculated from the period of a pendulum.  A pendulum can therefore be used as a [[gravimeter]] to measure the local [[gravity]], which varies by over 0.5% across the surface of the Earth.<ref>{{cite web
|url = http://www.ucl.ac.uk/EarthSci/people/lidunka/GEOL2014/Geophysics2%20-%20Gravity/gravity.htm
|title = Gravity, the shape of the Earth, isostasy, moment of inertia
|first1 = Lidunka
|last1 = Vočadlo
|access-date = 5 November 2012
}}</ref><ref group = Note>The value of "g" (acceleration due to gravity) at the [[equator]] is 9.780&nbsp;m/s<sup>2</sup> and at the [[North Pole|poles]] is 9.832&nbsp;m/s<sup>2</sup>, a difference of 0.53%.</ref> The pendulum in a clock is disturbed by the pushes it receives from the clock movement, so freeswinging pendulums were used, and were the standard instruments of [[gravimetry]] up to the 1930s.

The difference between clock pendulums and gravimeter pendulums is that to measure gravity, the pendulum's length as well as its period has to be measured. The period of freeswinging pendulums could be found to great precision by comparing their swing with a precision clock that had been adjusted to keep correct time by the passage of stars overhead. In the early measurements, a weight on a cord was suspended in front of the clock pendulum, and its length adjusted until the two pendulums swung in exact synchronism. Then the length of the cord was measured. From the length and the period, ''g'' could be calculated from equation (1).

=== The seconds pendulum ===
[[File:Pendulum2secondclock.gif|thumb|The seconds pendulum, a pendulum with a period of two seconds so each swing takes one second]] 
The [[seconds pendulum]], a pendulum with a period of two seconds so each swing takes one second, was widely used to measure gravity, because its period could be easily measured by comparing it to precision [[regulator clock]]s, which all had seconds pendulums.  By the late 17th century, the length of the seconds pendulum became the standard measure of the strength of gravitational acceleration at a location. By 1700 its length had been measured with submillimeter accuracy at several cities in Europe. For a seconds pendulum, ''g'' is proportional to its length:
<math display="block">g \propto L. </math>

=== Early observations ===
* '''1620''': British scientist [[Francis Bacon]] was one of the first to propose using a pendulum to measure gravity, suggesting taking one up a mountain to see if gravity varies with altitude.<ref>{{cite web
  | last = Baker
  | first = Lyman A.
  | title = Chancellor Bacon
  | website = English 233 – Introduction to Western Humanities
  | publisher = English Dept., Kansas State Univ.
  | date = Spring 2000
  | url = http://www-personal.ksu.edu/~lyman/english233/Voltaire-Bacon.htm
  | access-date = 2009-02-20}}</ref>
* '''1644''': Even before the pendulum clock, French priest [[Marin Mersenne]] first determined the length of the seconds pendulum was {{convert|39.1|in|mm}}, by comparing the swing of a pendulum to the time it took a weight to fall a measured distance.  He also was first to discover the dependence of the period on amplitude of swing. 
* '''1669''': [[Jean Picard]] determined the length of the seconds pendulum at Paris, using a {{convert|1|in|adj=on}} copper ball suspended by an aloe fiber, obtaining {{convert|39.09|in}}.<ref name="Poynting & Thompson 1907, p.9">[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA9 Poynting & Thompson 1907, p.9]</ref>  He also did the first experiments on thermal expansion and contraction of pendulum rods with temperature.
* '''1672''': The first observation that gravity varied at different points on Earth was made in 1672 by [[Jean Richer]], who took a [[pendulum clock]] to [[Cayenne]], [[French Guiana]] and found that it lost {{frac|2|1|2}} minutes per day; its seconds pendulum had to be shortened by {{frac|1|1|4}} ''[[ligne]]s'' (2.6&nbsp;mm) shorter than at Paris, to keep correct time.<ref>{{cite book
  | last = Poynting
  | first = John Henry
  |author2=Joseph John Thompson
  | title = A Textbook of Physics, 4th Ed
  | publisher = Charles Griffin & Co.
  | year = 1907
  | location = London
  | page = [https://archive.org/details/bub_gb_TL4KAAAAIAAJ/page/n30 20]
  | url = https://archive.org/details/bub_gb_TL4KAAAAIAAJ
  }}</ref><ref name="Lenzen1964">{{cite conference
  | first = Lenzen
  | last = Victor F.
  |author2=Robert P. Multauf
  | title = Paper 44: Development of gravity pendulums in the 19th century
  | book-title = United States National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
  | pages = 307
  | publisher = Smithsonian Institution Press
  | year = 1964
  | location = Washington
  | url = https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA307
  | access-date = 2009-01-28}}</ref>  In 1687 [[Isaac Newton]] in ''[[Principia Mathematica (Newton)|Principia Mathematica]]'' showed this was because the Earth had a slightly  [[Oblate spheroid|oblate]] shape (flattened at the poles) caused by the [[centrifugal force]] of its rotation.  At higher latitudes the surface was closer to the center of the Earth, so gravity increased with latitude.<ref name="Lenzen1964" />  From this time on, pendulums began to be taken to distant lands to measure gravity, and tables were compiled of the length of the seconds pendulum at different locations on Earth. In 1743 [[Alexis Claude Clairaut]] created the first hydrostatic model of the Earth, [[Clairaut's theorem]],<ref name="Poynting & Thompson 1907, p.9" /> which allowed the [[ellipticity]] of the Earth to be calculated from gravity measurements. Progressively more accurate models of the shape of the Earth followed.
* '''1687''': Newton experimented with pendulums (described in ''Principia'') and found that equal length pendulums with bobs made of different materials had the same period, proving that the gravitational force on different substances was exactly proportional to their [[mass]] (inertia).  This principle, called the [[equivalence principle]], confirmed to greater accuracy in later experiments, became the foundation on which [[Albert Einstein]] based his [[general theory of relativity]]. 
[[File:Borda and Cassini pendulum experiment.png|thumb|190px|Borda & Cassini's 1792 measurement of the length of the seconds pendulum]]
* '''1737''': French mathematician [[Pierre Bouguer]] made a sophisticated series of pendulum observations in the [[Andes]] mountains, Peru.<ref name="Poynting & Thompson, 1907, p.10">[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA10 Poynting & Thompson, 1907, p.10]</ref>  He used a copper pendulum bob in the shape of a double pointed cone suspended by a thread; the bob could be reversed to eliminate the effects of nonuniform density. He calculated the length to the center of oscillation of thread and bob combined, instead of using the center of the bob. He corrected for thermal expansion of the measuring rod and barometric pressure, giving his results for a pendulum swinging in vacuum. Bouguer swung the same pendulum at three different elevations, from sea level to the top of the high Peruvian ''[[altiplano]]''. Gravity should fall with the inverse square of the distance from the center of the Earth. Bouguer found that it fell off slower, and correctly attributed the 'extra' gravity to the gravitational field of the huge Peruvian plateau. From the density of rock samples he calculated an estimate of the effect of the ''altiplano'' on the pendulum, and comparing this with the gravity of the Earth was able to make the first rough estimate of the [[Mass of the Earth|density of the Earth]].
* '''1747''': [[Daniel Bernoulli]] showed how to correct for the lengthening of the period due to a finite angle of swing ''θ''<sub>0</sub> by using the first order correction ''θ''<sub>0</sub><sup>2</sup>/16, giving the period of a pendulum with an extremely small swing.<ref name="Poynting & Thompson, 1907, p.10" />
* '''1792''': To define a pendulum standard of length for use with the new [[metric system]], in 1792 [[Jean-Charles de Borda]] and [[Dominique, comte de Cassini|Jean-Dominique Cassini]] made a precise measurement of the seconds pendulum at Paris. They used a {{frac|1|1|2}}-inch (14&nbsp;mm){{clarify|reason=1.5" is not 14mm|date=September 2019}} platinum ball suspended by a {{convert|12|ft|adj=on}} iron wire. Their main innovation was a technique called the "''method of coincidences''" which allowed the period of pendulums to be compared with great precision.  (Bouguer had also used this method). The time interval Δ''t'' between the recurring instants when the two pendulums swung in synchronism was timed. From this the difference between the periods of the pendulums, ''T''<sub>1</sub> and ''T''<sub>2</sub>, could be calculated: <math display="block">\frac {1}{\Delta t} = \frac {1}{T_1} - \frac {1}{T_2}</math>
* '''1821''': [[Francesco Carlini]] made pendulum observations on top of Mount Cenis, Italy, from which, using methods similar to Bouguer's, he calculated the density of the Earth.<ref>{{cite book
  | last = Poynting
  | first = John Henry
  | title = The Mean Density of the Earth
  | publisher = Charles Griffin
  | year = 1894
  | location = London
  | pages = [https://archive.org/details/meandensityeart00poyngoog/page/n44 22]–24
  | url = https://archive.org/details/meandensityeart00poyngoog
  }}</ref>  He compared his measurements to an estimate of the gravity at his location assuming the mountain wasn't there, calculated from previous nearby pendulum measurements at sea level. His measurements showed 'excess' gravity, which he allocated to the effect of the mountain. Modeling the mountain as a segment of a sphere {{convert|11|mi}} in diameter and {{convert|1|mi}} high, from rock samples he calculated its gravitational field, and estimated the density of the Earth at 4.39 times that of water. Later recalculations by others gave values of 4.77 and 4.95, illustrating the uncertainties in these geographical methods.

=== Kater's pendulum ===
{{Main|Kater's pendulum}}
{| style="float:right;"
|- 
|[[File:PenduloCaminos.jpg|thumb|115px|Kater's pendulum and stand]]
||[[File:Kater pendulum use.png|thumb|200px|Measuring gravity with Kater's reversible pendulum, from Kater's 1818 paper]]
|}
[[File:Kater pendulum vertical.png|thumb|upright=0.4|A Kater's pendulum]]
{{anchor|huygens-law}}<!-- anchor used by (at least) [[Huygens' law of the pendulum]] -->
The precision of the early gravity measurements above was limited by the difficulty of measuring the length of the pendulum, ''L'' .  ''L'' was the length of an idealized simple gravity pendulum (described at top), which has all its mass concentrated in a point at the end of the cord. In 1673 Huygens had shown that the period of a rigid bar pendulum (called a ''compound pendulum'') was equal to the period of a simple pendulum with a length equal to the distance between the [[wikt:pivot|pivot]] point and a point called the [[Center of percussion|center of oscillation]], located under the [[center of gravity]], that depends on the mass distribution along the pendulum. But there was no accurate way of determining the center of oscillation in a real pendulum. Huygens' discovery is sometimes referred to as ''Huygens' law of the (cycloidal) pendulum''.<ref>{{cite book |doi=10.1093/acprof:oso/9780199570409.003.0005 |title=Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology |chapter=Christiaan Huygens: A Great Natural Philosopher Who Measured Gravity and an Illuminating Foil for Newton on Method |last=Harper |first=William L. |date=Dec 2011|pages=194–219 |isbn=978-0-19-957040-9 }}</ref>

To get around this problem, the early researchers above approximated an ideal simple pendulum as closely as possible by using a metal sphere suspended by a light wire or cord. If the wire was light enough, the center of oscillation was close to the center of gravity of the ball, at its geometric center. This "ball and wire" type of pendulum wasn't very accurate, because it didn't swing as a rigid body, and the elasticity of the wire caused its length to change slightly as the pendulum swung.

However Huygens had also proved that in any pendulum, the pivot point and the center of oscillation were interchangeable.<ref name="HuygensReciprocity" />  That is, if a pendulum were turned upside down and hung from its center of oscillation, it would have the same period as it did in the previous position, and the old pivot point would be the new center of oscillation.

British physicist and army captain [[Henry Kater]] in 1817 realized that Huygens' principle could be used to find the length of a simple pendulum with the same period as a real pendulum.<ref name="Kater1818" />  If a pendulum was built with a second adjustable pivot point near the bottom so it could be hung upside down, and the second pivot was adjusted until the periods when hung from both pivots were the same, the second pivot would be at the center of oscillation, and the distance between the two pivots would be the length ''L'' of a simple pendulum with the same period.

Kater built a reversible pendulum (''see drawing'') consisting of a brass bar with two opposing pivots made of short triangular "knife" blades ''<span style="color:red;">(a)</span>'' near either end. It could be swung from either pivot, with the knife blades supported on agate plates. Rather than make one pivot adjustable, he attached the pivots a meter apart and instead adjusted the periods with a moveable weight on the pendulum rod ''<span style="color:red;">(b,c)</span>''. In operation, the pendulum is hung in front of a precision clock, and the period timed, then turned upside down and the period timed again. The weight is adjusted with the adjustment screw until the periods are equal. Then putting this period and the distance between the pivots into equation (1) gives the gravitational acceleration ''g'' very accurately.

Kater timed the swing of his pendulum using the "''method of coincidences''" and measured the distance between the two pivots with a micrometer. After applying corrections for the finite amplitude of swing, the buoyancy of the bob, the barometric pressure and altitude, and temperature, he obtained a value of 39.13929&nbsp;inches for the seconds pendulum at London, in vacuum, at sea level, at 62&nbsp;°F. The largest variation from the mean of his 12 observations was 0.00028 in.<ref>{{cite book
  | last = Cox
  | first = John
  | title = Mechanics
  | publisher = Cambridge Univ. Press
  | year = 1904
  | location = Cambridge, UK
  | pages = [https://archive.org/details/mechanics00coxgoog/page/n341 311]–312
  | url = https://archive.org/details/mechanics00coxgoog
  }}</ref> representing a precision of gravity measurement of 7×10<sup>−6</sup> (7 [[mGal]] or 70 [[Metre per second squared|μm/s<sup>2</sup>]]). Kater's measurement was used as Britain's official standard of length (see [[#Britain and Denmark|below]]) from 1824 to 1855.

Reversible pendulums (known technically as "convertible" pendulums) employing Kater's principle were used for absolute gravity measurements into the 1930s.

=== Later pendulum gravimeters ===
The increased accuracy made possible by Kater's pendulum helped make [[gravimetry]] a standard part of [[geodesy]]. Since the exact location (latitude and longitude) of the 'station' where the gravity measurement was made was necessary, gravity measurements became part of [[surveying]], and pendulums were taken on the great [[geodetic surveying|geodetic surveys]] of the 18th century, particularly the [[Great Trigonometric Survey]] of India.

[[File:Using Kater pendulum in India.png|thumb|250px|Measuring gravity with an invariable pendulum, Madras, India, 1821]]
* '''Invariable pendulums:'''  Kater introduced the idea of ''relative'' gravity measurements, to supplement the ''absolute'' measurements made by a Kater's pendulum.<ref>[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA23 Poynting & Thomson 1904, p.23]</ref>  Comparing the gravity at two different points was an easier process than measuring it absolutely by the Kater method. All that was necessary was to time the period of an ordinary (single pivot) pendulum at the first point, then transport the pendulum to the other point and time its period there. Since the pendulum's length was constant, from (1) the ratio of the gravitational accelerations was equal to the inverse of the ratio of the periods squared, and no precision length measurements were necessary. So once the gravity had been measured absolutely at some central station, by the Kater or other accurate method, the gravity at other points could be found by swinging pendulums at the central station and then taking them to the other location and timing their swing there. Kater made up a set of "invariable" pendulums, with only one knife edge pivot, which were taken to many countries after first being swung at a central station at [[Kew Observatory]], UK.
* '''Airy's coal pit experiments''': Starting in 1826, using methods similar to Bouguer, British astronomer [[George Airy]] attempted to determine the density of the Earth by pendulum gravity measurements at the top and bottom of a coal mine.<ref>{{cite book
  | last = Poynting
  | first = John Henry
  | title = The Mean Density of the Earth
  | publisher = Charles Griffin & Co.
  | year = 1894
  | location = London
  | pages = [https://archive.org/details/meandensityeart00poyngoog/page/n46 24]–29
  | url = https://archive.org/details/meandensityeart00poyngoog
  }}</ref><ref>{{cite EB1911|wstitle= Gravitation |volume= 12 |last= Poynting |first= John Henry |author-link= John Henry Poynting| pages = 384&ndash;389; see page 386 |quote= Airy's Experiment.—In 1854 Sir G. B. Airy....}}</ref>  The gravitational force below the surface of the Earth decreases rather than increasing with depth, because by [[Gauss's law for gravity|Gauss's law]] the mass of the spherical shell of crust above the subsurface point does not contribute to the gravity. The 1826 experiment was aborted by the flooding of the mine, but in 1854 he conducted an improved experiment at the Harton coal mine, using seconds pendulums swinging on agate plates, timed by precision chronometers synchronized by an electrical circuit. He found the lower pendulum was slower by 2.24 seconds per day. This meant that the gravitational acceleration at the bottom of the mine, 1250&nbsp;ft below the surface, was 1/14,000 less than it should have been from the inverse square law; that is the attraction of the spherical shell was 1/14,000 of the attraction of the Earth. From samples of surface rock he estimated the mass of the spherical shell of crust, and from this estimated that the density of the Earth was 6.565 times that of water. Von Sterneck attempted to repeat the experiment in 1882 but found inconsistent results.

[[File:Repsold pendulum.png|thumb|left|65px|Repsold pendulum, 1864]]
* '''Repsold-Bessel pendulum:''' It was time-consuming and error-prone to repeatedly swing the Kater's pendulum and adjust the weights until the periods were equal.  [[Friedrich Bessel]] showed in 1835 that this was unnecessary.<ref>[https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA320 Lenzen & Multauf 1964, p.320]</ref>  As long as the periods were close together, the gravity could be calculated from the two periods and the center of gravity of the pendulum.<ref>[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA18 Poynting & Thompson 1907, p.18]</ref>  So the reversible pendulum didn't need to be adjustable, it could just be a bar with two pivots. Bessel also showed that if the pendulum was made symmetrical in form about its center, but was weighted internally at one end, the errors due to air drag would cancel out. Further, another error due to the finite diameter of the knife edges could be made to cancel out if they were interchanged between measurements. Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold, under contract by the Swiss Geodetic Commission made a pendulum along these lines.  The Repsold pendulum was about 56&nbsp;cm long and had a period of about {{frac|3|4}} second. It was used extensively by European geodetic agencies, and with the Kater pendulum in the Survey of India. Similar pendulums of this type were designed by Charles Pierce and C. Defforges.

[[File:Mendenhall gravimeter pendulums.jpg|thumb|250px|Pendulums used in Mendenhall gravimeter, 1890]]
* '''Von Sterneck and Mendenhall gravimeters:''' In 1887 Austro-Hungarian scientist Robert von Sterneck developed a small gravimeter pendulum mounted in a temperature-controlled vacuum tank to eliminate the effects of temperature and air pressure. It used a "half-second pendulum," having a period close to one second, about 25&nbsp;cm long.  The pendulum was nonreversible, so the instrument was used for relative gravity measurements, but their small size made them small and portable. The period of the pendulum was picked off by reflecting the image of an [[electric spark]] created by a precision chronometer off a mirror mounted at the top of the pendulum rod. The Von Sterneck instrument, and a similar instrument developed by Thomas C. Mendenhall of the [[United States Coast and Geodetic Survey]] in 1890,<ref name="NOAA">{{cite web
  | title = The downs and ups of gravity surveys
  | website = NOAA Celebrates 200 Years
  | publisher = US National Oceanographic and Atmospheric Administration
  | date = 2007-07-09
  | url = http://celebrating200years.noaa.gov/foundations/gravity_surveys/welcome.html#at
  }}</ref> were used extensively for surveys into the 1920s.

:The Mendenhall pendulum was actually a more accurate timekeeper than the highest precision clocks of the time, and as the 'world's best clock' it was used by [[Albert A. Michelson]] in his 1924 measurements of the [[speed of light]] on Mt. Wilson, California.<ref name="NOAA" />
* '''Double pendulum gravimeters:''' Starting in 1875, the increasing accuracy of pendulum measurements revealed another source of error in existing instruments: the swing of the pendulum caused a slight swaying of the tripod stand used to support portable pendulums, introducing error. In 1875 Charles S Peirce calculated that measurements of the length of the seconds pendulum made with the Repsold instrument required a correction of 0.2&nbsp;mm due to this error.<ref>[https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA324 Lenzen & Multauf 1964, p.324]</ref>  In 1880 C. Defforges used a [[Michelson interferometer]] to measure the sway of the stand dynamically, and interferometers were added to the standard Mendenhall apparatus to calculate sway corrections.<ref>[https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA329 Lenzen & Multauf 1964, p.329]</ref>  A method of preventing this error was first suggested in 1877 by Hervé Faye and advocated by Peirce, Cellérier and Furtwangler: mount two identical pendulums on the same support, swinging with the same amplitude, 180° out of phase. The opposite motion of the pendulums would cancel out any sideways forces on the support. The idea was opposed due to its complexity, but by the start of the 20th century the Von Sterneck device and other instruments were modified to swing multiple pendulums simultaneously.

[[File:Quartz gravimeter pendulums.jpg|thumb|250px|Quartz pendulums used in Gulf gravimeter, 1929]]
* '''Gulf gravimeter''': One of the last and most accurate pendulum gravimeters was the apparatus developed in 1929 by the Gulf Research and Development Co.<ref name="Woolard">{{cite conference
  | first = George P.
  | last = Woolard
  | title = Gravity observations during the IGY
  | book-title = Geophysics and the IGY: Proceedings of the symposium at the opening of the International Geophysical Year
  | pages = 200
  | publisher = American Geophysical Union, Nat'l Academy of Sciences
  | date = June 28–29, 1957
  | location = Washington, D.C.
  | url = https://books.google.com/books?id=dUIrAAAAYAAJ&pg=PA200
  | access-date = 2009-05-27}}</ref><ref>[https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA336 Lenzen & Multauf 1964, p.336, fig.28]</ref>  It used two pendulums made of [[fused quartz]], each {{convert|10.7|in|mm}} in length with a period of 0.89 second, swinging on pyrex knife edge pivots, 180° out of phase. They were mounted in a permanently sealed temperature and humidity controlled vacuum chamber. Stray electrostatic charges on the quartz pendulums had to be discharged by exposing them to a radioactive salt before use. The period was detected by reflecting a light beam from a mirror at the top of the pendulum, recorded by a chart recorder and compared to a precision [[crystal oscillator]] calibrated against the [[WWV (radio station)|WWV]] radio time signal. This instrument was accurate to within (0.3–0.5)×10<sup>−7</sup> (30–50 [[microgal]]s or 3–5&nbsp;nm/s<sup>2</sup>).<ref name="Woolard" /> It was used into the 1960s.

Relative pendulum gravimeters were superseded by the simpler LaCoste zero-length spring gravimeter, invented in 1934 by [[Lucien LaCoste]].<ref name="NOAA" />  Absolute (reversible) pendulum gravimeters were replaced in the 1950s by free fall gravimeters, in which a weight is allowed to fall in a vacuum tank and its acceleration is measured by an optical [[interferometer]].<ref name="Torge" />