Editing Pendulum (section)

=== 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" />