Seconds pendulum

Source: Wikipedia, the free encyclopedia.

The second pendulum, with a period of two seconds so each swing takes one second
simple pendulum
exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude.

A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.[1]

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing.

For a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without resistance, the length of the string of a seconds pendulum is equal to L = g/π2 where g is the acceleration due to gravity, with units of length per second squared, and L is the length of the string in the same units. Using the SI recommended acceleration due to gravity of g0 = 9.80665 m/s2, the length of the string will be approximately 993.6 millimetres, i.e. less than a centimetre short of one metre everywhere on Earth. This is because the value of g, expressed in m/s2, is very close to π2.

Defining the second

The second pendulum clock built around 1673 by Christiaan Huygens, inventor of the pendulum clock. Drawing is from his treatise Horologium Oscillatorium, published 1673, Paris, and it records improvements to the mechanism that Huygens had illustrated in the 1658 publication of his invention, titled Horologium. It is a weight-driven clock (the weight chain is removed) with a verge escapement (K,L), with the 1 second pendulum (X) suspended on a cord (V). The large metal plate (T) in front of the pendulum cord is the first illustration of Huygens' 'cycloidal cheeks', an attempt to improve accuracy by forcing the pendulum to follow a cycloidal path, making its swing isochronous. Huygens claimed it achieved an accuracy of 10 seconds per day.

The

verge and foliot
' clocks were retrofitted with pendulums.

These early clocks, due to their

grandfather clocks. The increased accuracy resulting from these developments caused the minute hand, previously rare, to be added to clock faces beginning around 1690.[7]
: 190 

The 18th- and 19th-century wave of

mercury pendulum by George Graham in 1721 and the gridiron pendulum by John Harrison in 1726.[7]
: 193–195  With these improvements, by the mid-18th century precision pendulum clocks achieved accuracies of a few seconds per week.

At the time the second was defined as a fraction of the Earth's rotation time or mean solar day and determined by clocks whose precision was checked by astronomical observations.[8][9] Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time (sundial time) and mean solar time (clock time).

The delay curve—above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow.

Mean solar time is the hour angle of the mean Sun plus 12 hours. This 12 hour offset comes from the decision to make each day start at midnight for civil purposes whereas the hour angle or the mean sun is measured from the zenith (noon).[10] The duration of daylight varies during the year but the length of a mean solar day is nearly constant, unlike that of an apparent solar day.[11] An apparent solar day can be 20 seconds shorter or 30 seconds longer than a mean solar day.[12] Long or short days occur in succession, so the difference builds up until mean time is ahead of apparent time by about 14 minutes near February 6 and behind apparent time by about 16 minutes near November 3. The equation of time is this difference, which is cyclical and does not accumulate from year to year.

Mean time follows the mean sun. Jean Meeus describes the mean sun as follows:

"Consider a first fictitious Sun travelling along the ecliptic with a constant speed and coinciding with the true sun at the perigee and apogee (when the Earth is in perihelion and aphelion, respectively). Then consider a second fictitious Sun travelling along the celestial equator at a constant speed and coinciding with the first fictitious Sun at the equinoxes. This second fictitious sun is the mean Sun..."[13]

In 1936 French and German astronomers found that Earth's rotation speed is irregular. Since 1967 atomic clocks define the second.[14][Note 1]

Usage in metrology

The length of a seconds pendulum was determined (in toises) by Marin Mersenne in 1644. In 1660, the Royal Society proposed that it be the standard unit of length. In 1671 Jean Picard measured this length at the Paris observatory. He found the value of 440.5 lignes of the Toise of Châtelet which had been recently renewed. He proposed a universal toise (French: Toise universelle) which was twice the length of the seconds pendulum.[8][15] However, it was soon discovered that the length of a seconds pendulum varies from place to place: French astronomer Jean Richer had measured the 0.3% difference in length between Cayenne (in what is now French Guiana) and Paris.[16]

Relationship to the figure of the Earth

See also: History of the metre § Meridional definition, and Meridian arc § History of measurement

Jean Richer and Giovanni Domenico Cassini measured the parallax of Mars between Paris and Cayenne in French Guiana when Mars was at its closest to Earth in 1672. They arrived at a figure for the solar parallax of 9.5 arcseconds, equivalent to an Earth–Sun distance of about 22000 Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague Jean Picard in 1669 as 3269 thousand toises. Picard's geodetic observations had been confined to the determination of the magnitude of the Earth considered as a sphere, but the discovery made by Jean Richer turned the attention of mathematicians to its deviation from a spherical form. Christiaan Huygens found out the centrifugal force which explained variations of gravitational acceleration depending on latitude. He also discovered that the seconds pendulum length was a means to measure gravitational acceleration. In the 18th century, in addition of its significance for cartography, geodesy grew in importance as a means of empirically demonstrating the theory of gravity, which Émilie du Châtelet promoted in France in combination with Leibniz's mathematical work and because the radius of the Earth was the unit to which all celestial distances were to be referred. Indeed, Earth proved to be an oblate spheroid through geodetic surveys in Ecuador and Lapland and this new data called into question the value of Earth radius as Picard had calculated it.[17][18][19][20][8][21][22][23][24]

The English physicist

Principia Mathematica (1687) in which he outlined his theory and calculations on the shape of the Earth. Newton theorised correctly that the Earth was not precisely a sphere but had an oblate ellipsoidal shape, slightly flattened at the poles due to the centrifugal force of its rotation. Since the surface of the Earth is closer to its centre at the poles than at the equator, gravity is stronger there. Using geometric calculations, he gave a concrete argument as to the hypothetical ellipsoid shape of the Earth.[26]

The goal of

Principia
. Clairaut's article did not provide a valid equation to back up his argument. This created much controversy in the scientific community.

It was not until Clairaut wrote Théorie de la figure de la terre in 1743 that a proper answer was provided. In it, he promulgated what is more formally known today as

Clairaut's theorem. By applying Clairaut's theorem, Laplace found from 15 gravity values that the flattening of the Earth was 1/330. A modern estimate is 1/298.25642.[29]

In 1790, one year before the

Talleyrand proposed that the metre be the length of the seconds pendulum at a latitude of 45°.[1] This option, with one-third of this length defining the foot, was also considered by Thomas Jefferson and others for redefining the yard in the United States shortly after gaining independence from the British Crown.[30]

Drawing of pendulum experiment to determine the length of the seconds pendulum at Paris, conducted in 1792 by Jean-Charles de Borda and Jean-Dominique Cassini. From their original paper. They used a pendulum that consisted of a 1+1⁄2-inch (3.8 cm) platinum ball suspended by a 12-foot (3.97 m) iron wire (F,Q). It was suspended in front of the pendulum (B) of a precision clock (A).

Instead of the seconds pendulum method, the commission of the

Spanish-French geodetic mission combined with an earlier measurement of the Paris meridian arc and the Lapland geodetic mission had confirmed that the Earth was an oblate spheroid.[21] Moreover, observations were made with a pendulum to determine the local acceleration due to local gravity and centrifugal acceleration; and these observations coincided with the geodetic results in proving that the Earth is flattened at the poles. The acceleration of a body near the surface of the Earth, which is measured with the seconds pendulum, is due to the combined effects of local gravity and centrifugal acceleration. The gravity diminishes with the distance from the center of the Earth while the centrifugal force augments with the distance from the axis of the Earth's rotation, it follows that the resulting acceleration towards the ground is 0.5% greater at the poles than at the Equator and that the polar diameter of the Earth is smaller than its equatorial diameter.[21][31][32][33][Note 2]

The

Spanish-French Geodesic Mission, conducted in actual Ecuador from 1735 to 1744.[36]

Jean-Baptiste Biot and François Arago published in 1821 their observations completing those of Delambre and Mechain. It was an account of the length's variation of the degrees of latitude along the Paris meridian as well as the account of the variation of the seconds pendulum's length along the same meridian between Shetland and the Baleares. The seconds pendulum's length is a mean to measure g, the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity).[37][38][39]

The task of surveying the Paris

Mètre des Archives
).

See also

Notes

  1. atomic time
    .
  2. ^ Gravity diminishes proportionally to the square of the distance from the centre of the Earth. Centrifugal force is a pseudo force corresponding to inertia and is related to the speed of rotation of an object situated at the surface of the Earth, which is proportional to the distance from the axis of the Earth's rotation: v = 2πR/T.

References

  1. ^ a b Seconds pendulum
  2. ^ "Huygens' Clocks". Stories. Science Museum, London, UK. Retrieved 14 November 2007.
  3. ^ "Pendulum Clock". The Galileo Project. Rice Univ. Retrieved 3 December 2007.
  4. ^ A modern reconstruction can be seen at "Pendulum clock designed by Galileo, Item #1883-29". Time Measurement. Science Museum, London, UK. Retrieved 14 November 2007.
  5. ^ Bennet, Matthew; et al. (2002). "Huygens' Clocks" (PDF). Georgia Institute of Technology. Archived from the original (PDF) on 10 April 2008. Retrieved 4 December 2007., p. 3, also published in Proceedings of the Royal Society of London, A 458, 563–579
  6. ^ Headrick, Michael (2002). "Origin and Evolution of the Anchor Clock Escapement". Control Systems Magazine. 22 (2). Archived from the original on 25 October 2009. Retrieved 6 June 2007.
  7. ^
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  8. ^
    Gallica
    .
  9. ^ Alain Bernard (15 April 2018), Le système solaire 2 : La révolution de la Terre, archived from the original on 14 December 2021, retrieved 12 October 2018
  10. ^ "Solar Apparent Time and Mean Solar Time" (PDF). Archived (PDF) from the original on 28 March 2018. Retrieved 28 March 2018.
  11. ^ For a discussion of the slight changes that affect the mean solar day, see the ΔT article.
  12. ^ "The duration of the true solar day" Archived 2009-08-26 at the Wayback Machine. Pierpaolo Ricci. pierpaoloricci.it. (Italy)
  13. ^ Meeus, J. (1998). Astronomical Algorithms. 2nd ed. Richmond VA: Willmann-Bell. p. 183.
  14. ^ "Revivre notre histoire | Les 350 ans de l'Observatoire de Paris". 350ans.obspm.fr (in French). Retrieved 28 September 2018.
  15. ^ Bigourdan, Guillaume (1901). Le système métrique des poids et mesures; son établissement et sa propagation graduelle, avec l'histoire des opérations qui ont servi à déterminer le mètre et le kilogramme. University of Ottawa. Paris : Gauthier-Villars. pp. 6–8.
  16. ^ Poynting, John Henry; Thomson, Joseph John (1907). A Textbook of Physics. C. Griffin. pp. 20.
  17. OCLC 894499177
    .
  18. ^ "Première détermination de la distance de la Terre au Soleil | Les 350 ans de l'Observatoire de Paris". 350ans.obspm.fr (in French). Retrieved 2 October 2018.
  19. ^ "1967LAstr..81..234G Page 234". adsbit.harvard.edu. Retrieved 2 October 2018.
  20. ^ "INRP - CLEA - Archives : Fascicule N° 137, Printemps 2012 Les distances". clea-astro.eu (in French). Retrieved 2 October 2018.
  21. ^ a b c Clarke, Alexander Ross; Helmert, Friedrich Robert (1911). "Earth, Figure of the" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 08 (11th ed.). Cambridge University Press.
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  25. Gallica
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  26. ^ Newton, Isaac. Principia, Book III, Proposition XIX, Problem III.
  27. ISBN 978-0-521-38541-1
    .
  28. JSTOR 103921
    .
  29. ^ Table 1.1 IERS Numerical Standards (2003))
  30. ^ Cochrane, Rexmond (1966). "Appendix B: The metric system in the United States". Measures for progress: a history of the National Bureau of Standards. U.S. Department of Commerce. p. 532. Archived from the original on 27 April 2011. Retrieved 5 March 2011.
  31. Gallica
    .
  32. ^ Alain Bernard (29 December 2017), Le système solaire 1: la rotation de la Terre, archived from the original on 14 December 2021, retrieved 12 October 2018
  33. OCLC 895784336
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  34. Bibcode:1986CRASG...3..261L
    . Retrieved 4 September 2018 – via Gallica.
  35. ^ "Histoire du mètre". Direction Générale des Entreprises (DGE) (in French). Retrieved 28 September 2018.
  36. S2CID 109333769
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  37. ^ Larousse, Pierre (1874). Larousse, Pierre, ed. (1874), "Métrique", Grand dictionnaire universel du XIXe siècle, 11. Paris. pp. 163–164.{{cite book}}: CS1 maint: location missing publisher (link)
  38. OCLC 314175913
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  39. Gallica
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    pp. 10–11. Retrieved 5 April 2011.
  41. ^ Public Domain Larousse, Pierre, ed. (1874), "Métrique", Grand dictionnaire universel du XIXe siècle, vol. 11, Paris: Pierre Larousse, pp. 163–164
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