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A seconds pendulum is a
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 ...
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.Seconds pendulum
/ref>


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 centimeter 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
pendulum clock A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. The advantage of a pendulum for timekeeping is that it is a harmonic oscillator: It swings back and forth in a precise time interval dependent on i ...
was invented in 1656 by Dutch scientist and inventor
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists ...
, and patented the following year. Huygens contracted the construction of his clock designs to clockmaker
Salomon Coster Salomon Coster (c. 1620–1659) was a Dutch clockmaker of the Hague, who in 1657 was the first to make a pendulum clock, which had been invented by Christiaan Huygens (1629-1695). Coster died a sudden death in 1659. Coster's earliest pendulum ...
, who actually built the clock. Huygens was inspired by investigations of pendulums by Galileo Galilei beginning around 1602. Galileo discovered the key property that makes pendulums useful timekeepers: isochronism, which means that the
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
of swing of a pendulum is approximately the same for different sized swings. Galileo had the idea for a pendulum clock in 1637, which was partly constructed by his son in 1649, but neither lived to finish it. The introduction of the pendulum, the first harmonic oscillator used in timekeeping, increased the accuracy of clocks enormously, from about 15 minutes per day to 15 seconds per day leading to their rapid spread as existing '
verge and foliot The verge (or crown wheel) escapement is the earliest known type of mechanical escapement, the mechanism in a mechanical clock that controls its rate by allowing the gear train to advance at regular intervals or 'ticks'. Its origin is unknown. V ...
' clocks were retrofitted with pendulums. These early clocks, due to their verge escapements, had wide pendulum swings of 80–100°. In his 1673 analysis of pendulums, ''
Horologium Oscillatorium (English: ''The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks'') is a book published by Dutch physicist Christiaan Huygens in 1673 and his major work on pendulums and horology. It is regarde ...
'', Huygens showed that wide swings made the pendulum inaccurate, causing its
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
, and thus the rate of the clock, to vary with unavoidable variations in the driving force provided by the
movement Movement may refer to: Common uses * Movement (clockwork), the internal mechanism of a timepiece * Motion, commonly referred to as movement Arts, entertainment, and media Literature * "Movement" (short story), a short story by Nancy Fu ...
. Clockmakers' realisation that only pendulums with small swings of a few degrees are isochronous motivated the invention of the
anchor escapement In horology, the anchor escapement is a type of escapement used in pendulum clocks. The escapement is a mechanism in a mechanical clock that maintains the swing of the pendulum by giving it a small push each swing, and allows the clock's wheels ...
around 1670, which reduced the pendulum's swing to 4–6°. The anchor became the standard escapement used in pendulum clocks. In addition to increased accuracy, the anchor's narrow pendulum swing allowed the clock's case to accommodate longer, slower pendulums, which needed less power and caused less wear on the movement. The seconds pendulum (also called the Royal pendulum), 0.994 m (39.1 in) long, in which each swing takes one second, became widely used in quality clocks. The long narrow clocks built around these pendulums, first made by William Clement around 1680, became known as
grandfather clock A grandfather clock (also a longcase clock, tall-case clock, grandfather's clock, or floor clock) is a tall, freestanding, weight-driven pendulum clock with the pendulum held inside the tower or waist of the case. Clocks of this style are common ...
s. The increased accuracy resulting from these developments caused the minute hand, previously rare, to be added to clock faces beginning around 1690. The 18th- and 19th-century wave of
horological Horology (; related to Latin '; ; , interfix ''-o-'', and suffix ''-logy''), . is the study of the measurement of time. Clocks, watches, clockwork, sundials, hourglasses, clepsydras, timers, time recorders, marine chronometers, and atomic cl ...
innovation that followed the invention of the pendulum brought many improvements to pendulum clocks. The
deadbeat escapement In horology, the anchor escapement is a type of escapement used in pendulum clocks. The escapement is a mechanism in a mechanical clock that maintains the swing of the pendulum by giving it a small push each swing, and allows the clock's wheels ...
invented in 1675 by Richard Towneley and popularised by George Graham around 1715 in his precision "regulator" clocks gradually replaced the anchor escapement and is now used in most modern pendulum clocks. The observation that pendulum clocks slowed down in summer brought the realisation that thermal expansion and contraction of the pendulum rod with changes in temperature was a source of error. This was solved by the invention of temperature-compensated pendulums; the
mercury 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 ...
by George Graham in 1721 and the gridiron pendulum by John Harrison in 1726. 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. Solar time is a calculation of the passage of time based on the
position of the Sun The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the cel ...
in the
sky The sky is an unobstructed view upward from the surface of the Earth. It includes the atmosphere and outer space. It may also be considered a place between the ground and outer space, thus distinct from outer space. In the field of astronomy, ...
. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time (
sundial A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
time) and mean solar time (clock time). 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). 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. An apparent solar day can be 20 seconds shorter or 30 seconds longer than a mean solar day. 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 Jean Meeus (born 12 December 1928) is a Belgian meteorologist and amateur astronomer specializing in celestial mechanics, spherical astronomy, and mathematical astronomy. Meeus studied mathematics at the University of Leuven in Belgium, w ...
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''..."
In 1936 French and German astronomers found that Earth's rotation speed is irregular. Since 1967
atomic clock An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s define the second.For further informations see
atomic time International Atomic Time (abbreviated TAI, from its French name ) is a high-precision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid. TAI is a weighted average of the time kept by over 450 atomi ...
.


Usage in metrology

The length of a seconds pendulum was determined (in
toise A toise (; symbol: T) is a unit of measure for length, area and volume originating in pre-revolutionary France. In North America, it was used in colonial French establishments in early New France, French Louisiana (''Louisiane''), Acadia (''Acad ...
s) by Marin Mersenne in 1644. In 1660, the Royal Society proposed that it be the standard unit of length. In 1671
Jean Picard Jean Picard (21 July 1620 – 12 July 1682) was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He is principally notable for his accurate measure of the size of the Earth, bas ...
measured this length at the Paris observatory. He found the value of 440.5 lines 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. 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 Cayenne (; ; gcr, Kayenn) is the capital city of French Guiana, an overseas region and department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic coast. The city's mot ...
(in what is now French Guiana) and Paris.


Relationship to the figure of the Earth

Jean Richer and Giovanni Domenico Cassini measured the parallax of Mars between Paris and
Cayenne Cayenne (; ; gcr, Kayenn) is the capital city of French Guiana, an overseas region and department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic coast. The city's mot ...
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 Jean Picard (21 July 1620 – 12 July 1682) was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He is principally notable for his accurate measure of the size of the Earth, bas ...
in 1669 as 3269 thousand ''
toise A toise (; symbol: T) is a unit of measure for length, area and volume originating in pre-revolutionary France. In North America, it was used in colonial French establishments in early New France, French Louisiana (''Louisiane''), Acadia (''Acad ...
s''. 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. The determination of the
figure of the Earth Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...
became a problem of the highest importance in astronomy, inasmuch as the diameter of the Earth was the unit to which all celestial distances had to be referred. The English physicist
Sir Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the gre ...
, who used Picard's Earth measurement for establishing his law of universal gravity, explained this variation of the seconds pendulum's length in his ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' (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 In Christianity (especially in the Roman Catholic, Orthodox, Anglican and Methodist traditions), an oblate is a person who is specifically dedicated to God or to God's service. Oblates are individuals, either laypersons or clergy, normally li ...
ellipsoidal shape, slightly flattened at the poles due to the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...
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. The goal of '' Principia'' was not to provide exact answers for natural phenomena, but to theorise potential solutions to these unresolved factors in science. Newton pushed for scientists to look further into the unexplained variables. Two prominent researchers whom he inspired were
Alexis Clairaut Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had out ...
and
Pierre Louis Maupertuis Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
. They both sought to prove the validity of Newton's theory on the shape of the Earth. In order to do so, they went on an expedition to Lapland in an attempt to accurately measure the meridian arc. From such measurements they could calculate the eccentricity of the Earth, its degree of departure from a perfect sphere. Clairaut confirmed that Newton's theory that the Earth was ellipsoidal was correct, but his calculations were in error; he wrote a letter to the
Royal Society of London The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, r ...
with his findings. The society published an article in ''Philosophical Transactions'' the following year in 1737 that revealed his discovery. Clairaut showed how Newton's equations were incorrect, and did not prove an ellipsoid shape to the Earth. However, he corrected problems with the theory, that in effect would prove Newton's theory correct. Clairaut believed that Newton had reasons for choosing the shape that he did, but he did not support it in '' 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 Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise ...
. By applying Clairaut's theorem, Laplace found from 15 gravity values that the flattening of the Earth was . A modern estimate is . In 1790, one year before the metre was ultimately based on a quadrant of the Earth, Talleyrand proposed that the metre be the length of the seconds pendulum at a
latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
of 45°. 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. Instead of the seconds pendulum method, the commission of the French Academy of Sciences – whose members included Lagrange, Laplace,
Monge Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During ...
and
Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher and mathematician. His ideas, including support for a liberal economy, free and equal p ...
– decided that the new measure should be equal to one ten-millionth of the distance from the North Pole to the Equator (the quadrant of the Earth's circumference), measured along the
meridian Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...
passing through Paris. Apart from the obvious consideration of safe access for French surveyors, the Paris meridian was also a sound choice for scientific reasons: a portion of the quadrant from Dunkirk to Barcelona (about 1000 km, or one-tenth of the total) could be surveyed with start- and end-points at sea level, and that portion was roughly in the middle of the quadrant, where the effects of the Earth's oblateness were expected to be the largest. 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. 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 In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parall ...
. The gravity diminishes with the distance from the center of the Earth while the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...
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. Gravity diminishes proportionally to the square of the distance from the centre of the Earth.
Centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...
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.
The
Academy of Sciences An academy of sciences is a type of learned society or academy (as special scientific institution) dedicated to sciences that may or may not be state funded. Some state funded academies are tuned into national or royal (in case of the Unite ...
planned to infer the flattening of the Earth from the length's differences between meridional portions corresponding to one degree of
latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
.
Pierre Méchain Pierre François André Méchain (; 16 August 1744 – 20 September 1804) was a French astronomer and surveyor who, with Charles Messier, was a major contributor to the early study of deep-sky objects and comets. Life Pierre Méchain was born i ...
and
Jean-Baptiste Delambre Jean Baptiste Joseph, chevalier Delambre (19 September 1749 – 19 August 1822) was a French mathematician, astronomer, historian of astronomy, and geodesist. He was also director of the Paris Observatory, and author of well-known books on t ...
combined their measurements with the results of the Spanish-French geodetic mission and found a value of 1/334 for the Earth's flattening, and they then extrapolated from their measurement of the Paris meridian arc between Dunkirk and Barcelona the distance from the
North Pole The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Magn ...
to the Equator which was 5 130 740
toise A toise (; symbol: T) is a unit of measure for length, area and volume originating in pre-revolutionary France. In North America, it was used in colonial French establishments in early New France, French Louisiana (''Louisiane''), Acadia (''Acad ...
s. As the metre had to be equal to one ten-millionth of this distance, it was defined as 0.513074
toise A toise (; symbol: T) is a unit of measure for length, area and volume originating in pre-revolutionary France. In North America, it was used in colonial French establishments in early New France, French Louisiana (''Louisiane''), Acadia (''Acad ...
or 3
feet The foot ( : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg made ...
and 11.296 lines of the Toise of Peru. The Toise of Peru had been constructed in 1735 as the standard of reference in the Spanish-French Geodesic Mission, conducted in actual Ecuador from 1735 to 1744.
Jean-Baptiste Biot Jean-Baptiste Biot (; ; 21 April 1774 – 3 February 1862) was a French physicist, astronomer, and mathematician who co-discovered the Biot–Savart law of magnetostatics with Félix Savart, established the reality of meteorites, made an early b ...
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). The task of surveying the Paris meridian arc took more than six years (1792–1798). The technical difficulties were not the only problems the surveyors had to face in the convulsed period of the aftermath of the French Revolution: Méchain and Delambre, and later Arago, were imprisoned several times during their surveys, and Méchain died in 1804 of yellow fever, which he contracted while trying to improve his original results in northern Spain. In the meantime, the commission of the French Academy of Sciences calculated a provisional value from older surveys of 443.44 ''
ligne The ''ligne'' ( ), or line or Paris line, is a historic unit of length used in France and elsewhere prior to the adoption of the metric system in the late 18th century, and used in various sciences after that time. The ''loi du 19 frimaire an ...
s''. This value was set by legislation on 7 April 1795. While Méchain and Delambre were completing their survey, the commission had ordered a series of
platinum Platinum is a chemical element with the symbol Pt and atomic number 78. It is a dense, malleable, ductile, highly unreactive, precious, silverish-white transition metal. Its name originates from Spanish , a diminutive of "silver". Platin ...
bars to be made based on the provisional metre. When the final result was known, the bar whose length was closest to the meridional definition of the metre was selected and placed in the National Archives on 22 June 1799 (4 messidor An VII in the Republican calendar) as a permanent record of the result. This standard metre bar became known as the Committee metre (French : '' Mètre des Archives'').


See also

*
Pendulum (mathematics) A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gr ...
* Kater's pendulum *
Metre Convention The Metre Convention (french: link=no, Convention du Mètre), also known as the Treaty of the Metre, is an international treaty that was signed in Paris on 20 May 1875 by representatives of 17 nations (Argentina, Austria-Hungary, Belgium, Braz ...


Notes


References

{{DEFAULTSORT:Seconds Pendulum Units of time Units of length Timekeeping components Pendulums