![Simple gravity pendulum](https://upload.wikimedia.org/wikipedia/commons/b/b2/Simple_gravity_pendulum.svg)
A pendulum is a weight suspended from a
pivot
Pivot may refer to:
*Pivot, the point of rotation in a lever system
*More generally, the center point of any rotational system
*Pivot joint, a kind of joint between bones in the body
*Pivot turn, a dance move
Companies
*Incitec Pivot, an Austra ...
so that it can swing freely. When a pendulum is displaced sideways from its resting,
equilibrium position, it is subject to a
restoring force
In physics, the restoring force is a force that acts to bring a body to its equilibrium position. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the s ...
due to
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to
oscillate
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
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
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 ...
. The period depends on the length of the pendulum and also to a slight degree on the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
, the width of the pendulum's swing.
From the first scientific investigations of the pendulum around 1602 by
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
, the regular motion of pendulums was used for timekeeping and was the world's most accurate timekeeping technology until the 1930s.
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 it ...
invented by
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 of ...
in 1658 became the world's standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by the
quartz clock
Quartz clocks and quartz watches are timepieces that use an electronic oscillator regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks and watches are at least an ...
in the 1930s. Pendulums are also used in
scientific instrument
A scientific instrument is a device or tool used for scientific purposes, including the study of both natural phenomena and theoretical research.
History
Historically, the definition of a scientific instrument has varied, based on usage, laws, an ...
s such as
accelerometer
An accelerometer is a tool that measures proper acceleration. Proper acceleration is the acceleration (the rate of change of velocity) of a body in its own instantaneous rest frame; this is different from coordinate acceleration, which is accele ...
s and
seismometer
A seismometer is an instrument that responds to ground noises and shaking such as caused by earthquakes, volcanic eruptions, and explosions. They are usually combined with a timing device and a recording device to form a seismograph. The outpu ...
s. Historically they were used as
gravimeter
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
s to measure the
acceleration of gravity in geo-physical surveys, and even as a standard of length. The word ''pendulum'' is
new Latin
New Latin (also called Neo-Latin or Modern Latin) is the revival of Literary Latin used in original, scholarly, and scientific works since about 1500. Modern scholarly and technical nomenclature, such as in zoological and botanical taxonomy ...
, from the Latin ''pendulus'', meaning ''hanging''.
Simple gravity pendulum
The ''simple gravity pendulum'' is an idealized mathematical model of a pendulum.
This is a weight (or
bob
Bob, BOB, or B.O.B. may refer to:
Places
*Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname) ...
) on the end of a massless cord suspended from a
pivot
Pivot may refer to:
*Pivot, the point of rotation in a lever system
*More generally, the center point of any rotational system
*Pivot joint, a kind of joint between bones in the body
*Pivot turn, a dance move
Companies
*Incitec Pivot, an Austra ...
, without
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
. When given an initial push, it will swing back and forth at a constant
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
. Real pendulums are subject to friction and
air drag
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
, so the amplitude of their swings declines.
Period of oscillation
The period of swing of a
simple gravity 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 ...
depends on its
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
, the local
strength of gravity, and to a small extent on the maximum
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
that the pendulum swings away from vertical, ''θ''
0, called the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
.
[, p.188-194] It is independent of the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of the bob. If the amplitude is limited to small swings,
[A "small" swing is one in which the angle is small enough that can be approximated by when is measured in radians] 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 a simple pendulum, the time taken for a complete cycle, is:
where
is the length of the pendulum and
is the local
acceleration of gravity.
For small swings the period of swing is approximately the same for different size swings: that is, ''the period is independent of amplitude''. This property, called
isochronism
A sequence of events is isochronous if the events occur regularly, or at equal time intervals. The term ''isochronous'' is used in several technical contexts, but usually refers to the primary subject maintaining a constant period or interval ( ...
, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
For larger
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
s, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of ''θ''
0 = 0.4 radians (23°) it is 1% larger than given by (1). The period increases asymptotically (to infinity) as ''θ''
0 approaches π radians (180°), because the value ''θ''
0 = π is an
unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see
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 ...
), one example being the
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
:
where
is in radians.
The difference between this true period and the period for small swings (1) above is called the ''circular error''. In the case of a typical
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 ...
whose pendulum has a swing of 6° and thus an amplitude of 3° (0.05 radians), the difference between the true period and the small angle approximation (1) amounts to about 15 seconds per day.
For small swings the pendulum approximates a
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
, and its motion as a function of time, ''t'', is approximately
simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
:
where
is a constant value, dependent on
initial conditions
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
.
For real pendulums, the period varies slightly with factors such as the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, and flexibility and stretching of the string.
In precision applications, corrections for these factors may need to be applied to eq. (1) to give the period accurately.
Compound pendulum
Any swinging
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum. The appropriate equivalent length
for calculating the period of any such pendulum is the distance from the pivot to the ''
center of oscillation The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
''.
[, Part 4, Proposition 5] This point is located under the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
at a distance from the pivot traditionally called the radius of oscillation, which depends on the mass distribution of the pendulum. If most of the mass is concentrated in a relatively small bob compared to the pendulum length, the center of oscillation is close to the center of mass.
The radius of oscillation or equivalent length
of any physical pendulum can be shown to be
where
is the
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
of the pendulum about the pivot point,
is the mass of the pendulum, and
is the distance between the pivot point and the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
.
Substituting this expression in (1) above, the period
of a compound pendulum is given by
for sufficiently small oscillations.
For example, a rigid uniform rod of length
pivoted about one end has moment of inertia
. The center of mass is located at the center of the rod, so
Substituting these values into the above equation gives
. This shows that a rigid rod pendulum has the same period as a simple pendulum of 2/3 its length.
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 of ...
proved in 1673 that the pivot point and the center of oscillation are interchangeable.
[Huygens (1673) Horologium Oscillatorium](_blank)
Part 4, Proposition 20 This means if any pendulum is turned upside down and swung from a pivot located at its previous center of oscillation, it will have the same period as before and the new center of oscillation will be at the old pivot point. In 1817
Henry Kater
Henry Kater FRS, FRAS (16 April 1777 – 26 April 1835) was a British physicist of German descent.
Early life
He was born at Bristol. At first he intended to study law; but he gave up the idea on his father's death in 1794. He entered the a ...
used this idea to produce a type of reversible pendulum, now known as a
Kater pendulum
A Kater's pendulum is a reversible free swinging pendulum invented by British physicist and army captain Henry Kater in 1817 for use as a gravimeter instrument to measure the local acceleration of gravity. Its advantage is that, unlike previous ...
, for improved measurements of the acceleration due to gravity.
History
![EastHanSeismograph](https://upload.wikimedia.org/wikipedia/commons/a/ae/EastHanSeismograph.JPG)
One of the earliest known uses of a pendulum was a 1st-century
seismometer
A seismometer is an instrument that responds to ground noises and shaking such as caused by earthquakes, volcanic eruptions, and explosions. They are usually combined with a timing device and a recording device to form a seismograph. The outpu ...
device of
Han Dynasty
The Han dynasty (, ; ) was an imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...
Chinese scientist
Zhang Heng
Zhang Heng (; AD 78–139), formerly romanized as Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, ma ...
.
[Morton, W. Scott and Charlton M. Lewis (2005). China: Its History and Culture. New York: McGraw-Hill, Inc., p. 70] Its function was to sway and activate one of a series of levers after being disturbed by the tremor of an
earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
far away.
[Needham, Volume 3, 627-629] Released by a lever, a small ball would fall out of the urn-shaped device into one of eight metal toad's mouths below, at the eight points of the compass, signifying the direction the earthquake was located.
Many sources claim that the 10th-century Egyptian astronomer
Ibn Yunus
Abu al-Hasan 'Ali ibn 'Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi al-Misri (Arabic: ابن يونس; c. 950 – 1009) was an important Egyptians, Egyptian astronomer and Islamic mathematics, mathematician, whose works are noted for being ahead o ...
used a pendulum for time measurement, but this was an error that originated in 1684 with the British historian
Edward Bernard
Edward Bernard (1638 – 12 January 1697) was an English scholar and Savilian professor of astronomy at the University of Oxford, from 1673 to 1691.
Life
He was born at Paulerspury, Northamptonshire. He was educated at Merchant Taylors' School, ...
.
During the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
, large hand-pumped pendulums were used as sources of power for manual reciprocating machines such as saws, bellows, and pumps.
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
made many drawings of the motion of pendulums, though without realizing its value for timekeeping.
1602: Galileo's research
Italian scientist
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
was the first to study the properties of pendulums, beginning around 1602.
The earliest extant report of his research is contained in a letter to Guido Ubaldo dal Monte, from Padua, dated November 29, 1602.
His biographer and student,
Vincenzo Viviani
Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo. , claimed his interest had been sparked around 1582 by the swinging motion of a chandelier in
Pisa Cathedral
Pisa Cathedral ( it, Cattedrale Metropolitana Primaziale di Santa Maria Assunta; Duomo di Pisa) is a medieval Roman Catholic cathedral dedicated to the Assumption of the Virgin Mary, in the Piazza dei Miracoli in Pisa, Italy, the oldest of the th ...
. Galileo discovered the crucial property that makes pendulums useful as timekeepers, called isochronism; the period of the pendulum is approximately independent of the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
or width of the swing.
He also found that the period is independent of the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of the bob, and proportional to the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
of the length of the pendulum. He first employed freeswinging pendulums in simple timing applications. His physician friend,
Santorio Santorii
Santorio Santori (29 March, 1561 – 25 February, 1636) also called Santorio Santorio, Santorio de' Sanctoriis, or Sanctorius of Padua and various combinations of these names, was an Italian physiologist, physician, and professor, who introduce ...
, invented a device which measured a patient's
pulse
In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...
by the length of a pendulum; the ''pulsilogium''.
In 1641 Galileo dictated to his son
Vincenzo
Vincenzo is an Italian male given name, derived from the Latin name Vincentius (the verb ''vincere'' means to win or to conquer). Notable people with the name include:
Art
* Vincenzo Amato (born 1966), Italian actor and sculptor
*Vincenzo Bell ...
a design for a pendulum clock;
Vincenzo began construction, but had not completed it when he died in 1649.
1656: The pendulum clock
In 1656 the Dutch scientist
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 of ...
built the first
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 it ...
. This was a great improvement over existing mechanical clocks; their best accuracy was improved from around 15 minutes deviation a day to around 15 seconds a day. Pendulums spread over Europe as existing clocks were
retrofitted
Retrofitting is the addition of new technology or features to older systems. Retrofits can happen for a number of reasons, for example with big capital expenditures like naval vessels, military equipment or manufacturing plants, businesses or go ...
with them.
The English scientist
Robert Hooke
Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
studied the
conical pendulum
A conical pendulum consists of a weight (or bob) fixed on the end of a string or rod suspended from a pivot. Its construction is similar to an ordinary pendulum; however, instead of swinging back and forth, the bob of a conical pendulum moves at ...
around 1666, consisting of a pendulum that is free to swing in two dimensions, with the bob rotating in a circle or ellipse.
He used the motions of this device as a model to analyze the
orbital motion
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
s of the
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s. Hooke suggested to
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 grea ...
in 1679 that the components of orbital motion consisted of inertial motion along a tangent direction plus an attractive motion in the radial direction. This played a part in Newton's formulation of the
law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
. Robert Hooke was also responsible for suggesting as early as 1666 that the pendulum could be used to measure the force of gravity.
During his expedition to
Cayenne
Cayenne (; ; gcr, Kayenn) is the capital city of French Guiana, an overseas region and Overseas department, department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic Oc ...
,
French Guiana
French Guiana ( or ; french: link=no, Guyane ; gcr, label=French Guianese Creole, Lagwiyann ) is an overseas departments and regions of France, overseas department/region and single territorial collectivity of France on the northern Atlantic ...
in 1671,
Jean Richer
Jean Richer (1630–1696) was a French astronomer and assistant (''élève astronome'') at the French Academy of Sciences, under the direction of Giovanni Domenico Cassini.
Between 1671 and 1673 he performed experiments and carried out celestial ...
found that a
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 it ...
was minutes per day slower at Cayenne than at Paris. From this he deduced that the force of gravity was lower at Cayenne. In 1687,
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 grea ...
in ''
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. ...
'' showed that this was because the Earth was not a true sphere but slightly
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 livi ...
(flattened at the poles) from the effect of
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 paralle ...
due to its rotation, causing gravity to increase with
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 pol ...
. Portable pendulums began to be taken on voyages to distant lands, as precision
gravimeter
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
s to measure the
acceleration of gravity at different points on Earth, eventually resulting in accurate models of the
shape 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 approxim ...
.
1673: Huygens' ''Horologium Oscillatorium''
In 1673, 17 years after he invented the pendulum clock,
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 of ...
published his theory of the pendulum, ''
Horologium Oscillatorium sive de motu pendulorum''.
Marin Mersenne
Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
and
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
had discovered around 1636 that the pendulum was not quite isochronous; its period increased somewhat with its amplitude.
Huygens analyzed this problem by determining what curve an object must follow to descend by gravity to the same point in the same time interval, regardless of starting point; the so-called ''
tautochrone curve
A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independe ...
''. By a complicated method that was an early use of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, he showed this curve was a
cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
, rather than the circular arc of a pendulum, confirming that the pendulum was not isochronous and Galileo's observation of isochronism was accurate only for small swings. Huygens also solved the problem of how to calculate the period of an arbitrarily shaped pendulum (called a ''compound pendulum''), discovering the ''
center of oscillation The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
'', and its interchangeability with the pivot point.
The existing clock movement, the
verge escapement
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 ...
, made pendulums swing in very wide arcs of about 100°.
Huygens showed this was a source of inaccuracy, causing the period to vary with amplitude changes caused by small unavoidable variations in the clock's drive force. To make its period isochronous, Huygens mounted cycloidal-shaped metal guides next to the pivots in his clocks, that constrained the suspension cord and forced the pendulum to follow a cycloid arc (see
cycloidal pendulum
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curv ...
).
[Andrewes, W.J.H]
''Clocks and Watches: The leap to precision''
in This solution didn't prove as practical as simply limiting the pendulum's swing to small angles of a few degrees. The realization that only small swings were
isochronous
A sequence of events is isochronous if the events occur regularly, or at equal time intervals. The term ''isochronous'' is used in several technical contexts, but usually refers to the primary subject maintaining a constant period or interval ( ...
motivated the development 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 swing in clocks to 4°–6°.
This became the standard escapement used in pendulum clocks.
1721: Temperature compensated pendulums
![Foucault pendulum animated](https://upload.wikimedia.org/wikipedia/commons/a/a1/Foucault_pendulum_animated.gif)
During the 18th and 19th century, 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 it ...
's role as the most accurate timekeeper motivated much practical research into improving pendulums. It was found that a major source of error was that the pendulum rod expanded and contracted with changes in ambient temperature, changing the period of swing.
This was solved with the invention of temperature compensated pendulums, the mercury pendulum in 1721
[ cited in ] and the
gridiron pendulum
The gridiron pendulum was a temperature-compensated clock pendulum invented by British clockmaker John Harrison around 1726 and later modified by John Ellicott. It was used in precision clocks. In ordinary clock pendulums, the pendulum rod expan ...
in 1726, reducing errors in precision pendulum clocks to a few seconds per week.
The accuracy of gravity measurements made with pendulums was limited by the difficulty of finding the location of their
center of oscillation The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
. Huygens had discovered in 1673 that a pendulum has the same period when hung from its center of oscillation as when hung from its pivot,
and the distance between the two points was equal to the length of a simple gravity pendulum of the same period.
In 1818 British Captain
Henry Kater
Henry Kater FRS, FRAS (16 April 1777 – 26 April 1835) was a British physicist of German descent.
Early life
He was born at Bristol. At first he intended to study law; but he gave up the idea on his father's death in 1794. He entered the a ...
invented the reversible
Kater's pendulum
A Kater's pendulum is a reversible free swinging pendulum invented by British physicist and army captain Henry Kater in 1817 for use as a gravimeter instrument to measure the local acceleration of gravity. Its advantage is that, unlike previous ...
which used this principle, making possible very accurate measurements of gravity. For the next century the reversible pendulum was the standard method of measuring absolute gravitational acceleration.
1851: Foucault pendulum
In 1851,
Jean Bernard Léon Foucault showed that the plane of oscillation of a pendulum, like a
gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rota ...
, tends to stay constant regardless of the motion of the pivot, and that this could be used to demonstrate the
rotation of the Earth
Earth's rotation or Earth's spin is the rotation of planet Earth around its own axis, as well as changes in the orientation of the rotation axis in space. Earth rotates eastward, in prograde motion. As viewed from the northern polar star Pola ...
. He suspended a pendulum free to swing in two dimensions (later named the
Foucault pendulum
The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist Léon Foucault, conceived as an experiment to demonstrate the Earth's rotation. A long and heavy pendulum suspended from the high roof above a circular a ...
) from the dome of the
Panthéon
The Panthéon (, from the Classical Greek word , , ' empleto all the gods') is a monument in the 5th arrondissement of Paris, France. It stands in the Latin Quarter, atop the , in the centre of the , which was named after it. The edifice was b ...
in Paris. The length of the cord was . Once the pendulum was set in motion, the plane of swing was observed to
precess
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
or rotate 360° clockwise in about 32 hours.
This was the first demonstration of the Earth's rotation that didn't depend on celestial observations, and a "pendulum mania" broke out, as Foucault pendulums were displayed in many cities and attracted large crowds.
1930: Decline in use
Around 1900 low-
thermal-expansion materials began to be used for pendulum rods in the highest precision clocks and other instruments, first
invar
Invar, also known generically as FeNi36 (64FeNi in the US), is a nickel–iron alloy notable for its uniquely low coefficient of thermal expansion (CTE or α). The name ''Invar'' comes from the word ''invariable'', referring to its relative lac ...
, a nickel steel alloy, and later
fused quartz
Fused quartz, fused silica or quartz glass is a glass consisting of almost pure silica (silicon dioxide, SiO2) in amorphous (non-crystalline) form. This differs from all other commercial glasses in which other ingredients are added which change ...
, which made temperature compensation trivial.
Precision pendulums were housed in low pressure tanks, which kept the air pressure constant to prevent changes in the period due to changes in
buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...
of the pendulum due to changing
atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, 7 ...
.
The best pendulum clocks achieved accuracy of around a second per year.
The timekeeping accuracy of the pendulum was exceeded by the
quartz
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...
crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses a piezoelectric crystal as a frequency-selective element. The oscillator frequency is often used to keep track of time, as in quartz wristwatches, to provide a stable cloc ...
, invented in 1921, and
quartz clock
Quartz clocks and quartz watches are timepieces that use an electronic oscillator regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks and watches are at least an ...
s, invented in 1927, replaced pendulum clocks as the world's best timekeepers.
Pendulum clocks were used as time standards until World War 2, although the French Time Service continued using them in their official time standard ensemble until 1954. Pendulum
gravimeter
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
s were superseded by "free fall" gravimeters in the 1950s,
but pendulum instruments continued to be used into the 1970s.
Use for time measurement
For 300 years, from its discovery around 1582 until development of the
quartz clock
Quartz clocks and quartz watches are timepieces that use an electronic oscillator regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks and watches are at least an ...
in the 1930s, the pendulum was the world's standard for accurate timekeeping.
In addition to clock pendulums, freeswinging
seconds pendulum
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.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
s were widely used as precision timers in scientific experiments in the 17th and 18th centuries. Pendulums require great mechanical stability: a length change of only 0.02%, 0.2 mm in a grandfather clock pendulum, will cause an error of a minute per week.
Clock pendulums
Pendulums in clocks (see example at right) are usually made of a weight or
bob
Bob, BOB, or B.O.B. may refer to:
Places
*Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname) ...
''
(b)'' suspended by a rod of wood or metal ''
(a)''.
To reduce
air resistance
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
(which accounts for most of the energy loss in precision clocks) the bob is traditionally a smooth disk with a lens-shaped cross section, although in antique clocks it often had carvings or decorations specific to the type of clock. In quality clocks the bob is made as heavy as the suspension can support and the movement can drive, since this improves the regulation of the clock (see
Accuracy
Accuracy and precision are two measures of ''observational error''.
''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...
below). A common weight for
seconds pendulum
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.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
bobs is . Instead of hanging from a
pivot
Pivot may refer to:
*Pivot, the point of rotation in a lever system
*More generally, the center point of any rotational system
*Pivot joint, a kind of joint between bones in the body
*Pivot turn, a dance move
Companies
*Incitec Pivot, an Austra ...
, clock pendulums are usually supported by a short straight
spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a ...
''
(d)'' of flexible metal ribbon. This avoids the friction and 'play' caused by a pivot, and the slight bending force of the spring merely adds to the pendulum's
restoring force
In physics, the restoring force is a force that acts to bring a body to its equilibrium position. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the s ...
. The highest precision clocks have pivots of 'knife' blades resting on agate plates. The impulses to keep the pendulum swinging are provided by an arm hanging behind the pendulum called the ''crutch'', ''
(e)'', which ends in a ''fork'', ''
(f)'' whose prongs embrace the pendulum rod. The crutch is pushed back and forth by the clock's
escapement
An escapement is a mechanical linkage in mechanical watches and clocks that gives impulses to the timekeeping element and periodically releases the gear train to move forward, advancing the clock's hands. The impulse action transfers energy to ...
, ''
(g,h)''.
Each time the pendulum swings through its centre position, it releases one tooth of the ''escape wheel'' ''
(g)''. The force of the clock's
mainspring
A mainspring is a spiral torsion spring of metal ribbon—commonly spring steel—used as a power source in mechanical watches, some clocks, and other clockwork mechanisms. ''Winding'' the timepiece, by turning a knob or key, stores energy in ...
or a driving weight hanging from a pulley, transmitted through the clock's
gear train
A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage.
Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission ...
, causes the wheel to turn, and a tooth presses against one of the pallets ''
(h)'', giving the pendulum a short push. The clock's wheels, geared to the escape wheel, move forward a fixed amount with each pendulum swing, advancing the clock's hands at a steady rate.
The pendulum always has a means of adjusting the period, usually by an adjustment nut ''
(c)'' under the bob which moves it up or down on the rod.
Moving the bob up decreases the pendulum's length, causing the pendulum to swing faster and the clock to gain time. Some precision clocks have a small auxiliary adjustment weight on a threaded shaft on the bob, to allow finer adjustment. Some
tower clocks and precision clocks use a tray attached near to the midpoint of the pendulum rod, to which small weights can be added or removed. This effectively shifts the centre of oscillation and allows the rate to be adjusted without stopping the clock.
The pendulum must be suspended from a rigid support.
During operation, any elasticity will allow tiny imperceptible swaying motions of the support, which disturbs the clock's period, resulting in error. Pendulum clocks should be attached firmly to a sturdy wall.
The most common pendulum length in quality clocks, which is always used in
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, is the
seconds pendulum
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.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
, about long. In
mantel clock
Mantel clocks—or shelf clocks—are relatively small house clocks traditionally placed on the shelf, or mantel, above the fireplace. The form, first developed in France in the 1750s, can be distinguished from earlier chamber clocks of simila ...
s, half-second pendulums, long, or shorter, are used. Only a few large
tower clocks use longer pendulums, the 1.5 second pendulum, long, or occasionally the two-second pendulum,
which is used in
Big Ben
Big Ben is the nickname for the Great Bell of the Great Clock of Westminster, at the north end of the Palace of Westminster in London, England, and the name is frequently extended to refer also to the clock and the clock tower. The officia ...
.
Temperature compensation
![Howard astronomical regulator clock](https://upload.wikimedia.org/wikipedia/commons/1/18/Howard_astronomical_regulator_clock.png)
The largest source of error in early pendulums was slight changes in length due to thermal expansion and contraction of the pendulum rod with changes in ambient temperature. This was discovered when people noticed that pendulum clocks ran slower in summer, by as much as a minute per week
(one of the first was
Godefroy Wendelin
Godfried Wendelen or Govaert Wendelen, Latinized Godefridus Wendelinus, or sometimes Vendelinus and in French-language sources referred to as Godefroy Wendelin (6 June 1580 – 24 October 1667) was an astronomer and Catholic priest from Lièg ...
, as reported by Huygens in 1658). Thermal expansion of pendulum rods was first studied by
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, base ...
in 1669.
A pendulum with a steel rod will expand by about 11.3
parts per million
In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they ...
(ppm) with each degree Celsius increase, causing it to lose about 0.27 seconds per day for every degree Celsius increase in temperature, or 9 seconds per day for a change. Wood rods expand less, losing only about 6 seconds per day for a change, which is why quality clocks often had wooden pendulum rods. The wood had to be varnished to prevent water vapor from getting in, because changes in humidity also affected the length.
Mercury pendulum
The first device to compensate for this error was the mercury pendulum, invented by
George Graham
George Graham (born 30 November 1944), nicknamed "Stroller", is a Scottish former Association football, football player and manager (association football), manager.
In his successful playing career, he made 455 appearances in England's Football ...
in 1721.
The liquid metal
mercury
Mercury commonly refers to:
* Mercury (planet), the nearest planet to the Sun
* Mercury (element), a metallic chemical element with the symbol Hg
* Mercury (mythology), a Roman god
Mercury or The Mercury may also refer to:
Companies
* Merc ...
expands in volume with temperature. In a mercury pendulum, the pendulum's weight (bob) is a container of mercury. With a temperature rise, the pendulum rod gets longer, but the mercury also expands and its surface level rises slightly in the container, moving its
centre of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
closer to the pendulum pivot. By using the correct height of mercury in the container these two effects will cancel, leaving the pendulum's centre of mass, and its period, unchanged with temperature. Its main disadvantage was that when the temperature changed, the rod would come to the new temperature quickly but the mass of mercury might take a day or two to reach the new temperature, causing the rate to deviate during that time.
[Matthys 2004](_blank)
p.7-12 To improve thermal accommodation several thin containers were often used, made of metal. Mercury pendulums were the standard used in precision regulator clocks into the 20th century.
Gridiron pendulum
The most widely used compensated pendulum was the
gridiron pendulum
The gridiron pendulum was a temperature-compensated clock pendulum invented by British clockmaker John Harrison around 1726 and later modified by John Ellicott. It was used in precision clocks. In ordinary clock pendulums, the pendulum rod expan ...
, invented in 1726 by
John Harrison
John Harrison ( – 24 March 1776) was a self-educated English Carpentry, carpenter and clockmaker who invented the marine chronometer, a long-sought-after device for solving the History of longitude, problem of calculating longitude while at s ...
.
This consists of alternating rods of two different metals, one with lower thermal expansion (
CTE),
steel
Steel is an alloy made up of iron with added carbon to improve its strength and fracture resistance compared to other forms of iron. Many other elements may be present or added. Stainless steels that are corrosion- and oxidation-resistant ty ...
, and one with higher thermal expansion,
zinc
Zinc is a chemical element with the symbol Zn and atomic number 30. Zinc is a slightly brittle metal at room temperature and has a shiny-greyish appearance when oxidation is removed. It is the first element in group 12 (IIB) of the periodi ...
or
brass
Brass is an alloy of copper (Cu) and zinc (Zn), in proportions which can be varied to achieve different mechanical, electrical, and chemical properties. It is a substitutional alloy: atoms of the two constituents may replace each other with ...
. The rods are connected by a frame, as shown in the drawing at the right, so that an increase in length of the zinc rods pushes the bob up, shortening the pendulum. With a temperature increase, the low expansion steel rods make the pendulum longer, while the high expansion zinc rods make it shorter. By making the rods of the correct lengths, the greater expansion of the zinc cancels out the expansion of the steel rods which have a greater combined length, and the pendulum stays the same length with temperature.
Zinc-steel gridiron pendulums are made with 5 rods, but the thermal expansion of brass is closer to steel, so brass-steel gridirons usually require 9 rods. Gridiron pendulums adjust to temperature changes faster than mercury pendulums, but scientists found that friction of the rods sliding in their holes in the frame caused gridiron pendulums to adjust in a series of tiny jumps.
In high precision clocks this caused the clock's rate to change suddenly with each jump. Later it was found that zinc is subject to
creep. For these reasons mercury pendulums were used in the highest precision clocks, but gridirons were used in quality regulator clocks.
Gridiron pendulums became so associated with good quality that, to this day, many ordinary clock pendulums have decorative 'fake' gridirons that don't actually have any temperature compensation function.
Invar and fused quartz
Around 1900, low thermal expansion materials were developed which could be used as pendulum rods in order to make elaborate temperature compensation unnecessary.
These were only used in a few of the highest precision clocks before the pendulum became obsolete as a time standard. In 1896
Charles Édouard Guillaume
Charles Édouard Guillaume (15 February 1861, in Fleurier, Switzerland – 13 May 1938, in Sèvres, France) was a Swiss physicist who received the Nobel Prize in Physics in 1920 in recognition of the service he had rendered to precision measuremen ...
invented the
nickel
Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow to ...
steel
Steel is an alloy made up of iron with added carbon to improve its strength and fracture resistance compared to other forms of iron. Many other elements may be present or added. Stainless steels that are corrosion- and oxidation-resistant ty ...
alloy
An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductility, ...
Invar
Invar, also known generically as FeNi36 (64FeNi in the US), is a nickel–iron alloy notable for its uniquely low coefficient of thermal expansion (CTE or α). The name ''Invar'' comes from the word ''invariable'', referring to its relative lac ...
. This has a
CTE of around 0.5 µin/(in·°F), resulting in pendulum temperature errors over 71 °F of only 1.3 seconds per day, and this residual error could be compensated to zero with a few centimeters of aluminium under the pendulum bob
(this can be seen in the Riefler clock image above). Invar pendulums were first used in 1898 in the
Riefler regulator clock
The Riefler escapement is a mechanical escapement for precision pendulum clocks invented and patented by German instrument maker Sigmund Riefler in 1889. It was used in the astronomical regulator clocks made by his German firm Clemens Riefler f ...
which achieved accuracy of 15 milliseconds per day. Suspension springs of
Elinvar
Elinvar is a nickel–iron–chromium alloy notable for having a modulus of elasticity which does not change much with temperature changes. The name is a contraction of the French ('invariable elasticity'). It was invented by Charles Édouard Guil ...
were used to eliminate temperature variation of the spring's
restoring force
In physics, the restoring force is a force that acts to bring a body to its equilibrium position. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the s ...
on the pendulum. Later
fused quartz
Fused quartz, fused silica or quartz glass is a glass consisting of almost pure silica (silicon dioxide, SiO2) in amorphous (non-crystalline) form. This differs from all other commercial glasses in which other ingredients are added which change ...
was used which had even lower CTE. These materials are the choice for modern high accuracy pendulums.
Atmospheric pressure
The effect of the surrounding air on a moving pendulum is complex and requires
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
to calculate precisely, but for most purposes its influence on the period can be accounted for by three effects:
* By
Archimedes' principle
Archimedes' principle (also spelled Archimedes's principle) states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Archimede ...
the effective
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
of the
bob
Bob, BOB, or B.O.B. may refer to:
Places
*Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname) ...
is reduced by the buoyancy of the air it displaces, while the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
(
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
) remains the same, reducing the pendulum's acceleration during its swing and increasing the period. This depends on the air pressure and the density of the pendulum, but not its shape.
* The pendulum carries an amount of air with it as it swings, and the mass of this air increases the inertia of the pendulum, again reducing the acceleration and increasing the period. This depends on both its density and shape.
* Viscous
air resistance
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
slows the pendulum's velocity. This has a negligible effect on the period, but dissipates energy, reducing the amplitude. This reduces the pendulum's
Q factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
, requiring a stronger drive force from the clock's mechanism to keep it moving, which causes increased disturbance to the period.
Increases in
barometric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, 7 ...
increase a pendulum's period slightly due to the first two effects, by about 0.11 seconds per day per kilopascal (0.37 seconds per day per
inch of mercury
Inch of mercury (inHg and ″Hg) is a non- SI unit of measurement for pressure. It is used for barometric pressure in weather reports, refrigeration and aviation in the United States.
It is the pressure exerted by a column of mercury in heigh ...
or 0.015 seconds per day per
torr
The torr (symbol: Torr) is a unit of pressure based on an absolute scale, defined as exactly of a standard atmosphere (). Thus one torr is exactly (≈ ).
Historically, one torr was intended to be the same as one "millimeter of mercury ...
).
Researchers using pendulums to measure the
acceleration of gravity had to correct the period for the air pressure at the altitude of measurement, computing the equivalent period of a pendulum swinging in vacuum. A pendulum clock was first operated in a constant-pressure tank by Friedrich Tiede in 1865 at the
Berlin Observatory
The Berlin Observatory (Berliner Sternwarte) is a German astronomical institution with a series of observatories and related organizations in and around the city of Berlin in Germany, starting from the 18th century. It has its origins in 1700 w ...
, and by 1900 the highest precision clocks were mounted in tanks that were kept at a constant pressure to eliminate changes in atmospheric pressure. Alternatively, in some a small
aneroid barometer
A barometer is a scientific instrument that is used to measure air pressure in a certain environment. Pressure tendency can forecast short term changes in the weather. Many measurements of air pressure are used within surface weather analysis ...
mechanism attached to the pendulum compensated for this effect.
Gravity
Pendulums are affected by changes in gravitational acceleration, which varies by as much as 0.5% at different locations on Earth, so precision pendulum clocks have to be recalibrated after a move. Even moving a pendulum clock to the top of a tall building can cause it to lose measurable time from the reduction in gravity.
Accuracy of pendulums as timekeepers
The timekeeping elements in all clocks, which include pendulums,
balance wheel
A balance wheel, or balance, is the timekeeping device used in mechanical watches and small clocks, analogous to the pendulum in a pendulum clock. It is a weighted wheel that rotates back and forth, being returned toward its center position by a ...
s, the
quartz crystals
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...
used in
quartz watch
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen Tetrahedral molecular geometry, tetrahedra, with each oxygen being shared between two tetrahedra, ...
es, and even the vibrating atoms in
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, are in physics called
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
s. The reason harmonic oscillators are used in clocks is that they vibrate or oscillate at a specific
resonant frequency
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
or period and resist oscillating at other rates. However, the resonant frequency is not infinitely 'sharp'. Around the resonant frequency there is a narrow natural band of
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(or periods), called the
resonance width
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
or
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
, where the harmonic oscillator will oscillate.
[ p.39] In a clock, the actual frequency of the pendulum may vary randomly within this resonance width in response to disturbances, but at frequencies outside this band, the clock will not function at all. The resonance width is determined by the
damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
, the
frictional energy loss per swing of the pendulum.
''Q'' factor
![Shortt Synchronome free pendulum clock](https://upload.wikimedia.org/wikipedia/commons/e/e4/Shortt_Synchronome_free_pendulum_clock.jpg)
The measure of a harmonic oscillator's resistance to disturbances to its oscillation period is a dimensionless parameter called the
''Q'' factor equal to the resonant frequency divided by the
resonance width
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
.
The higher the ''Q'', the smaller the resonance width, and the more constant the frequency or period of the oscillator for a given disturbance. The reciprocal of the Q is roughly proportional to the limiting accuracy achievable by a harmonic oscillator as a time standard.
The ''Q'' is related to how long it takes for the oscillations of an oscillator to die out. The
''Q'' of a pendulum can be measured by counting the number of oscillations it takes for the amplitude of the pendulum's swing to decay to 1/''e'' = 36.8% of its initial swing, and multiplying by 2''π''.
In a clock, the pendulum must receive pushes from the clock's
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 ...
to keep it swinging, to replace the energy the pendulum loses to friction. These pushes, applied by a mechanism called the
escapement
An escapement is a mechanical linkage in mechanical watches and clocks that gives impulses to the timekeeping element and periodically releases the gear train to move forward, advancing the clock's hands. The impulse action transfers energy to ...
, are the main source of disturbance to the pendulum's motion. The ''Q'' is equal to 2''π'' times the energy stored in the pendulum, divided by the energy lost to friction during each oscillation period, which is the same as the energy added by the escapement each period. It can be seen that the smaller the fraction of the pendulum's energy that is lost to friction, the less energy needs to be added, the less the disturbance from the escapement, the more 'independent' the pendulum is of the clock's mechanism, and the more constant its period is. The ''Q'' of a pendulum is given by:
where ''M'' is the mass of the bob, is the pendulum's radian frequency of oscillation, and Γ is the frictional
damping force on the pendulum per unit velocity.
''ω'' is fixed by the pendulum's period, and ''M'' is limited by the load capacity and rigidity of the suspension. So the ''Q'' of clock pendulums is increased by minimizing frictional losses (Γ). Precision pendulums are suspended on low friction pivots consisting of triangular shaped 'knife' edges resting on agate plates. Around 99% of the energy loss in a freeswinging pendulum is due to air friction, so mounting a pendulum in a vacuum tank can increase the ''Q'', and thus the accuracy, by a factor of 100.
The ''Q'' of pendulums ranges from several thousand in an ordinary clock to several hundred thousand for precision regulator pendulums swinging in vacuum.
A quality home pendulum clock might have a ''Q'' of 10,000 and an accuracy of 10 seconds per month. The most accurate commercially produced pendulum clock was the
Shortt-Synchronome free pendulum clock, invented in 1921.
Its
Invar
Invar, also known generically as FeNi36 (64FeNi in the US), is a nickel–iron alloy notable for its uniquely low coefficient of thermal expansion (CTE or α). The name ''Invar'' comes from the word ''invariable'', referring to its relative lac ...
master pendulum swinging in a vacuum tank had a ''Q'' of 110,000
and an error rate of around a second per year.
Their Q of 10
3–10
5 is one reason why pendulums are more accurate timekeepers than the
balance wheel
A balance wheel, or balance, is the timekeeping device used in mechanical watches and small clocks, analogous to the pendulum in a pendulum clock. It is a weighted wheel that rotates back and forth, being returned toward its center position by a ...
s in watches, with ''Q'' around 100–300, but less accurate than the
quartz crystals
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...
in
quartz clock
Quartz clocks and quartz watches are timepieces that use an electronic oscillator regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks and watches are at least an ...
s, with ''Q'' of 10
5–10
6.
Escapement
Pendulums (unlike, for example, quartz crystals) have a low enough ''Q'' that the disturbance caused by the impulses to keep them moving is generally the limiting factor on their timekeeping accuracy. Therefore, the design of the
escapement
An escapement is a mechanical linkage in mechanical watches and clocks that gives impulses to the timekeeping element and periodically releases the gear train to move forward, advancing the clock's hands. The impulse action transfers energy to ...
, the mechanism that provides these impulses, has a large effect on the accuracy of a clock pendulum. If the impulses given to the pendulum by the escapement each swing could be exactly identical, the response of the pendulum would be identical, and its period would be constant. However, this is not achievable; unavoidable random fluctuations in the force due to friction of the clock's pallets, lubrication variations, and changes in the torque provided by the clock's power source as it runs down, mean that the force of the impulse applied by the escapement varies.
If these variations in the escapement's force cause changes in the pendulum's width of swing (amplitude), this will cause corresponding slight changes in the period, since (as discussed at top) a pendulum with a finite swing is not quite isochronous. Therefore, the goal of traditional escapement design is to apply the force with the proper profile, and at the correct point in the pendulum's cycle, so force variations have no effect on the pendulum's amplitude. This is called an ''isochronous escapement''.
The Airy condition
Clockmakers had known for centuries that the disturbing effect of the escapement's drive force on the period of a pendulum is smallest if given as a short impulse as the pendulum passes through its bottom
equilibrium position.
If the impulse occurs before the pendulum reaches bottom, during the downward swing, it will have the effect of shortening the pendulum's natural period, so an increase in drive force will decrease the period. If the impulse occurs after the pendulum reaches bottom, during the upswing, it will lengthen the period, so an increase in drive force will increase the pendulum's period. In 1826 British astronomer
George Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
proved this; specifically, he proved that if a pendulum is driven by an impulse that is
symmetrical
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
about its bottom equilibrium position, the pendulum's period will be unaffected by changes in the drive force. The most accurate escapements, such as the
deadbeat, approximately satisfy this condition.
Gravity measurement
The presence of the
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
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
to measure the local
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, which varies by over 0.5% across the surface of the Earth.
[The value of "g" (acceleration due to gravity) at the ]equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
is 9.780 m/s2 and at the poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
is 9.832 m/s2, a difference of 0.53%. 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
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
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
![Pendulum2secondclock](https://upload.wikimedia.org/wikipedia/commons/f/f2/Pendulum2secondclock.gif)
The
seconds pendulum
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.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
, 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
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 it ...
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:
Early observations
* 1620: British scientist
Francis Bacon
Francis Bacon, 1st Viscount St Alban (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England. Bacon led the advancement of both ...
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.
* 1644: Even before the pendulum clock, French priest
Marin Mersenne
Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
first determined the length of the seconds pendulum was , 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
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, base ...
determined the length of the seconds pendulum at Paris, using a copper ball suspended by an aloe fiber, obtaining .
[Poynting & Thompson 1907, p.9](_blank)
/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
Jean Richer (1630–1696) was a French astronomer and assistant (''élève astronome'') at the French Academy of Sciences, under the direction of Giovanni Domenico Cassini.
Between 1671 and 1673 he performed experiments and carried out celestial ...
, who took a 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 it ...
to Cayenne
Cayenne (; ; gcr, Kayenn) is the capital city of French Guiana, an overseas region and Overseas department, department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic Oc ...
, French Guiana
French Guiana ( or ; french: link=no, Guyane ; gcr, label=French Guianese Creole, Lagwiyann ) is an overseas departments and regions of France, overseas department/region and single territorial collectivity of France on the northern Atlantic ...
and found that it lost minutes per day; its seconds pendulum had to be shortened by ''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 V ...
s'' (2.6 mm) shorter than at Paris, to keep correct time. In 1687 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 grea ...
in ''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. ...
'' showed this was because the Earth had a slightly 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 livi ...
shape (flattened at the poles) caused by 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 paralle ...
of its rotation. At higher latitudes the surface was closer to the center of the Earth, so gravity increased with latitude. 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
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 ...
created the first hydrostatic model of the Earth, 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 treatis ...
, which allowed the ellipticity
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
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
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
(inertia). This principle, called the equivalence principle
In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...
, confirmed to greater accuracy in later experiments, became the foundation on which Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
based his general theory of relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
.
* 1737: French mathematician Pierre Bouguer
Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture".
Career
Bouguer's father, Jean Bouguer, one ...
made a sophisticated series of pendulum observations in the Andes
The Andes, Andes Mountains or Andean Mountains (; ) are the longest continental mountain range in the world, forming a continuous highland along the western edge of South America. The range is long, wide (widest between 18°S – 20°S ...
mountains, Peru.[Poynting & Thompson, 1907, p.10](_blank)
/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
The Altiplano (Spanish for "high plain"), Collao (Quechua and Aymara: Qullaw, meaning "place of the Qulla") or Andean Plateau, in west-central South America, is the most extensive high plateau on Earth outside Tibet. The plateau is located at the ...
''. 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 density of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...
.
* 1747: Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
showed how to correct for the lengthening of the period due to a finite angle of swing ''θ''0 by using the first order correction ''θ''02/16, giving the period of a pendulum with an extremely small swing.
* 1792: To define a pendulum standard of length for use with the new metric system
The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
, in 1792 Jean-Charles de Borda
Jean-Charles, chevalier de Borda (4 May 1733 – 19 February 1799) was a French mathematician, physicist, and Navy officer.
Biography
Borda was born in the city of Dax to Jean‐Antoine de Borda and Jeanne‐Marie Thérèse de Lacroix.
In 175 ...
and Jean-Dominique Cassini made a precise measurement of the seconds pendulum at Paris. They used a -inch (14 mm) platinum ball suspended by a 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''1 and ''T''2, could be calculated:
* 1821: Francesco Carlini
Francesco Carlini (January 7, 1783 – August 29, 1862) was an Italian astronomer. Born in Milan, he became director of the Brera Astronomical Observatory there in 1832. He published ''Nuove tavole de moti apparenti del sole'' in 1832. In 1810, ...
made pendulum observations on top of Mount Cenis, Italy, from which, using methods similar to Bouguer's, he calculated the density of the Earth. 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 in diameter and 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
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 pivot
Pivot may refer to:
*Pivot, the point of rotation in a lever system
*More generally, the center point of any rotational system
*Pivot joint, a kind of joint between bones in the body
*Pivot turn, a dance move
Companies
*Incitec Pivot, an Austra ...
point and a point called the center of oscillation The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
, located under the center of gravity
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...
, 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.
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. 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
Henry Kater FRS, FRAS (16 April 1777 – 26 April 1835) was a British physicist of German descent.
Early life
He was born at Bristol. At first he intended to study law; but he gave up the idea on his father's death in 1794. He entered the a ...
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. 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 (shown at right) consisting of a brass bar with two opposing pivots made of short triangular "knife" blades ''(a)'' 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 ''(b,c)''. 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 inches for the seconds pendulum at London, in vacuum, at sea level, at 62 °F. The largest variation from the mean of his 12 observations was 0.00028 in. representing a precision of gravity measurement of 7×10−6 (7 mGal or 70 µm/s2). Kater's measurement was used as Britain's official standard of length (see below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
) 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
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
a standard part of geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
. Since the exact location (latitude and longitude) of the 'station' where the gravity measurement was made was necessary, gravity measurements became part of surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
, and pendulums were taken on the great geodetic surveys
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiva ...
of the 18th century, particularly the Great Trigonometric Survey
The Great Trigonometrical Survey was a project that aimed to survey the entire Indian subcontinent with scientific precision. It was begun in 1802 by the British infantry officer William Lambton, under the auspices of the East India Company.Gil ...
of India.
* Invariable pendulums: Kater introduced the idea of ''relative'' gravity measurements, to supplement the ''absolute'' measurements made by a Kater's pendulum. 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
The King's Observatory (called for many years the Kew Observatory) is a Grade I listed building in Richmond, London. Now a private dwelling, it formerly housed an astronomical and terrestrial magnetic observatory founded by King George III. T ...
, UK.
* Airy's coal pit experiments: Starting in 1826, using methods similar to Bouguer, British astronomer George Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
attempted to determine the density of the Earth by pendulum gravity measurements at the top and bottom of a coal mine. The gravitational force below the surface of the Earth decreases rather than increasing with depth, because by Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
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 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.
* 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
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ...
showed in 1835 that this was unnecessary. 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. 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 cm long and had a period of about 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.
* 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 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
An electric spark is an abrupt electrical discharge that occurs when a sufficiently high electric field creates an ionized, electrically conductive channel through a normally-insulating medium, often air or other gases or gas mixtures. Michael F ...
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
The United States Coast and Geodetic Survey (abbreviated USC&GS), known from 1807 to 1836 as the Survey of the Coast and from 1836 until 1878 as the United States Coast Survey, was the first scientific agency of the United States Government. It ...
in 1890, 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
Albert Abraham Michelson Royal Society of London, FFRS HFRSE (surname pronunciation anglicized as "Michael-son", December 19, 1852 – May 9, 1931) was a German-born American physicist of Polish/Jewish origin, known for his work on measuring the ...
in his 1924 measurements of the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
on Mt. Wilson, California.
* 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 mm due to this error. In 1880 C. Defforges used a Michelson interferometer
The Michelson interferometer is a common configuration for optical interferometry and was invented by the 19/20th-century American physicist Albert Abraham Michelson. Using a beam splitter, a light source is split into two arms. Each of those li ...
to measure the sway of the stand dynamically, and interferometers were added to the standard Mendenhall apparatus to calculate sway corrections. 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.
* Gulf gravimeter: One of the last and most accurate pendulum gravimeters was the apparatus developed in 1929 by the Gulf Research and Development Co. It used two pendulums made of fused quartz
Fused quartz, fused silica or quartz glass is a glass consisting of almost pure silica (silicon dioxide, SiO2) in amorphous (non-crystalline) form. This differs from all other commercial glasses in which other ingredients are added which change ...
, each 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
A crystal oscillator is an electronic oscillator circuit that uses a piezoelectric crystal as a frequency-selective element. The oscillator frequency is often used to keep track of time, as in quartz wristwatches, to provide a stable cloc ...
calibrated against the WWV radio time signal. This instrument was accurate to within (0.3–0.5)×10−7 (30–50 microgal
The gal (symbol: Gal), sometimes called galileo after Galileo Galilei, is a unit of acceleration sometimes used in gravimetry.BIPM ''SI brochure'', 8th ed. 2006Table 9: Non-SI units associated with the CGS and the CGS-Gaussian system of units. T ...
s or 3–5 nm/s2). 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 Lucien LaCoste (1908 – 1995) was a physicist and metrologist. He coinvented the modern gravimeter and invented the zero-length spring and vehicle-mounted gravimeters. He was also co-founder of LaCoste Romberg, a prominent company selling grav ...
. 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
Interferometry is a technique which uses the ''interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber op ...
.
Standard of length
Because the acceleration of gravity is constant at a given point on Earth, the period of a simple pendulum at a given location depends only on its length. Additionally, gravity varies only slightly at different locations. Almost from the pendulum's discovery until the early 19th century, this property led scientists to suggest using a pendulum of a given 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 ...
as a standard of length.
Until the 19th century, countries based their systems of length measurement on prototypes, metal bar primary standard A primary standard in metrology is a standard that is sufficiently accurate such that it is not calibrated by or subordinate to other standards. Primary standards are defined via other quantities like length, mass and time. Primary standards are u ...
s, such as the standard yard
The yard (symbol: yd) is an English unit of length in both the British imperial and US customary systems of measurement equalling 3 feet or 36 inches. Since 1959 it has been by international agreement standardized as exactly 0.914 ...
in Britain kept at the Houses of Parliament, and the standard ''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 (''Acadi ...
'' in France, kept at Paris. These were vulnerable to damage or destruction over the years, and because of the difficulty of comparing prototypes, the same unit often had different lengths in distant towns, creating opportunities for fraud. During the Enlightenment scientists argued for a length standard that was based on some property of nature that could be determined by measurement, creating an indestructible, universal standard. The period of pendulums could be measured very precisely by timing them with clocks that were set by the stars. A pendulum standard amounted to defining the unit of length by the gravitational force of the Earth, for all intents constant, and the second, which was defined by the rotation rate of the Earth, also constant. The idea was that anyone, anywhere on Earth, could recreate the standard by constructing a pendulum that swung with the defined period and measuring its length.
Virtually all proposals were based on the seconds pendulum
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.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
, in which each swing (a half 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 ...
) takes one second, which is about a meter (39 inches) long, because by the late 17th century it had become a standard for measuring gravity (see previous section). By the 18th century its length had been measured with sub-millimeter accuracy at a number of cities in Europe and around the world.
The initial attraction of the pendulum length standard was that it was believed (by early scientists such as Huygens and Wren) that gravity was constant over the Earth's surface, so a given pendulum had the same period at any point on Earth. So the length of the standard pendulum could be measured at any location, and would not be tied to any given nation or region; it would be a truly democratic, worldwide standard. Although Richer found in 1672 that gravity varies at different points on the globe, the idea of a pendulum length standard remained popular, because it was found that gravity only varies with 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 pol ...
. Gravitational acceleration increases smoothly from the equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
to the poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
, due to the 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 livi ...
shape of the Earth, so at any given latitude (east–west line), gravity was constant enough that the length of a seconds pendulum was the same within the measurement capability of the 18th century. Thus the unit of length could be defined at a given latitude and measured at any point along that latitude. For example, a pendulum standard defined at 45° north latitude, a popular choice, could be measured in parts of France, Italy, Croatia, Serbia, Romania, Russia, Kazakhstan, China, Mongolia, the United States and Canada. In addition, it could be recreated at any location at which the gravitational acceleration had been accurately measured.
By the mid 19th century, increasingly accurate pendulum measurements by Edward Sabine
Sir Edward Sabine ( ; 14 October 1788 – 26 June 1883) was an Irish astronomer, geophysicist, ornithologist, explorer, soldier and the 30th president of the Royal Society.
He led the effort to establish a system of magnetic observatories in ...
and Thomas Young revealed that gravity, and thus the length of any pendulum standard, varied measurably with local geologic features such as mountains and dense subsurface rocks. So a pendulum length standard had to be defined at a single point on Earth and could only be measured there. This took much of the appeal from the concept, and efforts to adopt pendulum standards were abandoned.
Early proposals
One of the first to suggest defining length with a pendulum was Flemish scientist Isaac Beeckman
Isaac Beeckman (10 December 1588van Berkel, p10 – 19 May 1637) was a Dutch philosopher and scientist, who, through his studies and contact with leading natural philosophers, may have "virtually given birth to modern atomism".Harold J. Cook, in ...
who in 1631 recommended making the seconds pendulum "the invariable measure for all people at all times in all places". Marin Mersenne
Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
, who first measured the seconds pendulum in 1644, also suggested it. The first official proposal for a pendulum standard was made by the British Royal Society
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, re ...
in 1660, advocated by 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 of ...
and Ole Rømer
Ole Christensen Rømer (; 25 September 1644 – 19 September 1710) was a Danish astronomer who, in 1676, made the first measurement of the speed of light.
Rømer also invented the modern thermometer showing the temperature between two fix ...
, basing it on Mersenne's work, and Huygens in ''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 ...
'' proposed a "horary foot" defined as 1/3 of the seconds pendulum. Christopher Wren
Sir Christopher Wren PRS FRS (; – ) was one of the most highly acclaimed English architects in history, as well as an anatomist, astronomer, geometer, and mathematician-physicist. He was accorded responsibility for rebuilding 52 churches ...
was another early supporter. The idea of a pendulum standard of length must have been familiar to people as early as 1663, because Samuel Butler satirizes it in ''Hudibras
''Hudibras'' is a vigorous satirical poem, written in a mock-heroic style by Samuel Butler (1613–1680), and published in three parts in 1663, 1664 and 1678. The action is set in the last years of the Interregnum, around 1658–60, immediately b ...
'':
:Upon the bench I will so handle ‘em
:That the vibration of this pendulum
:Shall make all taylors’ yards of one
:Unanimous opinion
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, base ...
proposed a pendulum-defined 'universal foot' in his influential ''Mesure de la Terre''. Gabriel Mouton
Gabriel Mouton (1618 – 28 September 1694) was a French abbot and scientist. He was a doctor of theology from Lyon, but was also interested in mathematics and astronomy. His 1670 book, the ''Observationes diametrorum solis et lunae apparenti ...
around 1670 suggested defining the ''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 (''Acadi ...
'' either by a seconds pendulum or a minute of terrestrial degree. A plan for a complete system of units based on the pendulum was advanced in 1675 by Italian polymath Tito Livio Burratini. In France in 1747, geographer Charles Marie de la Condamine
Charles Marie de La Condamine (28 January 1701 – 4 February 1774) was a French explorer, geographer, and mathematician. He spent ten years in territory which is now Ecuador, measuring the length of a degree of latitude at the equator and prepar ...
proposed defining length by a seconds pendulum at the equator; since at this location a pendulum's swing wouldn't be distorted by the Earth's rotation. James Steuart (1780) and George Skene Keith were also supporters.
By the end of the 18th century, when many nations were reforming their weight and measure systems, the seconds pendulum
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.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
was the leading choice for a new definition of length, advocated by prominent scientists in several major nations. In 1790, then US Secretary of State Thomas Jefferson
Thomas Jefferson (April 13, 1743 – July 4, 1826) was an American statesman, diplomat, lawyer, architect, philosopher, and Founding Fathers of the United States, Founding Father who served as the third president of the United States from 18 ...
proposed to Congress a comprehensive decimalized US 'metric system' based on the seconds pendulum at 38° North latitude, the mean latitude of the United States. No action was taken on this proposal. In Britain the leading advocate of the pendulum was politician John Riggs Miller. When his efforts to promote a joint British–French–American metric system fell through in 1790, he proposed a British system based on the length of the seconds pendulum at London. This standard was adopted in 1824 (below).
The metre
In the discussions leading up to the French adoption of the metric system
The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
in 1791, the leading candidate for the definition of the new unit of length, the metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
, was the seconds pendulum at 45° North latitude. It was advocated by a group led by French politician Talleyrand and mathematician Antoine Nicolas Caritat de Condorcet. This was one of the three final options considered by the French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
committee. However, on March 19, 1791, the committee instead chose to base the metre on the length of 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
* ...
through Paris. A pendulum definition was rejected because of its variability at different locations, and because it defined length by a unit of time. (However, since 1983 the metre has been officially defined in terms of the length of the second and the speed of light.) A possible additional reason is that the radical French Academy didn't want to base their new system on the second, a traditional and nondecimal unit from the ''ancien regime
''Ancien'' may refer to
* the French word for "ancient, old"
** Société des anciens textes français
* the French for "former, senior"
** Virelai ancien
** Ancien Régime
** Ancien Régime in France
''Ancien'' may refer to
* the French word fo ...
''.
Although not defined by the pendulum, the final length chosen for the metre, 10−7 of the pole-to-equator meridian arc
In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length.
The purpose of measuring meridian arcs is to de ...
, was very close to the length of the seconds pendulum (0.9937 m), within 0.63%. Although no reason for this particular choice was given at the time, it was probably to facilitate the use of the seconds pendulum as a secondary standard, as was proposed in the official document. So the modern world's standard unit of length is certainly closely linked historically with the seconds pendulum.
Britain and Denmark
Britain and Denmark appear to be the only nations that (for a short time) based their units of length on the pendulum. In 1821 the Danish inch was defined as 1/38 of the length of the mean solar seconds pendulum at 45° latitude at the meridian of Skagen
Skagen () is Denmark's northernmost town, on the east coast of the Skagen Odde peninsula in the far north of Jutland, part of Frederikshavn Municipality in Nordjylland, north of Frederikshavn and northeast of Aalborg. The Port of Skagen is ...
, at sea level, in vacuum. The British parliament passed the ''Imperial Weights and Measures Act'' in 1824, a reform of the British standard system which declared that if the prototype standard yard
The yard (symbol: yd) is an English unit of length in both the British imperial and US customary systems of measurement equalling 3 feet or 36 inches. Since 1959 it has been by international agreement standardized as exactly 0.914 ...
was destroyed, it would be recovered by defining the inch
Measuring tape with inches
The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth") ...
so that the length of the solar seconds pendulum at London, at sea level
Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
, in a vacuum, at 62 °F was 39.1393 inches. This also became the US standard, since at the time the US used British measures. However, when the prototype yard was lost in the 1834 Houses of Parliament fire, it proved impossible to recreate it accurately from the pendulum definition, and in 1855 Britain repealed the pendulum standard and returned to prototype standards.
Other uses
Seismometers
A pendulum in which the rod is not vertical but almost horizontal was used in early seismometers
A seismometer is an instrument that responds to ground noises and shaking such as caused by earthquakes, volcanic eruptions, and explosions. They are usually combined with a timing device and a recording device to form a seismograph. The output ...
for measuring Earth tremors. The bob of the pendulum does not move when its mounting does, and the difference in the movements is recorded on a drum chart.
Schuler tuning
As first explained by Maximilian Schuler in a 1923 paper, a pendulum whose period exactly equals the orbital period of a hypothetical satellite orbiting just above the surface of the Earth (about 84 minutes) will tend to remain pointing at the center of the Earth when its support is suddenly displaced. This principle, called Schuler tuning
Schuler tuning is a design principle for inertial navigation systems that accounts for the curvature of the Earth. An inertial navigation system, used in submarines, ships, aircraft, and other vehicles to keep track of position, determines directi ...
, is used in inertial guidance system
An inertial navigation system (INS) is a navigation device that uses motion sensors (accelerometers), rotation sensors ( gyroscopes) and a computer to continuously calculate by dead reckoning the position, the orientation, and the velocity (dire ...
s in ships and aircraft that operate on the surface of the Earth. No physical pendulum is used, but the control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
that keeps the inertial platform An inertial platform, also known as a gyroscopic platform or stabilized platform, is a system using gyroscopes to maintain a platform in a fixed orientation in space despite the movement of the vehicle that it is attached to. These can then be used ...
containing the gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rota ...
s stable is modified so the device acts as though it is attached to such a pendulum, keeping the platform always facing down as the vehicle moves on the curved surface of the Earth.
Coupled pendulums
In 1665 Huygens made a curious observation about pendulum clocks. Two clocks had been placed on his mantlepiece
The fireplace mantel or mantelpiece, also known as a chimneypiece, originated in medieval times as a hood that projected over a fire grate to catch the smoke. The term has evolved to include the decorative framework around the fireplace, and ca ...
, and he noted that they had acquired an opposing motion. That is, their pendulums were beating in unison but in the opposite direction; 180° out of phase
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
. Regardless of how the two clocks were started, he found that they would eventually return to this state, thus making the first recorded observation of a coupled oscillator
Oscillation is the repetitive or Periodic function, periodic variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. Familiar examples o ...
.
The cause of this behavior was that the two pendulums were affecting each other through slight motions of the supporting mantlepiece. This process is called entrainment
Entrainment may refer to:
* Air entrainment, the intentional creation of tiny air bubbles in concrete
* Brainwave entrainment, the practice of entraining one's brainwaves to a desired frequency
* Entrainment (biomusicology), the synchronization o ...
or mode locking
Mode locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes r ...
in physics and is observed in other coupled oscillators. Synchronized pendulums have been used in clocks and were widely used in gravimeter
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
s in the early 20th century. Although Huygens only observed out-of-phase synchronization, recent investigations have shown the existence of in-phase synchronization, as well as "death" states wherein one or both clocks stops.
Religious practice
Pendulum motion appears in religious ceremonies as well. The swinging incense
Incense is aromatic biotic material that releases fragrant smoke when burnt. The term is used for either the material or the aroma. Incense is used for aesthetic reasons, religious worship, aromatherapy, meditation, and ceremony. It may also be ...
burner called a censer
A censer, incense burner, perfume burner or pastille burner is a vessel made for burning incense or perfume in some solid form. They vary greatly in size, form, and material of construction, and have been in use since ancient times throughout t ...
, also known as a thurible
A thurible (via Old French from Medieval Latin ) is a metal censer suspended from chains, in which incense is burned during worship services. It is used in Christian churches including the Roman Catholic, Eastern Orthodox, Assyrian Church of th ...
, is an example of a pendulum. Pendulums are also seen at many gatherings in eastern Mexico where they mark the turning of the tides on the day which the tides are at their highest point. See also pendulums for divination and dowsing.
Education
Pendulums are widely used in science education
Science education is the teaching and learning of science to school children, college students, or adults within the general public. The field of science education includes work in science content, science process (the scientific method), some ...
as an example of a harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
, to teach dynamics and oscillatory motion. One use is to demonstrate the law of conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
. A heavy object such as a bowling ball
A bowling ball is a hard spherical ball used to knock down bowling pins in the sport of bowling.
Balls used in ten-pin bowling and American nine-pin bowling traditionally have holes for two fingers and the thumb. Balls used in five-pin bowling, c ...
or wrecking ball
A wrecking ball is a heavy steel ball, usually hung from a crane, that is used for demolishing large buildings. It was most commonly in use during the 1950s and 1960s. Several wrecking companies claim to have invented the wrecking ball. An e ...
[ is attached to a string. The weight is then moved to within a few inches of a volunteer's face, then released and allowed to swing and come back. In most instances, the weight reverses direction and then returns to (almost) the same position as the original release location — ''i.e.'' a small distance from the volunteer's face — thus leaving the volunteer unharmed. On occasion the volunteer is injured if either the volunteer does not stand still] or the pendulum is initially released with a push (so that when it returns it surpasses the release position).
Torture device
It is claimed that the pendulum was used as an instrument of torture
Torture is the deliberate infliction of severe pain or suffering on a person for reasons such as punishment, extracting a confession, interrogation for information, or intimidating third parties. Some definitions are restricted to acts c ...
and execution
Capital punishment, also known as the death penalty, is the State (polity), state-sanctioned practice of deliberately killing a person as a punishment for an actual or supposed crime, usually following an authorized, rule-governed process to ...
by the Spanish Inquisition
The Tribunal of the Holy Office of the Inquisition ( es, Tribunal del Santo Oficio de la Inquisición), commonly known as the Spanish Inquisition ( es, Inquisición española), was established in 1478 by the Catholic Monarchs, King Ferdinand ...
in the 18th century. The allegation is contained in the 1826 book ''The history of the Inquisition of Spain '' by the Spanish priest, historian and liberal
Liberal or liberalism may refer to:
Politics
* a supporter of liberalism
** Liberalism by country
* an adherent of a Liberal Party
* Liberalism (international relations)
* Sexually liberal feminism
* Social liberalism
Arts, entertainment and m ...
activist Juan Antonio Llorente
Juan Antonio Llorente, ORE (March 30, 1756 in Rincón de Soto (La Rioja), Spain – February 5, 1823 in Madrid) was a Spanish historian.
Biography
Llorente was raised by an uncle after his parents died. He studied at the University of Zaragoza, ...
. A swinging pendulum whose edge is a knife blade slowly descends toward a bound prisoner until it cuts into his body. This method of torture came to popular consciousness through the 1842 short story "The Pit and the Pendulum
"The Pit and the Pendulum" is a short story by American writer Edgar Allan Poe and first published in 1842 in the literary annual ''The Gift: A Christmas and New Year's Present for 1843''. The story is about the torments endured by a prisoner of ...
" by American author Edgar Allan Poe
Edgar Allan Poe (; Edgar Poe; January 19, 1809 – October 7, 1849) was an American writer, poet, editor, and literary critic. Poe is best known for his poetry and short stories, particularly his tales of mystery and the macabre. He is wide ...
but there is considerable skepticism that it actually was used.
Most knowledgeable sources are skeptical that this torture was ever actually used. The only evidence of its use is one paragraph in the preface to Llorente's 1826 ''History'', relating a second-hand account by a single prisoner released from the Inquisition's Madrid dungeon in 1820, who purportedly described the pendulum torture method. Modern sources point out that due to Jesus' admonition against bloodshed, Inquisitors were only allowed to use torture methods which did not spill blood, and the pendulum method would have violated this stricture. One theory is that Llorente misunderstood the account he heard; the prisoner was actually referring to another common Inquisition torture, the ''strappado
The strappado, also known as corda, is a form of torture in which the victim's hands are tied behind his back and the victim is suspended by a rope attached to the wrists, typically resulting in dislocated shoulders. Weights may be added to t ...
'' (garrucha), in which the prisoner has his hands tied behind his back and is hoisted off the floor by a rope tied to his hands. This method was also known as the "pendulum". Poe's popular horror tale, and public awareness of the Inquisition's other brutal methods, has kept the myth of this elaborate torture method alive.
See also
Notes
The value of g reflected by the period of a pendulum varies from place to place. The gravitational force varies with distance from the center of the Earth, i.e. with altitude - or because the Earth's shape is oblate, g varies with latitude.
A more important cause of this reduction in g at the equator is because the equator is spinning at one revolution per day, so the acceleration by the gravitational force is partially canceled there by 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 paralle ...
.
References
Further reading
* G. L. Baker and J. A. Blackburn (2009). ''The Pendulum: A Case Study in Physics'' (Oxford University Press).
* M. Gitterman (2010). ''The Chaotic Pendulum'' (World Scientific).
* Michael R. Matthews, Arthur Stinner, Colin F. Gauld (2005) ''The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives'', Springer
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* Schlomo Silbermann,(2014) "Pendulum Fundamental; The Path Of Nowhere" (Book)
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* L. P. Pook (2011). ''Understanding Pendulums: A Brief Introduction'' (Springer).
External links
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