Pendulum

# Pendulum

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A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

from its resting equilibrium position
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....

, it is subject to a restoring force
Restoring force
Restoring force, in a physics context, is a variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium....

due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. A pendulum swings with a specific period which depends (mainly) on its length.

From its discovery around 1602 by Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s. Pendulums are used to regulate pendulum clock
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 resonant device; it swings back and forth in a precise time interval dependent on its length, and resists swinging at other rates...

s, and are used in scientific instruments such as accelerometer
Accelerometer
An accelerometer is a device that measures proper acceleration, also called the four-acceleration. This is not necessarily the same as the coordinate acceleration , but is rather the type of acceleration associated with the phenomenon of weight experienced by a test mass that resides in the frame...

s and seismometer
Seismometer
Seismometers are instruments that measure motions of the ground, including those of seismic waves generated by earthquakes, volcanic eruptions, and other seismic sources...

s. Historically they were used as gravimeter
Gravimeter
A gravimeter or gravitometer is an instrument used in gravimetry for measuring the local gravitational field of the Earth. A gravimeter is a type of accelerometer, specialized for measuring the constant downward acceleration of gravity, which varies by about 0.5% over the surface of the Earth...

s to measure the acceleration of gravity in geophysical surveys, and even as a standard of length. The word 'pendulum' is new Latin
New Latin
The term New Latin, or Neo-Latin, is used to describe the Latin language used in original works created between c. 1500 and c. 1900. Among other uses, Latin during this period was employed in scholarly and scientific publications...

, from the Latin pendulus, meaning 'hanging'.

The simple gravity pendulum is an idealized mathematical model of a pendulum. This is a weight (or bob
Bob (physics)
A bob is the weight on the end of a pendulum most commonly, but not exclusively, found in pendulum clocks.- Reason for use :Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a...

) on the end of a massless cord suspended from a pivot, without friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

. When given an initial push, it will swing back and forth at a constant amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.

## Period of oscillation

The period of swing of a simple gravity pendulum depends on its length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

, the local strength of gravity, and to a small extent on the maximum angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

that the pendulum swings away from vertical, θ0, called the amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

. It is independent of the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

of the bob. If the amplitude is limited to small swings, the period
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g 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, 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
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

s, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ0 = 23° it is 1% larger than given by (1). The true period for an ideal simple gravity pendulum is given by
where K is a complete elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

of the first kind. The elliptic integral
can be evaluated using the arithmetic-geometric mean
method

or can be written using an infinite series to give:

The difference between this true period and the period for small swings (1) above is called the circular error.
For small swings the pendulum approximates a harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

, and its motion as a function of time, t, is approximately simple harmonic motion
Simple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....

:
For real pendulums, corrections to the period may be needed to take into account the presence of air, the mass of the string, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string, motion of the support, and local gravitational gradients.

## Compound or Physical pendulum

The length L of the ideal simple pendulum above, used for calculating the period, is the distance from the pivot point to the center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

of the bob. A pendulum consisting of any swinging rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

, which is free to rotate about a fixed horizontal axis is called a compound pendulum or a physical pendulum. For these pendulums the appropriate equivalent length is the distance from the pivot point to a point in the pendulum called the center of oscillation
Center of percussion
The center of percussion is the point on an object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point, so that the pivot will not be moving momentarily after the impulse....

. This is located under the center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

, at a distance called the radius of gyration
Radius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis....

, that depends on the mass distribution along the pendulum. However, for any pendulum in which most of the mass is concentrated in the bob, the center of oscillation is close to the center of mass.

Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable. 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
Henry Kater was an English 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 army, obtaining a commission in the 12th Regiment of Foot, then stationed in India, where he assisted...

used this idea to produce a type of reversible pendulum, now known as a Kater pendulum, for improved measurements of the acceleration due to gravity.

In general, the period for a physical pendulum can be found by replacing with in the equation for the simple pendulum, where is the distance between the pivot point and the center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

and is the moment of rotational inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

for rotations about the pivot point. For the ideal simple pendulum of length , In contrast, for a rigid rod of length which pivots about its end,

## History

One of the earliest known uses of a pendulum was in the 1st. century seismometer
Seismometer
Seismometers are instruments that measure motions of the ground, including those of seismic waves generated by earthquakes, volcanic eruptions, and other seismic sources...

device of Han Dynasty
Han Dynasty
The Han Dynasty was the second imperial dynasty of China, preceded by the Qin Dynasty and succeeded by the Three Kingdoms . It was founded by the rebel leader Liu Bang, known posthumously as Emperor Gaozu of Han. It was briefly interrupted by the Xin Dynasty of the former regent Wang Mang...

Chinese scientist Zhang Heng
Zhang Heng
Zhang Heng was a Chinese astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman, and literary scholar from Nanyang, Henan. He lived during the Eastern Han Dynasty of China. He was educated in the capital cities of Luoyang and Chang'an, and began his career as a...

. Its function was to sway and activate one of a series of levers after being disturbed by the tremor of an earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

far away. 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
Ibn Yunus
Ibn Yunus was an important Egyptian Muslim astronomer and mathematician, whose works are noted for being ahead of their time, having been based on meticulous calculations and attention to detail.The crater Ibn Yunus on the Moon is named after...

used a pendulum for time measurement, but this was an error that originated in 1684 with the British historian Edward Bernard.

During the Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...

, large pendulums were used as sources of power for manual reciprocating machines such as saws, bellows, and pumps. Leonardo da Vinci
Leonardo da Vinci
Leonardo di ser Piero da Vinci was an Italian Renaissance polymath: painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer whose genius, perhaps more than that of any other figure, epitomized the Renaissance...

made many drawings of the motion of pendulums, though without realizing its value for timekeeping.

### 1602: Galileo's research

Italian scientist Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

was the first to study the properties of pendulums, beginning around 1602. His first existent 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
Vincenzo Viviani was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo.-Biography:...

, claimed his interest had been sparked around 1582 by the swinging motion of a chandelier in the Pisa cathedral. Galileo discovered the crucial property that makes pendulums useful as timekeepers, called isochronism; the period of the pendulum is approximately independent of the amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

or width of the swing. He also found that the period is independent of the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

of the bob, and proportional to the square root of the length of the pendulum. He first employed freeswinging pendulums in simple timing applications, such as a metronome
Metronome
A metronome is any device that produces regular, metrical ticks — settable in beats per minute. These ticks represent a fixed, regular aural pulse; some metronomes also include synchronized visual motion...

for musicians. A physician friend used it as a timer to take patients' pulses
Pulse
In medicine, one's pulse represents the tactile arterial palpation of the heartbeat by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed against a bone, such as at the neck , at the wrist , behind the knee , on the inside of the elbow , and near the...

, the pulsilogium. In 1641 Galileo also conceived a design for a pendulum clock. The pendulum was the first harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

used by man.

### 1656: The pendulum clock

In 1656 the Dutch scientist Christiaan Huygens built the first pendulum clock
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 resonant device; it swings back and forth in a precise time interval dependent on its length, and resists swinging at other rates...

. This was a great improvement over existing mechanical clocks; their best accuracy was increased from around 15 minutes deviation a day to around 15 seconds a day. Pendulums spread over Europe as existing clocks were retrofitted with them.

The English scientist Robert Hooke
Robert Hooke
Robert Hooke FRS was an English natural philosopher, architect and polymath.His adult life comprised three distinct periods: as a scientific inquirer lacking money; achieving great wealth and standing through his reputation for hard work and scrupulous honesty following the great fire of 1666, but...

studied the conical pendulum
Conical pendulum
A conical pendulum is a weight fixed on the end of a string suspended from a pivot. Its construction is similar to an ordinary pendulum; however, instead of rocking back and forth, the bob of a conical pendulum moves at a constant speed in a circle with the string tracing out a cone...

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 motions of the planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s. Hooke suggested to Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

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. 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
Cayenne is the capital of French Guiana, an overseas region and department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic coast. The city's motto is "Ferit Aurum Industria" which means "Work brings wealth"...

, French Guiana
French Guiana
French Guiana is an overseas region of France, consisting of a single overseas department located on the northern Atlantic coast of South America. It has borders with two nations, Brazil to the east and south, and Suriname to the west...

in 1671, Jean Richer
Jean Richer
Jean Richer was a French astronomer and assistant of Giovanni Domenico Cassini.Between 1671 and 1673 he traveled to Cayenne at the request of the French Academy of Sciences to observe Mars during its perigee...

found that a pendulum clock
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 resonant device; it swings back and forth in a precise time interval dependent on its length, and resists swinging at other rates...

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
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

in Principia Mathematica showed that this was because the Earth was not a true sphere but slightly oblate (flattened at the poles) from the effect of centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

due to its rotation, causing gravity to increase with latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. Portable pendulums began to be taken on voyages to distant lands, as precision gravimeter
Gravimeter
A gravimeter or gravitometer is an instrument used in gravimetry for measuring the local gravitational field of the Earth. A gravimeter is a type of accelerometer, specialized for measuring the constant downward acceleration of gravity, which varies by about 0.5% over the surface of the Earth...

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
The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface...

.

### 1673: Huygens' Horologium Oscillatorium

In 1673, Christiaan Huygens published his theory of the pendulum, Horologium Oscillatorium sive de motu pendulorum. He demonstrated that for an object to descend down a curve under gravity in the same time interval, regardless of the starting point, it must follow a cycloid
Cycloid
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....

curve rather than the circular arc of a pendulum. This confirmed the earlier observation by Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

that the period of a pendulum does vary with its amplitude, and that Galileo's observation of isochronism was accurate only for small swings. Huygens also solved the issue of how to calculate the period of an arbitrarily shaped pendulum (called a compound pendulum), discovering the center of oscillation
Center of percussion
The center of percussion is the point on an object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point, so that the pivot will not be moving momentarily after the impulse....

, and its interchangeability with the pivot point.

The existing clock movement, the verge escapement
Verge escapement
The verge escapement is the earliest known type of mechanical escapement, the mechanism in a mechanical clock that controls its rate by advancing the gear train at regular intervals or 'ticks'. Its origin is unknown. Verge escapements were used from the 14th century until about 1800 in clocks...

, 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 'cheeks' next to the pivot in his 1673 clock, that constrained the suspension cord and forced the pendulum to follow a cycloid arc. 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
Isochronous
Isochronous : From Greek iso, equal + chronos, time. It literally means regularly, or at equal time intervals. In general English language, it refers to something that occurs at a regular interval, of the same duration; as opposed to synchronous which refers to more than one thing happening at the...

motivated the development of the anchor escapement
Anchor escapement
In horology, the recoil or anchor escapement is a type of escapement used in pendulum clocks. An escapement is the mechanism in a mechanical clock that maintains the swing of the pendulum and allows the clock's wheels to advance a fixed amount with each swing, moving the hands forward...

around 1670, which reduced the pendulum swing in clocks to 4°–6°.

### 1721: Temperature compensated pendulums

During the 18th and 19th century, the pendulum clock
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 resonant device; it swings back and forth in a precise time interval dependent on its length, and resists swinging at other rates...

'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 and the gridiron pendulum
Gridiron pendulum
The gridiron pendulum was an improved clock pendulum invented by British clockmaker John Harrison around 1726. It didn't change its effective length with temperature, so its period of swing stayed constant with changes in ambient temperature...

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
Center of percussion
The center of percussion is the point on an object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point, so that the pivot will not be moving momentarily after the impulse....

. 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
Henry Kater was an English 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 army, obtaining a commission in the 12th Regiment of Foot, then stationed in India, where he assisted...

invented the reversible Kater's pendulum
Kater's pendulum
A Kater's pendulum is a reversible freeswinging 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 pendulum gravimetry methods, the pendulum's centre of...

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
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

, tends to stay constant regardless of the motion of the pivot, and that this could be used to demonstrate the rotation of the Earth. He suspended a pendulum free to swing in two dimensions (later named the Foucault pendulum
Foucault pendulum
The Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...

) from the dome of the Panthéon
Panthéon, Paris
The Panthéon is a building in the Latin Quarter in Paris. It was originally built as a church dedicated to St. Genevieve and to house the reliquary châsse containing her relics but, after many changes, now functions as a secular mausoleum containing the remains of distinguished French citizens...

in Paris. The length of the cord was 67 metres (219.8 ft). Once the pendulum was set in motion, the plane of swing was observed to precess 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
Thermal expansion
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.When a substance is heated, its particles begin moving more and thus usually maintain a greater average separation. Materials which contract with increasing temperature are rare; this effect is...

materials began to be used for pendulum rods in the highest precision clocks and other instruments, first invar
Invar
Invar, also known generically as FeNi36 , is a nickel steel alloy notable for its uniquely low coefficient of thermal expansion . The name, Invar, comes from the word invariable, referring to its lack of expansion or contraction with temperature changes.It was invented in 1896 by Swiss scientist...

, a nickel steel alloy, and later fused quartz
Fused quartz
Fused quartz and fused silica are types of glass containing primarily silica in amorphous form. They are manufactured using several different processes...

, 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
In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...

of the pendulum due to changing atmospheric pressure
Atmospheric pressure
Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...

. The accuracy of the best pendulum clocks topped out at around a second per year.

The timekeeping accuracy of the pendulum was exceeded by the quartz
Quartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...

crystal oscillator
Crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...

, invented in 1921, and quartz clock
Quartz clock
A quartz clock is a clock that uses an electronic oscillator that is regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks are at least an order of magnitude more accurate than good mechanical clocks...

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
Gravimeter
A gravimeter or gravitometer is an instrument used in gravimetry for measuring the local gravitational field of the Earth. A gravimeter is a type of accelerometer, specialized for measuring the constant downward acceleration of gravity, which varies by about 0.5% over the surface of the Earth...

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 1602 until development of the quartz clock
Quartz clock
A quartz clock is a clock that uses an electronic oscillator that is regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks are at least an order of magnitude more accurate than good mechanical clocks...

in the 1930s, the pendulum was the world's standard for accurate timekeeping. In addition to clock pendulums, freeswinging seconds pendulum
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 1/2 Hz....

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 millimeters 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 (physics)
A bob is the weight on the end of a pendulum most commonly, but not exclusively, found in pendulum clocks.- Reason for use :Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a...

(b) suspended by a rod of wood or metal (a). To reduce air resistance (which accounts for most of the energy loss in 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 below). A common weight for seconds pendulum
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 1/2 Hz....

bobs is 15 pounds. (6.8 kg). Instead of hanging from a pivot, clock pendulums are usually supported by a short straight spring
Spring (device)
A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...

(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
Restoring force
Restoring force, in a physics context, is a variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium....

. A few 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
Escapement
In mechanical watches and clocks, an escapement is a device that transfers energy to the timekeeping element and enables counting the number of oscillations of the timekeeping element...

, (g,h).

Each time the pendulum swings through its center position, it releases one tooth of the escape wheel (g). The force of the clock's mainspring
Mainspring
A mainspring is a spiral torsion spring of metal ribbon that is the power source in mechanical watches and some clocks. Winding the timepiece, by turning a knob or key, stores energy in the mainspring by twisting the spiral tighter. The force of the mainspring then turns the clock's wheels as it...

or a driving weight hanging from a pulley, transmitted through the clock's gear train
Wheel train (horology)
In horology, a wheel train is the gear train of a mechanical watch or clock. Although the term is used for other types of gear trains, the long history of mechanical timepieces has created a traditional terminology for their gear trains which is not used in other applications of gears.Watch...

, 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
Turret clock
A Turret clock is a clock mounted in a tower or turret, usually to show the current time on a dial with hand or to announce the time by strike, or both. It can also have more than one dial to show days, moon phases, and other astronomical data.-Sundials:...

use a tray attached to the pendulum rod, to which small weights can be added or removed, to allow 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 clocks, is the seconds pendulum
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 1/2 Hz....

, about 1 meter (39 inches) long. In mantel clock
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 similar size due to a lack of carrying handles.These clocks...

s, half-second pendulums, 25 cm (10 in) long, or shorter, are used. Only a few large tower clocks
Turret clock
A Turret clock is a clock mounted in a tower or turret, usually to show the current time on a dial with hand or to announce the time by strike, or both. It can also have more than one dial to show days, moon phases, and other astronomical data.-Sundials:...

use longer pendulums, the 1.5 second pendulum, 2.25 m (7 ft) long, or occasionally the two-second pendulum, 4 m (13 ft).

### Temperature compensation

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
Godefroy Wendelin
Govaert Wendelen was a Flemish astronomer who was born in Herk-de-Stad. He is also known by the Latin name Vendelinus. His name is sometimes given as Godefroy Wendelin; his first name spelt Godefroid or Gottfried.Around 1630 he measured the distance between the Earth and the Sun using the method...

, as reported by Huygens in 1658). Thermal expansion of pendulum rods was first studied by Jean Picard
Jean Picard
Jean-Felix Picard was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He was the first person to measure the size of the Earth to a reasonable degree of accuracy in a survey conducted in 1669–70, for which he is honored with a...

in 1669. A pendulum with a steel rod will expand by about 11.3 parts per million (ppm) with each degree Celsius increase (6.3 ppm/°F), causing it to lose about 0.27 seconds per day, or 16 seconds per day for a 33 °C (60 °F) change. Wood rods expand less, losing only about 6 seconds per day for a 33 °C (60 °F) change, which is why quality clocks often had wooden pendulum rods.

#### Mercury pendulum

The first device to compensate for this error was the mercury pendulum, invented by George Graham
George Graham (clockmaker)
George Graham was an English clockmaker, inventor, and geophysicist, and a Fellow of the Royal Society.He was born to George Graham in Kirklinton, Cumberland. A Friend like his mentor Thomas Tompion, Graham left Cumberland in 1688 for London to work with Tompion...

in 1721. The liquid metal mercury
Mercury (element)
Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...

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 center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

closer to the pendulum pivot. By using the correct height of mercury in the container these two effects will cancel, leaving the pendulum's center 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. 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
Gridiron pendulum
The gridiron pendulum was an improved clock pendulum invented by British clockmaker John Harrison around 1726. It didn't change its effective length with temperature, so its period of swing stayed constant with changes in ambient temperature...

, invented in 1726 by John Harrison
John Harrison
John Harrison was a self-educated English clockmaker. He invented the marine chronometer, a long-sought device in solving the problem of establishing the East-West position or longitude of a ship at sea, thus revolutionising and extending the possibility of safe long distance sea travel in the Age...

. This consists of alternating rods of two different metals, one with lower thermal expansion (CTE), steel
Steel
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...

, and one with higher thermal expansion, zinc
Zinc
Zinc , or spelter , is a metallic chemical element; it has the symbol Zn and atomic number 30. It is the first element in group 12 of the periodic table. Zinc is, in some respects, chemically similar to magnesium, because its ion is of similar size and its only common oxidation state is +2...

or brass
Brass
Brass is an alloy of copper and zinc; the proportions of zinc and copper can be varied to create a range of brasses with varying properties.In comparison, bronze is principally an alloy of copper and tin...

. The rods are connected by a frame as shown, 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
Creep (deformation)
In materials science, creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses. It occurs as a result of long term exposure to high levels of stress that are below the yield strength of the material....

. For these reasons mercury pendulums were used in the highest precision clocks, but gridirons were used in quality regulator clocks. They became so associated with 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, when used as pendulum rods, made 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 Edouard Guillaume
Charles Edouard Guillaume
Charles Édouard Guillaume was a Swiss physicist who received the Nobel Prize in Physics in 1920 in recognition of the service he had rendered to precision measurements in physics by his discovery of anomalies in nickel steel alloys.Guillaume is known for his discovery of nickel-steel alloys he...

invented the nickel
Nickel
Nickel is a chemical element with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel belongs to the transition metals and is hard and ductile...

steel
Steel
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...

alloy
Alloy
An alloy is a mixture or metallic solid solution composed of two or more elements. Complete solid solution alloys give single solid phase microstructure, while partial solutions give two or more phases that may or may not be homogeneous in distribution, depending on thermal history...

Invar
Invar
Invar, also known generically as FeNi36 , is a nickel steel alloy notable for its uniquely low coefficient of thermal expansion . The name, Invar, comes from the word invariable, referring to its lack of expansion or contraction with temperature changes.It was invented in 1896 by Swiss scientist...

. 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 aluminum 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
Riefler escapement
The Riefler escapement is a mechanical escapement for precision pendulum clocks invented and patented by German instrument maker Sigmund Riefler in 1889...

which achieved accuracy of 15 milliseconds per day. Suspension springs of Elinvar
Elinvar
Elinvar is a nickel steel alloy with a modulus of elasticity which does not change much with temperature changes. The name is a contraction of the French elasticité invariable. It was invented in the late 1890s by Charles Édouard Guillaume, a Swiss physicist who also invented Invar, another...

were used to eliminate temperature variation of the spring's restoring force
Restoring force
Restoring force, in a physics context, is a variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium....

on the pendulum. Later fused quartz
Fused quartz
Fused quartz and fused silica are types of glass containing primarily silica in amorphous form. They are manufactured using several different processes...

was used which had even lower CTE. These materials are the choice for modern high accuracy pendulums.

### Atmospheric pressure

The presence of air around the pendulum has three effects on the period:
• By Archimedes principle the effective weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...

of the bob
Bob (physics)
A bob is the weight on the end of a pendulum most commonly, but not exclusively, found in pendulum clocks.- Reason for use :Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a...

is reduced by the buoyancy of the air it displaces, while the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

(inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

) remains the same, reducing the pendulum's acceleration during its swing and increasing the period. This depends on the density but not the shape of the pendulum.
• 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.
• Viscous air resistance 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
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

, requiring a stronger drive force from the clock's mechanism to keep it moving, which causes increased disturbance to the period.

So increases in barometric pressure 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
Inches of mercury, ' is a unit of measurement for pressure. It is still widely used for barometric pressure in weather reports, refrigeration and aviation in the United States, but is seldom used elsewhere....

or 0.015 seconds per day per torr
Torr
The torr is a non-SI unit of pressure with the ratio of 760 to 1 standard atmosphere, chosen to be roughly equal to the fluid pressure exerted by a millimetre of mercury, i.e., a pressure of 1 torr is approximately equal to 1 mmHg...

). 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
Berlin Observatory
The Berlin Observatory is a series of observatories and related organizations in and around the city of Berlin in Germany, starting from the 18th century...

, 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 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 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
Balance wheel
The balance wheel is the timekeeping device used in mechanical watches and some 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 spiral spring, the balance spring or hairspring...

s, the quartz crystal
Crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...

s used in quartz watches, and even the vibrating atoms in atomic clock
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...

s, are in physics called harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

s. The reason harmonic oscillators are used in clocks is that they vibrate or oscillate at a specific resonant frequency 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
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

(or periods), called the resonance width or bandwidth, where the harmonic oscillator will oscillate. In a clock, the actual frequency of the pendulum may vary randomly within this bandwidth in response to disturbances, but at frequencies outside this band, the clock will not function at all.

### Q factor

The measure of a harmonic oscillator's resistance to disturbances to its oscillation period is a dimensionless parameter called the Q factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

equal to the resonant frequency divided by the bandwidth. The higher the Q, the smaller the bandwidth, 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
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

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 2π.

In a clock, the pendulum must receive pushes from the clock's movement
Movement (clockwork)
In horology, a movement is the internal mechanism of a clock or watch, as opposed to the case, which encloses and protects the movement, and the face which displays the time. The term originated with mechanical timepieces, whose movements are made of many moving parts...

to keep it swinging, to replace the energy the pendulum loses to friction. These pushes, applied by a mechanism called the escapement
Escapement
In mechanical watches and clocks, an escapement is a device that transfers energy to the timekeeping element and enables counting the number of oscillations of the timekeeping element...

, are the main source of disturbance to the pendulum's motion. The Q is equal to 2π 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, ω = 2π/T is the pendulum's radian frequency of oscillation, and Γ is the frictional damping force
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

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
Shortt-synchronome clock
The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shortt in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK...

, invented in 1921. Its Invar
Invar
Invar, also known generically as FeNi36 , is a nickel steel alloy notable for its uniquely low coefficient of thermal expansion . The name, Invar, comes from the word invariable, referring to its lack of expansion or contraction with temperature changes.It was invented in 1896 by Swiss scientist...

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 103–105 explains why pendulums are more accurate timekeepers than the balance wheel
Balance wheel
The balance wheel is the timekeeping device used in mechanical watches and some 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 spiral spring, the balance spring or hairspring...

s in watches, with Q around 100-300, but less accurate than the quartz crystal
Crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...

s in quartz clock
Quartz clock
A quartz clock is a clock that uses an electronic oscillator that is regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks are at least an order of magnitude more accurate than good mechanical clocks...

s, with Q of 105–106.

### 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
Escapement
In mechanical watches and clocks, an escapement is a device that transfers energy to the timekeeping element and enables counting the number of oscillations of the timekeeping element...

, 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

In 1826 British astronomer George Airy proved what clockmakers had known for centuries; that the disturbing effect of a 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. Specifically, he proved that if a pendulum is driven by an impulse that is symmetrical about its bottom equilibrium position, the pendulum's amplitude 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
Gravimeter
A gravimeter or gravitometer is an instrument used in gravimetry for measuring the local gravitational field of the Earth. A gravimeter is a type of accelerometer, specialized for measuring the constant downward acceleration of gravity, which varies by about 0.5% over the surface of the Earth...

to measure the local gravity, which varies by about 0.5% at different points on the surface of the Earth. 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
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of gravitational field or the properties of matter responsible for its creation are of interest...

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 (1).

### The seconds pendulum

The seconds pendulum
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 1/2 Hz....

, a pendulum with a period of two seconds so each swing takes one second, was widely used to measure gravity, because most precision clocks 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
Francis Bacon, 1st Viscount St Albans, KC was an English philosopher, statesman, scientist, lawyer, jurist, author and pioneer of the scientific method. He served both as Attorney General and Lord Chancellor of England...

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
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

first determined the length of the seconds pendulum was 39.1 inches (993 mm), by comparing the swing of a pendulum to the time it took a weight to fall a measured distance.
• 1669: Jean Picard
Jean Picard
Jean-Felix Picard was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He was the first person to measure the size of the Earth to a reasonable degree of accuracy in a survey conducted in 1669–70, for which he is honored with a...

determined the length of the seconds pendulum at Paris, using a 1 inches (2.5 cm) copper ball suspended by an aloe fiber, obtaining 39.09 inches (99.3 cm).
• 1672: The first observation that gravity varied at different points on Earth was made in 1672 by Jean Richer
Jean Richer
Jean Richer was a French astronomer and assistant of Giovanni Domenico Cassini.Between 1671 and 1673 he traveled to Cayenne at the request of the French Academy of Sciences to observe Mars during its perigee...

, who took a pendulum clock
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 resonant device; it swings back and forth in a precise time interval dependent on its length, and resists swinging at other rates...

to Cayenne
Cayenne
Cayenne is the capital of French Guiana, an overseas region and department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic coast. The city's motto is "Ferit Aurum Industria" which means "Work brings wealth"...

, French Guiana
French Guiana
French Guiana is an overseas region of France, consisting of a single overseas department located on the northern Atlantic coast of South America. It has borders with two nations, Brazil to the east and south, and Suriname to the west...

and found that it lost minutes per day; its seconds pendulum had to be shortened by ligne
Ligne
The ligne is a unit of length that was in use prior to the French adoption of the metric system in the late 18th century, and is still used by French and Swiss wristwatch makers to measure the size of a watch movement.- Watchmakers' use :There are 12 lignes to one French inch...

s
(2.6 mm) shorter than at Paris, to keep correct time. In 1687 Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

in Principia Mathematica showed this was because the Earth had a slightly oblate shape (flattened at the poles) caused by the centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

of its rotation, 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 created the first hydrostatic model of the Earth, Clairaut's formula, which allowed the ellipticity of the Earth to be calculated from gravity measurements. Progressively more accurate models of the shape of the Earth followed.
• 1687: Newton experimented with pendulums (described in Principia) and found that equal length pendulums with bobs made of different materials had the same period, proving that the gravitational force on different substances was exactly proportional to their mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

(inertia).

• 1737: French mathematician Pierre Bouguer
Pierre Bouguer
Pierre Bouguer was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture"....

made a sophisticated series of pendulum observations in the Andes
Andes
The Andes is the world's longest continental mountain range. It is a continual range of highlands along the western coast of South America. This range is about long, about to wide , and of an average height of about .Along its length, the Andes is split into several ranges, which are separated...

mountains, Peru. 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
Altiplano
The Altiplano , in west-central South America, where the Andes are at their widest, is the most extensive area of high plateau on Earth outside of Tibet...

. 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.
• 1747: Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

showed how to correct for the lengthening of the period due to a finite angle of swing θ0 by using the first order correction θ02/16, giving the period of a pendulum with an infinitesimal swing.
• 1792: To define a pendulum standard of length for use with the new metric system
Metric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...

, in 1792 Jean-Charles de Borda
Jean-Charles de Borda
Jean-Charles, chevalier de Borda was a French mathematician, physicist, political scientist, and sailor.-Life history:...

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 12 feet (3.7 m) 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 ΔT between the recurring instants when the two pendulums swung in synchronism was timed. From this the difference between the periods of the pendulums, T1 and T2, could be calculated:
• 1821: Francesco Carlini
Francesco Carlini
Francesco Carlini was an Italian astronomer. Born in Milan, he became director of the observatory there in 1832. He published Nuove tavole de moti apparenti del sole in 1832. In 1810, he had already published Esposizione di un nuovo metodo di construire le taole astronomiche applicato alle...

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 11 miles (17.7 km) in diameter and 1 miles (1.6 km) 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 real pendulum (called a compound pendulum) was equal to the period of a simple pendulum with a length equal to the distance between the pivot point and a point called the center of oscillation
Center of percussion
The center of percussion is the point on an object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point, so that the pivot will not be moving momentarily after the impulse....

, located under the center of gravity
Center of gravity
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...

, that depends on the mass distribution along the pendulum. But there was no accurate way of determining the center of oscillation in a real pendulum.

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
Henry Kater was an English 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 army, obtaining a commission in the 12th Regiment of Foot, then stationed in India, where he assisted...

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 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 microscope. 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) 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
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of gravitational field or the properties of matter responsible for its creation are of interest...

a standard part of geodesy
Geodesy
Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...

. 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
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

, and pendulums were taken on the great geodetic surveys of the 18th century, particularly the Great Trigonometric Survey
Great Trigonometric Survey
The Great Trigonometric Survey was a project of the Survey of India throughout most of the 19th century. It was piloted in its initial stages by William Lambton, and later by George Everest. Among the many accomplishments of the Survey were the demarcation of the British territories in India and...

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 nearby point. 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
Kew Observatory
Kew Observatory was an astronomical and terrestrial magnetic observatoryfounded by King George III , located within the Old Deer Park of the former Richmond Palace in Richmond, Surrey, now within Greater London. The former royal manor of Kew lies to the immediate north...

, UK.

• Airy's coal pit experiments: Starting in 1826, using methods similar to Bouguer, British astronomer George Airy attempted to determine the density of the Earth by pendulum gravity measurements at the top and bottom of a coal mine. The gravitational force below the surface of the Earth decreases rather than increasing with depth, because by Gauss's law the mass of the spherical shell of crust above the subsurface point does not contribute to the gravity. The 1826 experiment was aborted by the flooding of the mine, but in 1854 he conducted an improved experiment at the Harton coal mine, using seconds pendulums swinging on agate plates, timed by precision chronometers synchronized by an electrical circuit. He found the lower pendulum was slower by 2.24 seconds per day. This meant that the gravitational acceleration at the bottom of the mine, 1250 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 Bessel
-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...

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. The pendulum had a half-second period, and was about 25 cm long. It was nonreversible, so it was used for relative gravity measurements, but its small size made the apparatus small and portable. The period of the pendulum was picked off by reflecting the image of an electric spark
Electric spark
An electric spark is a type of electrostatic discharge that occurs when an electric field creates an ionized electrically conductive channel in air producing a brief emission of light and sound. A spark is formed when the electric field strength exceeds the dielectric field strength of air...

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 US Coast and Geodetic Survey 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 A. A. Michelson in his 1924 measurements of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

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
Michelson interferometer
The Michelson interferometer is the most common configuration for optical interferometry and was invented by Albert Abraham Michelson. An interference pattern is produced by splitting a beam of light into two paths, bouncing the beams back and recombining them...

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 turn of the 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 quartz and fused silica are types of glass containing primarily silica in amorphous form. They are manufactured using several different processes...

, each 10.7 inches (272 mm) in length with a period of 0.89 second, swinging on pyrex knife edge pivots, 180° out of phase. They were mounted in a permanently sealed temperature and humidity controlled vacuum chamber. Stray electrostatic charges on the quartz pendulums had to be discharged by exposing them to a radioactive salt before use. The period was detected by reflecting a light beam from a mirror at the top of the pendulum, recorded by a chart recorder and compared to a precision crystal oscillator
Crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...

calibrated against the WWV radio time signal. This instrument was accurate to within (0.3–0.5)×10−7 (30–50 microgals 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. 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.

## 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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

as a standard of length
Unit of length
Many different units of length have been used across the world. The main units in modern use are U.S. customary units in the United States and the Metric system elsewhere. British Imperial units are still used for some purposes in the United Kingdom and some other countries...

.

Until the 19th century, countries based their systems of length measurement on prototypes, metal bar primary standard
Primary standard
A primary standard in metrology is a standard that is accurate enough 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 used to calibrate other standards referred to as working...

s, such as the standard yard in Britain kept at the Houses of Parliament, and the standard toise
Toise
A toise 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 , and Quebec...

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. Enlightenment
Age of Enlightenment
The Age of Enlightenment was an elite cultural movement of intellectuals in 18th century Europe that sought to mobilize the power of reason in order to reform society and advance knowledge. It promoted intellectual interchange and opposed intolerance and abuses in church and state...

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
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 1/2 Hz....

, in which each swing (a half period
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

) 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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. Gravitational acceleration increases smoothly from the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

to the poles
Geographical pole
A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...

, due to the oblate 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. So the unit of length could be defined at a given latitude and measured at any point at 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
Edward Sabine
General Sir Edward Sabine KCB FRS was an Irish astronomer, geophysicist, ornithologist and explorer.Two branches of Sabine's work in particular deserve very high credit: Determination of the length of the seconds pendulum, a simple pendulum whose time period on the surface of the Earth is two...

and Thomas Young
Thomas Young (scientist)
Thomas Young was an English polymath. He is famous for having partly deciphered Egyptian hieroglyphics before Jean-François Champollion eventually expanded on his work...

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
Isaac Beeckman was a Dutch philosopher and scientist, who, through his studies and contact with leading natural philosophers, may have "virtually given birth to modern atomism".-Biography:...

who in 1631 recommended making the seconds pendulum "the invariable measure for all people at all times in all places". Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

, 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
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

in 1660, advocated by Christiaan Huygens and Ole Rømer, basing it on Mersenne's work, and Huygens in Horologium Oscillatorum proposed a "horary foot" defined as 1/3 of the seconds pendulum. Christopher Wren
Christopher Wren
Sir Christopher Wren FRS is one of the most highly acclaimed English architects in history.He used to be accorded responsibility for rebuilding 51 churches in the City of London after the Great Fire in 1666, including his masterpiece, St. Paul's Cathedral, on Ludgate Hill, completed in 1710...

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
Samuel Butler (poet)
Samuel Butler was a poet and satirist. Born in Strensham, Worcestershire and baptised 14 February 1613, he is remembered now chiefly for a long satirical burlesque poem on Puritanism entitled Hudibras.-Biography:...

satirizes it in Hudibras
Hudibras
Hudibras is an English mock heroic narrative poem from the 17th century written by Samuel Butler.-Purpose:The work is a satirical polemic upon Roundheads, Puritans, Presbyterians and many of the other factions involved in the English Civil War...

:
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
Jean-Felix Picard was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He was the first person to measure the size of the Earth to a reasonable degree of accuracy in a survey conducted in 1669–70, for which he is honored with a...

proposed a pendulum defined 'universal foot' in his influential Mesure de la Terre. Gabriel Mouton
Gabriel Mouton
Gabriel Mouton was a French abbot and scientist. He was a doctor of theology from Lyon, but was also interested in mathematics and astronomy....

around 1670 suggested defining the toise
Toise
A toise 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 , and Quebec...

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
Charles Marie de La Condamine was a French explorer, geographer, and mathematician. He spent ten years in present-day Ecuador measuring the length of a degree latitude at the equator and preparing the first map of the Amazon region based on astronomical observations.-Biography:Charles Marie de La...

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. British politicians 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
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 1/2 Hz....

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
Thomas Jefferson was the principal author of the United States Declaration of Independence and the Statute of Virginia for Religious Freedom , the third President of the United States and founder of the University of Virginia...

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
John Riggs Miller
Sir John Riggs-Miller, 1st Baronet was an Anglo-Irish politician who championed reform of the customary system of weights and measures in favour of a scientifically founded system.-Early life:...

. 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
Metric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...

in 1791, the leading candidate for the definition of the new unit of length, the metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

, 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
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...

. This was one of the three final options considered by the French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

committee. However, on March 19, 1791 the committee instead chose to base the metre on the length of the meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

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.

Although not defined by the pendulum, the final length chosen for the metre, 10−7 of the pole-to-equator meridian arc
Meridian arc
In geodesy, a meridian arc measurement is a highly accurate determination of the distance between two points with the same longitude. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This...

, 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
Skagen is a projection of land and a town, with a population of 8,515 , in Region Nordjylland on the northernmost tip of Vendsyssel-Thy, a part of the Jutland peninsula in northern Denmark...

, 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 was destroyed, it would be recovered by defining the inch so that the length of the solar seconds pendulum at London, at sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

, 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
Burning of Parliament
Burning of Parliament is the popular name for the fire which destroyed the Palace of Westminster, the home of the Parliament of the United Kingdom, on 16 October 1834...

, it proved impossible to recreate it accurately from the pendulum definition, and in 1855 Britain repealed the pendulum standard and returned to prototype standards.

### Seismometers

A pendulum in which the rod is not vertical but almost horizontal was used in early seismometers 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
Max Schuler
The German engineer Maximilian Schuler is best known for discovering the principle known as Schuler tuning which is fundamental to the operation of a gyrocompass or inertial guidance system that will be operated near the surface of the earth....

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
Schuler tuning is a modification to the electronic control system used in 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 directions with respect...

, is used in inertial guidance systems in ships and aircraft that operate on the surface of the Earth. No physical pendulum is used, but the control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...

that keeps the inertial platform containing the gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

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, 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. 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.

The cause of this behavior was that the two pendulums were affecting each other through slight motions of the supporting mantlepiece. Many physical systems can be mathematically described as coupled oscillation. Under certain conditions these systems can also demonstrate chaotic motion
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

.

### Religious practice

Pendulum motion appears in religious ceremonies as well. The swinging incense
Incense
Incense is composed of aromatic biotic materials, which release fragrant smoke when burned. The term "incense" refers to the substance itself, rather than to the odor that it produces. It is used in religious ceremonies, ritual purification, aromatherapy, meditation, for creating a mood, and for...

burner called a censer
Censer
Censers are any type of vessels made for burning incense. These vessels vary greatly in size, form, and material of construction. They may consist of simple earthenware bowls or fire pots to intricately carved silver or gold vessels, small table top objects a few centimetres tall to as many as...

, also known as a thurible
Thurible
A thurible is a metal censer suspended from chains, in which incense is burned during worship services. It is used in the Catholic Church as well as in Anglican, Eastern Orthodox, Oriental Orthodox, Armenian Apostolic, some Lutheran, Old Catholic, and in various Gnostic Churches. It is also used...

, 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.

### Execution

During the Middle Ages, pendulums were used as a method of torture by the Spanish Inquisition. Using the basic principle of the pendulum, the weight (bob
Bob (physics)
A bob is the weight on the end of a pendulum most commonly, but not exclusively, found in pendulum clocks.- Reason for use :Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a...

) is replaced by an axe head. The victim is strapped to a table below, the device is activated, and the axe begins to swing back and forth through the air. With each pass, or return, the pendulum drops, gradually coming closer to the victim's torso, until finally cleaved. Because of the time required before the mortal action of the axe is complete, the pendulum is considered a method of torturing the victim before his or her demise.