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Encyclopedia > Pendulum
Simple gravity pendulum
An animation of a pendulum showing the velocity and acceleration vectors (v and A).

A pendulum is an object that is attached to a pivot point so it can swing freely. This object is subject to a restoring force that will accelerate it toward an equilibrium position. When the pendulum is displaced from its place of rest, the restoring force will cause the pendulum to oscillate about the equilibrium position. In other words, a weight attached to a string swings back and forth. Pendulum has several meanings: In physics, a pendulum can be either gravity pendulum or a torsion pendulum. ... Image File history File links Simple_pendulum. ... Image File history File links Simple_pendulum. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

A basic example is the simple gravity pendulum or bob pendulum. This is a weight (or bob) on the end of a massless string, which, when given an initial push, will swing back and forth under the influence of gravity over its central (lowest) point. A bob is the weight on the end of a pendulum. ... Gravity is a force of attraction that acts between bodies that have mass. ...

The regular motion of pendulums can be used for time keeping, and pendulums are used to regulate pendulum clocks. A pendulum clock uses a pendulum as its time base. ...

As recorded in the 20th century Chinese Book of Later Han, one of the earliest uses of the pendulum was in the seismometer device of the Han Dynasty (202 BC - 220 AD) scientist and inventor Zhang Heng (78-139).[1] Its function was to sway and activate a series of levers after being disturbed by the tremor of an earthquake far away.[2] After this was triggered, a small ball would fall out of the urn-shaped device into a metal toad's mouth below, signifying the cardinal direction of where the earthquake was located (and where government aid and assistance should be swiftly sent).[2] The Book of Later Han (Chinese:åŽæ±‰ä¹¦) is one of the official Chinese historical works which was compiled by Fan Ye in the 5th century, using a number of earlier histories and documents as sources. ... Seismometers (in Greek seismos = earthquake and metero = measure) are used by seismologists to measure and record the size and force of seismic waves. ... Han Dynasty in 87 BC Capital Changan (202 BCâ€“9 AD) Luoyang (25 ADâ€“190 AD) Language(s) Chinese Religion Taoism, Confucianism Government Monarchy History  - Establishment 206 BC  - Battle of Gaixia; Han rule of China begins 202 BC  - Interruption of Han rule 9 - 24  - Abdication to Cao Wei 220... Centuries: 2nd century BC - 3rd century BC - 4th century BC Decades: 230s BC 220s BC 210s BC - 200s BC - 190s BC 180s BC 170s BC Years: 207 BC 206 BC 205 BC 204 BC 203 BC - 202 BC - 201 BC 200 BC 199 BC 198 BC 197 BC Events October... Events Han Xiandi abdicates his throne to Cao Pi, symbolizing the end of the Han Dynasty and the beginning of the Three Kingdoms period in China. ... For other uses, see Zhang Heng (disambiguation). ... For other uses, see number 78. ... Events Births Deaths Zhang Heng, Chinese mathematician Categories: 139 ... This article is about the natural seismic phenomenon. ... This article or section does not cite its references or sources. ...

An Egyptian scholar, Ibn Yunus, is known to have described an early pendulum in the 10th century.[3] Some claimed that he used it for making measurements of time, but this is now believed to be a misinterpretation on the part of Edward Bernard, an English historian.[4][5] Ibn Yunus (Arabic: Ø§Ø¨Ù† ÙŠÙˆÙ†Ø³) (full name, Abu al-Hasan Ali abi Said Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi al-Misri) (c. ...

Among his scientific studies, Galileo Galilei performed a number of observations of all the properties of pendulums. His interest in pendulums may have been sparked by looking at the swinging motion of a chandelier in the Pisa cathedral. He began serious studies of the pendulum around 1602. Galileo noticed that period of the pendulum is independent of the bob mass or the amplitude of the swing. He also found a direct relationship between the square of the period and the length of the arm. The isochronism of the pendulum suggested a practical application for use as a metronome to aid musical students, and possibly for use in a clock.[6] Galileo redirects here. ... Isochronous means having an equal time difference or occurring simultaneously. ... A mechanical wind-up metronome in motion A digital metronome set to pulse at four beats per measure at a tempo of 130 BPM A metronome is any device that produces a regulated audible and/or visual pulse, usually used to establish a steady beat, or tempo, measured in beats...

Perhaps based upon the ideas of Galileo, in 1656 the Dutch scientist Christiaan Huygens patented a mechanical clock that employed a pendulum to regulate the movement.[7] This approached proved much more accurate than previous time pieces, such as the hourglass. Following an illness, in 1665 Huygens made a curious observation about pendulum clocks. Two such clocks had been placed on his mantlepiece, and he noted that they had acquired an opposing motion. That is, they were beating in unison but in the opposite direction—an anti-phase motion. Regardless of how the two clocks were adjusted, he found that they would eventually return to this state, thus making the first recorded observation of a coupled oscillator.[8] Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 â€“ July 8, 1698), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... A pendulum clock uses a pendulum as its time base. ... For other uses, see Hourglass (disambiguation). ... A mantelpiece or chimneypiece is the projecting hood which in medieval times was built over a fireplace to catch the smoke, and at a later date to the decorative framework, often carried up to the ceiling. ... Oscillation is the variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. ...

In his 1673 opus Horologium Oscillatorium sive de motu pendulorum,[11] Christian Huygens published his theory of the pendulum. 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 (rather than the circular arc of a pendulum). This confirmed the earlier observation by Marin Mersenne that the period of a pendulum does vary with amplitude, and that Galileo's observation was accurate only for small swings in the neighborhood of the center line.[12] Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ... Marin Mersenne, Marin Mersennus or le PÃ¨re Mersenne (September 8, 1588 â€“ September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ...

The English scientist Robert Hooke devised the conical pendulum, consisting of a pendulum that is free to swing in both directions. By analyzing the circular movements of the pendulum bob, he used it to analyze the orbital motions of the planets. Hooke would suggest to Isaac Newton in 1679 that the components of orbital motion consisted of inertial motion along a tangent direction plus an attractive motion in the radial direction. Isaac Newton was able to translate this idea into a mathematical form that described the movements of the planets with a central force that obeyed an inverse square lawNewton's law of universal gravitation.[13][14] Robert Hooke was also responsible for suggesting (as early as 1666) that the pendulum could be used to measure the force of gravity. Robert Hooke, FRS (July 18, 1635 â€“ March 3, 1703) was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work. ... Simple pendulum A pendulum is a body suspended from a fixed support that swings freely back and forth under the influence of gravity. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... In physics, an inverse-square law is any physical law stating that some quantity is inversely proportional to the square of the distance from a point. ... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ...

In 1851, Jean-Bernard-Leon Foucault suspended a pendulum (later named the Foucault pendulum) from the dome of the Panthéon in Paris. The mass of the pendulum was 28 kg and the length of the arm was 67 m. Once the pendulum was set in motion, the plane of motion was observed to precess 360° clockwise once per day. To Foucault, the precession was most easily explained by the rotation of the Earth.[15] J. B. LÃ©on Foucault Jean Bernard LÃ©on Foucault (name pronounced Foo-KOH) (18 September 1819 â€“ 11 February 1868) was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earths rotation. ... Foucaults Pendulum in the PanthÃ©on, Paris. ... The PanthÃ©on Interior Dome of the PanthÃ©on Entrance of the PanthÃ©on Voltaires statue and tomb in the crypt of the PanthÃ©on The PanthÃ©on (Latin Pantheon[1], from Greek Pantheon, meaning All the Gods) is a building in the Latin Quarter in Paris, France. ... This article is about the capital of France. ... Precession (also called gyroscopic precession) is the phenomenon by which the axis of a spinning object (e. ...

The National Institute of Standards and Technology based the U.S. national time standard on the Riefler Clock from 1904 until 1929. This pendulum clock maintained an accuracy of a few hundredths of a second per day. It was briefly replaced by the double-pendulum W. H. Shortt Clock before the NIST switched to an electronic time-keeping system.[16] NIST logo The National Institute of Standards and Technology (NIST, formerly known as The National Bureau of Standards) is a non-regulatory agency of the United States Department of Commerceâ€™s Technology Administration. ... A pendulum clock uses a pendulum as its time base. ...

## Basic principles

The mathematics of pendulums can be quite complex, but some formula and proofs are given below. ...

### Simple pendulum

If and only if the pendulum swings through a small angle (in the range where the function sin(θ) can be approximated as θ)[17] the motion may be approximated as simple harmonic motion. The period of a simple pendulum is significantly affected only by its length and the acceleration of gravity. The period of motion is independent of the mass of the bob or the angle at which the arm hangs at the moment of release. The period of the pendulum is the time taken for two swings (left to right and back again) of the pendulum. The formula for the period, T, is This article is about angles in geometry. ... Sine redirects here. ... Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... For other uses of this word, see Length (disambiguation). ... Gravity is a force of attraction that acts between bodies that have mass. ... For other uses, see Mass (disambiguation). ...

$T approx 2pi sqrtfrac{ell}{g},$

where $ell$ is the length of the pendulum measured from the pivot point to the bob's center of gravity and g is the gravitational acceleration.[18] This article or section may contain original research or unverified claims. ...

For larger amplitudes, the velocity of the pendulum can be derived for any point in its arc by observing that the total energy of the system is conserved. (Although, in a practical sense, the energy can slowly decline due to friction at the hinge and atmospheric drag.) Thus the sum of the potential energy of bob at some height above the equilibrium position, plus the kinetic energy of the moving bob at that point, is equal to the total energy. However, the total energy is also equal to maximum potential energy when the bob is at its peak height (at angle θmax). By this means it is possible to compute the velocity of the bob at each point along its arc, which in turn can be used to derive an exact period.[19] The resulting period is given by an infinite series: It has been suggested that pulse amplitude be merged into this article or section. ... This article is about velocity in physics. ... Look up conservation of energy in Wiktionary, the free dictionary. ... For other uses, see Friction (disambiguation). ... Atmospheric drag is a form of drag, which is the force that opposes an object moving through a liquid or gas. ... Potential energy can be thought of as energy stored within a physical system. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... In mathematics, a series is a sum of a sequence of terms. ...

[18]

Note that for small values of θmax, the value of the sine terms become negligible and the period can be approximated by a harmonic oscillator as shown earlier. In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...

### Physical pendulum

The simple pendulum assumes that the rod is massless and the bob has negligible angular momentum in itself.

A physical pendulum has significant extent, and the rod is massive and hence the pendulum has significant angular moment.

It turns out that a physical pendulum behaves like a simple pendulum but the expression for the period is modified.

where:

I is the moment of inertia of the pendulum about the pivot point
L is the distance from the center of mass to the pivot point
m is the mass

### Double pendulum

A more complex example is the double pendulum. This consists of a pendulum attached to the free end of another pendulum. Unlike a simple pendulum, the behavior of this system is much more complex. [20] For relatively small angles of displacement the behavior of this system can be simulated as a pair of springs that are attached end-to-end. As the angles increase, however, the double pendulum exhibits chaotic motion that is sensitive to the initial conditions. An example of a double pendulum. ... For other uses, see Chaos Theory (disambiguation). ...

## Use for measurement

The most widespread application is for timekeeping. A pendulum whose time period is 2 seconds is called the seconds pendulum since most clock escapements move the seconds hands on each swing. Clocks that keep time with the use of pendula lose accuracy due to friction. A pendulum clock uses a pendulum as its time base. ... A pendulum whose time period is precisely two seconds is called the seconds pendulum. ... A simple escapement. ...

The presence of g as a variable in the periodicity equation for a pendulum means that the frequency is different at various locations on Earth. So, for example, when an accurate pendulum clock in Glasgow, Scotland, (g = 9.815 63 m/s2) is transported to Cairo, Egypt, (g = 9.793 17 m/s2) the pendulum must be shortened by 0.23% to compensate. The pendulum can therefore be used in gravimetry to measure the local gravity at any point on the surface of the Earth. Note that g = 9.8 m/s² is a safe standard for acceleration due to gravity if locational accuracy is not a concern. For other uses, see Glasgow (disambiguation). ... This article is about the country. ... For other uses, see Cairo (disambiguation). ... Gravimetry is the measurement of a gravitational field. ... Gravity is a force of attraction that acts between bodies that have mass. ... g (also gee, g-force or g-load) is a non-SI unit of acceleration defined as exactly 9. ...

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. A seismometer is an instrument for measuring earthquakes and other ground motions. ...

## Other applications

### Schuler tuning

As first explained by Maximilian Schuler in his classic 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 is the basic principle of Schuler tuning that must be included in the design of any inertial guidance system that will be operated near the earth, such as in ships and aircraft. The German engineer (1882â€“1972) [1] Maximilian Schuler discovered 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. ... Schuler tuning describes the fundamental functional conditions for a gyrocompass. ... An inertial guidance system consists of an Inertial Measurement Unit (IMU) combined with a set of guidance algorithms and control mechanisms, allowing the path of a vehicle to be controlled according to the position determined by the inertial navigation system. ...

### Coupled pendula

Two coupled pendula form a double pendulum. Many physical systems can be mathematically described as coupled pendula. Under certain conditions these systems can also demonstrate chaotic motion. An example of a double pendulum. ... Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... For other uses, see Chaos (disambiguation). ...

### Religious practice

Pendulum motion appears in religious ceremonies as well. The swinging incense burner called a censer, also known as a thurible, is an example of a pendulum.[21] See also pendula for divination and dowsing. Incense is composed of aromatic organic materials. ... A censer is a vessel for burning incense. ... Stained glass window depiction of a thurible, St. ... For the English iconoclast, see William Dowsing. ...

A pendulum clock uses a pendulum as its time base. ... Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... Simple pendulum A pendulum is a body suspended from a fixed support that swings freely back and forth under the influence of gravity. ... A spherical pendulum is a generalization of the pendulum. ... An example of a double pendulum. ... Foucaults Pendulum in the PanthÃ©on, Paris. ... Katers pendulum is a reversible pendulum designed and built by Captain Henry Kater in 1817 to measure the acceleration of free fall so that gravity may be calculated without knowledge of the pendulums centre of gravity and radius of gyration. ... A harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. ... A mechanical wind-up metronome in motion A digital metronome set to pulse at four beats per measure at a tempo of 130 BPM A metronome is any device that produces a regulated audible and/or visual pulse, usually used to establish a steady beat, or tempo, measured in beats... A pendulum whose time period is precisely two seconds is called the seconds pendulum. ... This article does not cite any references or sources. ... A schematic drawing of the inverted pendulum on a cart. ...

## Notes

1. ^ Morton, 70.
2. ^ a b Needham, Volume 3, 627-629
3. ^ Piero Ariotti (Winter, 1968). "Galileo on the Isochrony of the Pendulum", Isis 59 (4), p. 414.
4. ^ O'Connor, J. J.; Robertson, E. F. (November 1999). Abu'l-Hasan Ali ibn Abd al-Rahman ibn Yunus. University of St Andrews. Retrieved on 2007-05-29.
5. ^ King, D. A. (1979). "Ibn Yunus and the pendulum: a history of errors". Archives Internationales d'Histoire des Sciences 29 (104): 35-52.
6. ^ Van Helden, Al (2005). Pendulum Clock. The Galileo Project. Retrieved on 2007-05-27.
7. ^ Huygens, Christiaan (1658). Horologium, 1st edition, Province of The Hague: Publishing House of Adrian Vlaqc.
8. ^ Toon, John (September 8, 2000). Out of Time: Researchers Recreate 1665 Clock Experiment to Gain Insights into Modern Synchronized Oscillators. Georgia Tech. Retrieved on 2007-05-31.
9. ^ Richer, Jean (1679). Observations astronomiques et physiques faites en l'isle de Caïenne. Mémoires de l'Académie Royale des Sciences.
10. ^ Mahoney, Michael S. (November 20, 1998). Charting the Globe and Tracking the Heavens: Navigation and the Sciences in the Early Modern Era. Princeton University. Retrieved on 2007-05-29.
11. ^ The constellation of Horologium was later named in honor of this book.
12. ^ Mahoney, Michael S. (March 19, 2007). Christian Huygens: The Measurement of Time and of Longitude at Sea. Princeton University. Retrieved on 2007-05-27.
13. ^ Nauenberg, Michael (2004). "Hooke and Newton: Divining Planetary Motions". Physics Today 57 (2): 13. Retrieved on 2007-05-30.
14. ^ The KGM Group, Inc. (2004). Heliocentric Models. Science Master. Retrieved on 2007-05-30.
15. ^ Giovannangeli, Françoise (November 1996). Spinning Foucault's Pendulum at the Panthéon. The Paris Pages. Retrieved on 2007-05-25.
16. ^ Staff (April 30, 2002). A Revolution in Timekeeping. NIST. Retrieved on 2007-05-29.
17. ^ For example, at the angle θ = 10°, θ is 0.1745 radians and sin θ equals 0.1736. So the approximation $theta approx sin theta$ has an error of 0.5% at this angle.
18. ^ a b Resnick, R.; Halliday, D. (1966). Physics. New York: John Wiley & Sons, Inc., 358. ISBN 0-471-71715-0.
19. ^ Symon, Keith R. (1971). Mechanics, Third edition, Reading, Massachusetts: Addison-Wesley Publishing Co.. ISBN 0-201-07392-7.
20. ^ Weisstein, Eric W. (2007). Double Pendulum. Wolfram Research. Retrieved on 2007-05-29.
21. ^ An interesting simulation of thurible motion can be found at this site.

Isis is an academic journal published by the University of Chicago devoted to the history of science, history of medicine, and the history of technology, as well as their cultural influences, featuring both original research articles as well as extensive book reviews and review essays. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 149th day of the year (150th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 147th day of the year (148th in leap years) in the Gregorian calendar. ... Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 â€“ July 8, 1698), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 151st day of the year (152nd in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 149th day of the year (150th in leap years) in the Gregorian calendar. ... Horologium (Latin for clock) is one of the lesser southern constellations (declination around -60 degrees). ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 147th day of the year (148th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 150th day of the year (151st in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 150th day of the year (151st in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 145th day of the year (146th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 149th day of the year (150th in leap years) in the Gregorian calendar. ... Some common angles, measured in radians. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 149th day of the year (150th in leap years) in the Gregorian calendar. ...

• Michael R.Matthews, Arthur Stinner, Colin F. Gauld. The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives. Springer, 2005.
• Michael R. Matthews, Colin Gauld and Arthur Stinner. The Pendulum: Its Place in Science, Culture and Pedagogy. Science & Education, 2005, 13, 261-277.
• Morton, W. Scott and Charlton M. Lewis (2005). China: It's History and Culture. New York: McGraw-Hill, Inc.
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.

Results from FactBites:

 Pendulum Physics Simulation (967 words) The pendulum is modeled as a point mass at the end of a massless rod. This is the equation of motion for the pendulum. The amplitude of the sine relationship is inversely proportional to length of the pendulum.
 Pendulum - Wikipedia, the free encyclopedia (564 words) The pendulum was discovered by Ibn Yunus during the 10th century, who was the first to study and document its oscillatory motion. A pendulum in which the rod is not vertical but almost horizontal was used in early seismometers for measuring earth tremors. Pendulums (these may be a crystal suspended on a chain, or a metal weight) can also be used in divination and dowsing.
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