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A pendulum is a mass that is attached to a pivot, from which it can swing freely. This object is subject to a restoring force due to gravity 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.

A basic example is the simple gravity pendulum or bob pendulum. This is a mass (or bob) on the end of a string of negligible mass, which, when initially displaced, will swing back and forth under the influence of gravity over its central (lowest) point.

The regular motion of the pendulum can be used for time keeping, and pendulums are used to regulate pendulum clocks.

History

As recorded in the 5th 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] An Egyptian scholar, Ibn Yunus, is known to have described an early pendulum in the 10th century.^[3]

Among his scientific studies, Galileo Galilei performed a number of observations of all the properties of pendulums. His interest in the pendulum 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 discovered the key property that makes pendulums useful timekeepers: isochronism, which means that the period of the pendulum is approximately independent of the amplitude (width) of its swing, so that successive swings of the pendulum will take the same amount of time even if the swing is decreasing due to friction. He also found that the period of a pendulum is independent of the mass of its bob, and proportional to the square root of its length. 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.^[4]

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.^[5] This approach 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 fireplace mantel, 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.^[6]

During his Académie des Sciences expedition to Cayenne, French Guiana in 1671, Jean Richer demonstrated that the periodicity of a pendulum was slower at Cayenne than at Paris. From this he deduced that the force of gravity was lower at Cayenne.^[7] Huygens reasoned that the centripetal force of the Earth's rotation modified the weight of the pendulum bob based on the latitude of the observer.^[8]

In his 1673 work Horologium Oscillatorium sive de motu pendulorum,^[9] Christiaan Huygens published his theory of the pendulum. He demonstrated that for an object to descend a curve under gravity in the same time interval, regardless of the starting point, it must follow a cycloid curve, rather than the circular arc that a pendulum follows. This confirmed the earlier observation by Marin Mersenne that the period of a pendulum does vary with amplitude, and that Galileo's observation of isochronism was accurate only for small swings in the neighborhood of the center line.^[10]

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 law—Newton's law of universal gravitation.^[11]^[12] Robert Hooke was also responsible for suggesting (as early as 1666) that the pendulum could be used to measure the force of gravity.

In 1851, Jean-Bernard-Leon Foucault suspended a pendulum (later named the Foucault pendulum) from the dome of the Panthéon in Paris. It was the third Foucault pendulum he constructed, the first one was constructed in his basement and the second one was a demonstration model with a length of 11 meters. The mass of the pendulum in Pantheon was 28 kg and the length of the arm was 67 m.^[13] The Foucault pendulum was a worldwide sensation: it was the first demonstration of the Earth's rotation with a purely indoors experiment. Once the Paris pendulum was set in motion the plane of motion was observed to precess about 270° clockwise per day. A pendulum located at either of the poles will precess 360° per day relative to the ground it is suspended above. There is a mathematical relation between the latitude where a Foucault pendulum is deployed and its rate of precession; the period of the precession is inversely proportional to the sine of the latitude.

For 270 years, from their invention until quartz crystal oscillators superseded them in the 1920s, pendulums were the world's most accurate timekeeping technology. The most accurate pendulum clocks, called astronomical regulators, were installed in astronomical observatories, and served as standards to set all other clocks. The National Institute of Standards and Technology based the U.S. national time standard on the Riefler clock made by the German firm Clemens Riefler, 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 quartz clocks in the 1930s.^[14]

Basic principles

Simple pendulum

Amplification of period factor of a pendulum for growing angular amplitude. For little oscilations factor is approximately 1 but it tends to infinity for angles near π (180º).

If, and only if, the pendulum swings through a small angle (in the range where the function sin(θ) can be approximated as θ)^[15] 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 one complete swing (left to right and back again) of the pendulum. The formula for the period, T, is

T\approx 2\pi {\sqrt {\frac {\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 local gravitational acceleration.^[16]

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 stationary at its peak height (at angle θ_max). 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.^[17] The resulting period is given by an infinite series:

T=2\pi {\sqrt {\frac {l}{g}}}\left(1+{\frac {1}{4}}\cdot \sin ^{2}{\frac {\theta _{max}}{2}}+{\frac {9}{64}}\cdot \sin ^{4}{\frac {\theta _{max}}{2}}+\cdots \right).

^[16]

Note that for small values of θ_max, the value of the sine terms become negligible and the period can be approximated by a harmonic oscillator as shown above.

Physical pendulum

The simple pendulum assumes that the rod is massless and that the bob is small, so has negligible angular momentum in itself. A physical pendulum (or compound pendulum) has significant size and mass, and hence a significant moment of inertia.

A physical pendulum behaves like a simple pendulum but the expression for the period is modified.

T=2\pi {\sqrt {\frac {I}{mgL}}}

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 of the pendulum

For a physical pendulum, since this has the same form as the simple pendulum, it is possible to define the center of oscillation. This is the distance from the pivot point, at which, if all the mass of the pendulum was concentrated (and thus forming a simple pendulum), would give the pendulum the same period.

\ell ={\frac {I}{mL}}

^[18]

Double pendulum

A double pendulum consists of one pendulum attached to the free end of another pendulum. The behaviour of this system is significantly more complicated than that of a single simple or physical pendulum. ^[19] For small angles of displacement this system is approximately linear and can be modelled by the theory of normal modes. As the angles increase, however, the double pendulum can exhibit chaotic motion that is sensitive to the initial conditions.

A special case of the compound double pendulum, known as Rott's Pendulum, is that where the pivots of the two pendulums are horizontal when the system is in static equilibrium, and the periods of the pendulums are a factor of two apart. In this case, the coupling between the pendulums is to first order non-linear which — for small angles — leads to periodic behaviour with a period much larger than that of either pendulum. ^[20]

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 pendulums lose accuracy due to friction. Pendulums are also widely used as metronomes for musicians.

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.

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.

Problems

Pendulums in air are affected by atmospheric and mechanical drag. These effects can be compensated for if they are known and constant. Atmospheric drag is affected by the density of air, which is in turn affected by its moisture content, temperature, and barometric pressure. Precise clocks used for the timing of astronomic observations were improved by operating the pendulum in a partially evacuated and temperature controlled chamber. Since the drag is proportional to the square of the velocity, a long pendulum or a pendulum with a high rotational moment of inertia about its pivot, which both produce slow oscillation, will be less affected by atmospheric drag than is a faster pendulum.

Simple pendulums in everyday clocks are affected by the ambient temperature, which thermal expansion of the material holding the bob will change the period of the pendulum. This change of length can be minimized by using special materials for the pendulum rod which exhibit little change with temperature or by using a more complex gridiron pendulum, sometimes called a "banjo" pendulum for its similarity in appearance to the musical instrument.

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. Pendulums are used in swing metronomes for pianists.

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]

Notes

^ Morton, 70.
^ ^a ^b Needham, Volume 3, 627-629
^ King, D. A. (1979). "Ibn Yunus and the pendulum: a history of errors". Archives Internationales d'Histoire des Sciences. 29 (104): 35–52.
^ Van Helden, Al (2005). "Pendulum Clock". The Galileo Project. Retrieved 2007-05-27.
^ Huygens, Christiaan (1658). Horologium (1st edition ed.). Province of The Hague: Publishing House of Adrian Vlaqc. {{cite book}}: |edition= has extra text (help)
^ Toon, John (September 8, 2000). "Out of Time: Researchers Recreate 1665 Clock Experiment to Gain Insights into Modern Synchronized Oscillators". Georgia Tech. Retrieved 2007-05-31.
^ Richer, Jean (1679). Observations astronomiques et physiques faites en l'isle de Caïenne. Mémoires de l'Académie Royale des Sciences.
^ 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 2007-05-29.
^ The constellation of Horologium was later named in honor of this book.
^ Mahoney, Michael S. (March 19, 2007). "Christian Huygens: The Measurement of Time and of Longitude at Sea". Princeton University. Retrieved 2007-05-27.
^ Nauenberg, Michael (2004). "Hooke and Newton: Divining Planetary Motions". Physics Today. 57 (2): 13. doi:10.1063/1.1688052. Retrieved 2007-05-30.
^ The KGM Group, Inc. (2004). "Heliocentric Models". Science Master. Retrieved 2007-05-30.
^ Giovannangeli, Françoise (November 1996). "Spinning Foucault's Pendulum at the Panthéon". The Paris Pages. Retrieved 2007-05-25.
^ "A Revolution in Timekeeping". A Walk Through Time. Time and Frequency Div., NIST. April 30, 2002. Retrieved 2007-05-29.
^ 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.
^ ^a ^b Resnick, R.; Halliday, D. (1966). Physics. New York: John Wiley & Sons, Inc. p. 358. ISBN 0-471-71715-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Symon, Keith R. (1971). Mechanics (Third edition ed.). Reading, Massachusetts: Addison-Wesley Publishing Co. ISBN 0-201-07392-7. {{cite book}}: |edition= has extra text (help)
^ What is the COP and does it matter?
^ Weisstein, Eric W. (2007). "Double Pendulum". Wolfram Research. Retrieved 2007-05-29.
^ Rott, N (1970). "A multiple pendulum for the demonstration of non-linear coupling". Zeitschrift für Angewandte Mathematik und Physik. 21 (4): 570–582.
^ An interesting simulation of thurible motion can be found at this site.

External links

[morton_70-1] Morton, 70.

[needham_volume_3_627_629-2] Needham, Volume 3, 627-629

[3] King, D. A. (1979). "Ibn Yunus and the pendulum: a history of errors". Archives Internationales d'Histoire des Sciences. 29 (104): 35–52.

[4] Van Helden, Al (2005). "Pendulum Clock". The Galileo Project. Retrieved 2007-05-27.

[5] Huygens, Christiaan (1658). Horologium (1st edition ed.). Province of The Hague: Publishing House of Adrian Vlaqc. {{cite book}}: |edition= has extra text (help)

[6] Toon, John (September 8, 2000). "Out of Time: Researchers Recreate 1665 Clock Experiment to Gain Insights into Modern Synchronized Oscillators". Georgia Tech. Retrieved 2007-05-31.

[7] Richer, Jean (1679). Observations astronomiques et physiques faites en l'isle de Caïenne. Mémoires de l'Académie Royale des Sciences.

[8] 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 2007-05-29.

[9] The constellation of Horologium was later named in honor of this book.

[10] Mahoney, Michael S. (March 19, 2007). "Christian Huygens: The Measurement of Time and of Longitude at Sea". Princeton University. Retrieved 2007-05-27.

[11] Nauenberg, Michael (2004). "Hooke and Newton: Divining Planetary Motions". Physics Today. 57 (2): 13. doi:10.1063/1.1688052. Retrieved 2007-05-30.

[12] The KGM Group, Inc. (2004). "Heliocentric Models". Science Master. Retrieved 2007-05-30.

[13] Giovannangeli, Françoise (November 1996). "Spinning Foucault's Pendulum at the Panthéon". The Paris Pages. Retrieved 2007-05-25.

[14] "A Revolution in Timekeeping". A Walk Through Time. Time and Frequency Div., NIST. April 30, 2002. Retrieved 2007-05-29.

[15] 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.

[rad66-16] Resnick, R.; Halliday, D. (1966). Physics. New York: John Wiley & Sons, Inc. p. 358. ISBN 0-471-71715-0.{{cite book}}: CS1 maint: multiple names: authors list (link)

[17] Symon, Keith R. (1971). Mechanics (Third edition ed.). Reading, Massachusetts: Addison-Wesley Publishing Co. ISBN 0-201-07392-7. {{cite book}}: |edition= has extra text (help)

[18] What is the COP and does it matter?

[19] Weisstein, Eric W. (2007). "Double Pendulum". Wolfram Research. Retrieved 2007-05-29.

[20] Rott, N (1970). "A multiple pendulum for the demonstration of non-linear coupling". Zeitschrift für Angewandte Mathematik und Physik. 21 (4): 570–582.

[21] An interesting simulation of thurible motion can be found at this site.

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 ==Other applications==
 ===Schuler tuning===
-As first explained by [[Max Schuler|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.
+As first explained by [[Max Schuler|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.  Pendulums are used in swing metronomes for pianists.
 ===Religious practice===