# User Contributed Dictionary

- Plural of pendulum

# Extensive Definition

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 massless string, 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 4th century Chinese Book of Taha hasan, 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). Its function was to sway and activate a series of levers after being disturbed by the tremor of an earthquake far away. 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).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 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.

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. 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.

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. 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.

In his 1673 opus Horologium Oscillatorium sive de
motu pendulorum, 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 (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.

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. 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. 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.

## Basic principles

### 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) 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\,

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.

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 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. The resulting period is given by an
infinite
series:

- T = 2\pi \sqrt \left ( 1 + \frac \cdot \sin^2 \frac + \frac \cdot \sin^4 \frac + \cdots \right ).

### 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. 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.

## 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 pianists.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.### 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.## See also

## Notes

## Further reading

- 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: Its 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.

## External links

pendulums in Arabic: بندول

pendulums in Bulgarian: Махало

pendulums in Catalan: Pèndol

pendulums in Czech: Kyvadlo

pendulums in Danish: Matematisk pendul

pendulums in German: Pendel

pendulums in Spanish: Péndulo

pendulums in Esperanto: Pendolo

pendulums in Basque: Pendulu

pendulums in Persian: آونگ

pendulums in French: Pendule (physique)

pendulums in Korean: 진자

pendulums in Italian: Pendolo

pendulums in Hebrew: מטוטלת מתמטית

pendulums in Hungarian: Matematikai inga

pendulums in Malay (macrolanguage): Bandul

pendulums in Dutch: Slinger (natuurkunde)

pendulums in Japanese: 振り子

pendulums in Polish: Wahadło

pendulums in Portuguese: Pêndulo

pendulums in Romanian: Pendul
gravitaţional

pendulums in Russian: Математический
маятник

pendulums in Slovak: Kyvadlo

pendulums in Slovenian: Nihalo

pendulums in Finnish: Heiluri

pendulums in Swedish: Pendel

pendulums in Ukrainian: Маятник

pendulums in Chinese: 擺