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"Simple gravity pendulum" assumes no air resistance and no friction of/at the nail/screw.
"Simple gravity pendulum" assumes no air resistance and no friction of/at the nail/screw.
An animation of a pendulum showing the velocity and acceleration vectors (v and A).
An animation of a pendulum showing the velocity and acceleration vectors (v and A).

A pendulum is a weight suspended from a pivot so 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. The time for one complete cycle, a left swing and a right swing, is called the period. The regular motion of pendulums has been used since the 17th century for timekeeping, and pendulums are used to regulate pendulum clocks.

An idealized mathematical model 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 at a constant amplitude forever.


Contents

[edit] History

One of the earliest uses of a pendulum was in the first century seismometer device of Han Dynasty China scientist and inventor Zhang Heng.[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 direction the earthquake was located.[2]

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]

Beginning in 1602, Galileo Galilei performed a number of observations of the properties of pendulums. His interest is said to have been sparked by looking at 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 independent of the amplitude or width of the swing. He also found that the period is independent of the mass of the bob, and proportional to the square root of the length of the pendulum. The isochronism of the pendulum suggested a practical application for use as a metronome to aid musical students, and he also .[6]

Perhaps inspired by Galileo, in 1656 the Dutch scientist Christiaan Huygens built and patented the first clock that employed a pendulum to regulate the movement.[7] The first harmonic oscillator used by man, this was a great improvement over previous mechanical clocks; their accuracy was increased from perhaps an half an hour a day to a minute a day. Pendulums spread over Europe as existing clocks were retrofitted with them.[8]

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.[9] 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.[10]

In his 1673 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 of isochronism was accurate only for small swings.[12] He also solved the problem of how to calculate the period of an arbitrarily shaped pendulum, and discovered the center of oscillation and its symmetry with the pivot point.

The existing clock movement, the verge escapement, indeed drove pendulums in very wide arcs of about 100°, causing the period to vary with the clock's drive force. In an attempt to make it's period isochronous, Huygens had mounted cycloidal-shaped 'cheeks' next to the pivot in his clock, that constrained the cord and forced the pendulum to follow a cycloid arc. This was a practical failure, but motivated the invention of the anchor escapement in 1670, which reduced the pendulum swing in clocks to a more isochronous 4°-6°.[13]

The English scientist Robert Hooke devised the conical pendulum, consisting of a pendulum that is free to swing in two dimensions. He used the motions of this device as a model to analyze the orbital motions of the planets. Hooke suggested 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.[14][15] Robert Hooke was also responsible for suggesting (as early as 1666) that the pendulum could be used to measure the force of gravity.

During the 18th century, the pendulum clock's role as the most accurate timekeeper motivated much practical research into improving pendulums.

In 1851, Jean Bernard Léon Foucault showed that the plane of oscillation of a pendulum, like a gyroscope, 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 (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 swing was observed to precess or rotate 360° clockwise in about 32 hours.[16] This was the first demonstration of the Earth's rotation that didn't depend on astronomical observations, and attracted large crowds.[17]


[edit] Basic principles

Main article: Pendulum (mathematics)

If and only if the pendulum swings through a small angle (in the range where the function sin(θ) can be approximated as θ)[18] the system mathematically approximates a harmonic oscillator, and the motion approximates 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 a complete cycle; two swings (left to right and back again). The formula for the period, T, is

T \approx 2\pi \sqrt\frac{\ell}{g} \qquad \qquad \qquad \qquad (1)\,

where \ell is the length of the pendulum measured from the pivot point to the bob's center of gravity.[19]

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 pivot 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.[20] 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 )[19]

For small values of θmax, the value of the sine terms become negligible and the period reduces to the approximation (1); the difference is called circular error.

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. [21] 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.

[edit] Use for time measurement

From it's invention in 1656 until the advent of the quartz clock in the 1930s, the pendulum was the world's most accurate timekeeping technology.[22][23] Accurate pendulum clocks called regulators, in places of business, were used to set other clocks. The most accurate, known as astronomical regulators, were used in observatories. Beginning in the Industrial Revolution, precision pendulum clocks at naval observatories served as primary standards for national time distribution services.[24] The National Bureau of Standards (now NIST) based the U.S. time standard on a 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 Shortt clock before the NIST switched to a quartz standard.[25] [26]

A pendulum whose period of oscillation is 2 seconds, that is with each swing taking 1 second, is called the seconds pendulum. Freeswinging seconds pendulums were widely used as precision timers in scientific experiments in the 17th through 19th centuries, before accurate portable clocks were available. In 1644, even before the pendulum clock, Marin Mersenne first determined that the length of the seconds pendulum was 39.1 inches (0.993m). Later when it was found to vary slightly with location, tables of the length of the seconds pendulum at various locales on the globe were compiled.

The need for more accurate pendulum clocks drove improvements in pendulum technology during the 18th and 19th century:

[edit] Temperature compensation

The largest cause of error in early pendulums was slight changes in length due to thermal expansion and contraction of the pendulum rod with changes in temperature. People noticed that pendulum clocks gained time in winter and lost time in summer. A pendulum with a steel rod will get about .0000063 longer with each 1° F temperature increase, causing it to lose about 0.27 seconds/day, or 16 seconds/day for a 60° F change. Wood rods expand less, losing only about 6 seconds/day for a 60° F change, which is why early clocks often had wooden pendulum rods.

[edit] Mercury pendulum

The first device to compensate for this error was the mercury pendulum, invented by George Graham in 1721. The liquid metal mercury expands in volume with temperature. In a mercury pendulum, the pendulum's weight (bob) is made of a container of mercury. With a temperature rise, the pendulum rod gets longer, but the mercury also expands and its surface level rises in the container, moving its center of mass 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 several days to reach the new temperature, causing the rate to deviate during that time. Mercury pendulums were used in precision clocks into the 1900s.

[edit] Gridiron pendulum

The most widely used compensated pendulum was the gridiron pendulum, invented in 1726 by John Harrison. This consists of alternating rods of two different metals, one with lower thermal expansion, steel, and one with higher thermal expansion, zinc or brass. The rods are connected at top and bottom as shown. With a temperature increase, the low expansion rods make the pendulum longer, while the high expansion rods make it shorter. By making the rods of the correct lengths, the changes cancel out 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 caused gridiron pendulums to adjust in a series of tiny jumps. In high precision clocks this caused the clock's rate to change suddenly with each jump. Later it was found that zinc is subject to creep. For these reasons mercury pendulums were used in the highest precision clocks. Gridiron pendulums were widely used in quality clocks and regulators.

[edit] Invar

Around 1900 low thermal expansion materials were developed which, when used as pendulum rods, made elaborate temperature compensation unnecessary. In 1896 Charles Edouard Guillaume invented the nickel steel alloy Invar. This has a thermal expansion of around 0.5 ppm/degree F, resulting in pendulum temperature errors over 60° F of only 1.3 seconds/day, and this residual error could be compensated to zero with a few centimeters of aluminum under the pendulum bob. Its

[edit] Other applications

[edit] Standard of length

[edit] Gravity measurement

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

[edit] Siesmometers

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.

[edit] 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.

[edit] Coupled pendulums

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.[27]

Two coupled pendulums form a double pendulum. Many physical systems can be mathematically described as coupled pendula. Under certain conditions these systems can also demonstrate chaotic motion.

[edit] 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.[28] See also pendula for divination and dowsing.

[edit] See also

[edit] 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. ^ Milham, Willis I. (1945). Time and Timekeepers. MacMillan. , p.144-145
  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. ^ Usher, Abbott Payson (1988). A History of Mechanical Inventions. Courier Dover. ISBN 048625593X. , p.312
  14. ^ Nauenberg, Michael (2004). "Hooke and Newton: Divining Planetary Motions". Physics Today 57 (2): 13. 
  15. ^ The KGM Group, Inc. (2004). Heliocentric Models. Science Master. Retrieved on 2007-05-30.
  16. ^ Rubin, Julian (Sept 2007). The Invention of the Foucault Pendulum. Following the Path of Discovery. Retrieved on 2007-10-31.
  17. ^ Giovannangeli, Françoise (November 1996). Spinning Foucault's Pendulum at the Panthéon. The Paris Pages. Retrieved on 2007-05-25.
  18. ^ 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.
  19. ^ a b Resnick, R.; Halliday, D. (1966). Physics. New York: John Wiley & Sons, Inc., 358. ISBN 0-471-71715-0. 
  20. ^ Symon, Keith R. (1971). Mechanics, Third edition, Reading, Massachusetts: Addison-Wesley Publishing Co.. ISBN 0-201-07392-7. 
  21. ^ Weisstein, Eric W. (2007). Double Pendulum. Wolfram Research. Retrieved on 2007-05-29.
  22. ^ Milham 1945, (p.330) precision regulators are the time standards of the world (p.334) they are all pendulum clocks
  23. ^ Marrison, Warren (1948). "The Evolution of the Quartz Crystal Clock". Bell System Technical Journal 27: 510-588. 
  24. ^ Milham, Willis I. (1945). Time and Timekeepers. New York: MacMillan. ISBN 0780800087. , p.83
  25. ^ Staff (April 30, 2002). A Revolution in Timekeeping. NIST. Retrieved on 2007-05-29.
  26. ^ Sullivan, D.B. (2001). "Time and frequency measurement at NIST: The first 100 years". . National Institute of Standards and Technology
  27. ^ 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.
  28. ^ An interesting simulation of thurible motion can be found at this site.

[edit] Further reading

  • Guillermo Ortuño Crespo. 365 ways to do it with your pc and USB. science and maths
  • 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.

[edit] External links


[edit] References