Pendulum

For other uses, see Pendulum (disambiguation).

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.

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.

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

History

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). 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).

An Egyptian scholar, Ibn Yunus, is known to have described an early pendulum in the 10th century. 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.

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.

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 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.

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, 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.

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. 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.

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.

Basic principles

Main article: Pendulum (mathematics)

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 two swings (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 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 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. The resulting period is given by an infinite series:

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

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 earlier.

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.

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

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

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. 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.

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.

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.

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.

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.

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 pendula for divination and dowsing.

See also

• Pendulum clock
• Simple harmonic motion
• Conical pendulum
• Spherical pendulum
• Double pendulum
• Foucault pendulum
• Kater's pendulum
• Harmonograph (a.k.a. "Lissajoux pendulum")
• Metronome
• Seconds pendulum
• Torsional pendulum
• Inverted pendulum

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: 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.

External links

• [1]
• Graphical derivation of the time period for a simple pendulum
• A more general explanation of pendula
• FORTRAN code for a numerical model of a simple pendulum
• FORTRAN code for modeling of a simple pendulum using the Euler and Euler-Cromer methods