A pendulum is a mechanical device consisting of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting mechanical equilibrium position, it is subject to a restoring force due to gravity, which will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, or frequency, is commonly known as its "period". The period depends both on the length of the pendulum and, somewhat, on the amplitude (the width of the pendulum's swing). The SI unit of the period of a pendulum is the second (s).
The regular motion of a pendulum lends itself to timekeeping. The pendulum clock was invented by Christiaan Huygens in 1656. It became the world's standard manner of timekeeping for several centuries. The Shortt-Synchronome clock achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically, pendulums were used as gravimeters to measure the acceleration of gravity in geo-physical surveys, and as a standard of length. The word pendulum is Neo-Latin, from the Latin , meaning .
Mechanics
Simple gravity pendulum
The simple gravity pendulumdefined by Christiaan Huygens: , Part 4, Definition 3, translated July 2007 by Ian Bruce is an idealized mathematical model of a pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.
Period of oscillation
The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude., p.188-194 It is independent of the mass of the bob. If the amplitude is limited to small swings,A "small" swing is one in which the angle is small enough that can be approximated by when is measured in radians the period of a simple pendulum, the time taken for a complete cycle, is:
where is the length of the pendulum and is the local acceleration of gravity.
For small swings the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ0 = 0.4 radians (23°) it is 1% larger than given by (1). The period increases asymptotically (to infinity) as θ0 approaches π radians (180°), because the value θ0 = π is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see pendulum (mechanics)), one example being the infinite series: includes a derivation
where is in radians.
The difference between this true period and the period for small swings (1) above is called the circular error. In the case of a typical grandfather clock whose pendulum has a swing of 6° and thus an amplitude of 3° (0.05 radians), the difference between the true period and the small angle approximation (1) amounts to about 15 seconds per day.
For small swings the pendulum approximates a harmonic oscillator, and its motion as a function of time, t, is approximately simple harmonic motion:
where is a constant value, dependent on initial conditions.
For real pendulums, the period varies slightly with factors such as the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, and flexibility and stretching of the string. In precision applications, corrections for these factors may need to be applied to eq. (1) to give the period accurately.
A damped, driven pendulum is a chaotic system.
Compound pendulum
Any swinging rigid body free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum. A compound pendulum has the same period as a simple gravity pendulum of length , called the equivalent length or radius of oscillation, equal to the distance from the pivot to a point called the center of oscillation., Part 4, Proposition 5 This point is located under the center of mass of the pendulum, at a distance which depends on the mass distribution of the pendulum. If most of the mass is concentrated in a relatively small bob compared to the pendulum length, the center of oscillation is close to the center of mass.
The radius of oscillation or equivalent length of any physical pendulum can be shown to be
where is the moment of inertia of the pendulum about the pivot point , is the total mass of the pendulum, and is the distance between the pivot point and the center of mass. Substituting this expression in (1) above, the period of a compound pendulum is given by
for sufficiently small oscillations.
For example, a rigid uniform rod of length pivoted about one end has moment of inertia . The center of mass is located at the center of the rod, so Substituting these values into the above equation gives . This shows that a rigid rod pendulum has the same period as a simple pendulum of two-thirds its length.
Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable.Huygens (1673) Horologium Oscillatorium, Part 4, Proposition 20 This means if any pendulum is turned upside down and swung from a pivot located at its previous center of oscillation, it will have the same period as before and the new center of oscillation will be at the old pivot point. In 1817 Henry Kater used this idea to produce a type of reversible pendulum, now known as a Kater pendulum, for improved measurements of the acceleration due to gravity.
Double pendulum
In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of coupled ordinary differential equations and is chaotic.
History
One of the earliest known uses of a pendulum was a 1st-century seismometer device of Han dynasty Chinese scientist Zhang Heng.Morton, W. Scott and Charlton M. Lewis (2005). China: Its History and Culture. New York: McGraw-Hill, Inc., p. 70 Its function was to sway and activate one of a series of levers after being disturbed by the tremor of an earthquake far away.Needham, Volume 3, 627-629 Released by a lever, a small ball would fall out of the urn-shaped device into one of eight metal toads' mouths below, at the eight points of the compass, signifying the direction the earthquake was located.
Many sources claim that the 10th-century Egyptian astronomer Ibn Yunus used a pendulum for time measurement, but this was an error that originated in 1684 with the British historian Edward Bernard., reprinted on the Muslim Heritage website.
During the Renaissance, large hand-pumped pendulums were used as sources of power for manual reciprocating machines such as saws, bellows, and pumps.
1602: Galileo's research
Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around 1602. The first recorded interest in pendulums made by Galileo was around 1588 in his posthumously published notes titled On Motion, in which he noted that heavier objects would continue to oscillate for a greater amount of time than lighter objects. The earliest extant report of his experimental research is contained in a letter to Guido Ubaldo dal Monte, from Padua, dated November 29, 1602. His biographer and student, Vincenzo Viviani, claimed his interest had been sparked around 1582 by the swinging motion of a chandelier in Pisa Cathedral.La Lampada di Galileo, by Francesco Malaguzzi Valeri, for Archivio storico dell'arte, Volume 6 (1893); Editor, Domenico Gnoli; Publisher Danesi, Rome; Page 215-218. Galileo discovered the crucial property that makes pendulums useful as timekeepers, called isochronism; the period of the pendulum is approximately 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. He first employed freeswinging pendulums in simple timing applications. Santorio Santori in 1602 invented a device which measured a patient's pulse by the length of a pendulum; the pulsilogium. In 1641 Galileo dictated to his son Vincenzo a design for a mechanism to keep a pendulum swinging, which has been described as the first pendulum clock; Vincenzo began construction, but had not completed it when he died in 1649.Drake 2003, p.419–420
1656: The pendulum clock
In 1656 the Dutch scientist Christiaan Huygens built the first pendulum clock.although there are unsubstantiated references to prior pendulum clocks made by others: This was a great improvement over existing mechanical clocks; their best accuracy was improved from around 15 minutes deviation a day to around 15 seconds a day. Pendulums spread over Europe as existing clocks were retrofitted with them.Milham 1945, p.145
The English scientist Robert Hooke studied the conical pendulum around 1666, consisting of a pendulum that is free to swing in two dimensions, with the bob rotating in a circle or ellipse. 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. This played a part in Newton's formulation of the 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.
During his expedition to Cayenne, French Guiana in 1671, Jean Richer found that a pendulum clock was minutes per day slower at Cayenne than at Paris. From this he deduced that the force of gravity was lower at Cayenne. cited in Lenzen & Multauf, 1964, p.307 In 1687, Isaac Newton in Principia Mathematica showed that this was because the Earth was not a true sphere but slightly oblate (flattened at the poles) from the effect of centrifugal force due to its rotation, causing gravity to increase with latitude.Lenzen & Multauf, 1964, p.307 Portable pendulums began to be taken on voyages to distant lands, as precision gravimeters to measure the acceleration of gravity at different points on Earth, eventually resulting in accurate models of the shape of the Earth.
1673: Huygens's Horologium Oscillatorium