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For small displacements, the movement of an ideal pendulum can be described mathematically as simple harmonic motion, as the change in potential energy at the bottom of a circular arc is nearly proportional to the square of the displacement. Real pendulums do not have infinitesimal displacements, so their behavior is actually non-linear. Real pendulums will also lose energy as they swing, and so their motion will be damped, with the size of the oscillation decreasing approximately exponentially with time. In the case of a pendulum with a mass
- <math>
This equation only applies when the amplitude of the swing is small; a complete description of a pendulum's behavior is not mathematically simple. Two coupled pendulums form a double pendulum.
If
- <math>T = 2 \pi \sqrt{\frac{I}{K}}</math>
Both
- <math>T' = 2\pi \sqrt{\frac{I+I'}{K}}</math>
and then solving the two equations to get
- <math>K = \frac{4\pi^2I'}{T'^2 - T^2}</math>
- <math>I = \frac{T^2I'}{T'^2 - T^2}</math>
The oscillating balance wheel of a watch is in effect a torsion pendulum, with the suspending fiber replaced by hairspring and pivots. The watch is regulated, first roughly by adjusting |

A When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with 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, is called the period. From its discovery around 1602 by Galileo Galilei the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s. The ## Period of oscillationThe period of swing of a simple gravity pendulum depends on its length, the acceleration of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, ^{[8]} It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:^{[9]}where For small swings, the period of swing is approximately the same for different size swings: that is, This formula is strictly accurate only for tiny infinitesimal swings. For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of ^{[11]}^{[12]}The difference between this true period and the period for small swings (1) above is called the Mathematically, for small swings the pendulum approximates a harmonic oscillator, and its motion approximates to simple harmonic motion: ## Compound pendulumThe length Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable. ## HistoryOne of the earliest known uses of a pendulum was in the first century seismometer device of Han Dynasty Chinese scientist Zhang Heng. Many recent encyclopaedias During the Renaissance, large pendulums were used as sources of power for manual reciprocating machines such as saws, bellows, and pumps. Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around 1602. In 1656 the Dutch scientist Christiaan Huygens built the first pendulum clock. 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. During his expedition to Cayenne, French Guiana in 1671, Jean Richer found that a pendulum clock was 2+ In 1673, Christiaan Huygens published his theory of the pendulum, The existing clock movement, the verge escapement, made pendulums swing in very wide arcs of about 100°. During the 18th and 19th century, the pendulum clock's role as the most accurate timekeeper motivated much practical research into improving pendulums. It was found that a major source of error was that the pendulum rod expanded and contracted with changes in ambient temperature, changing the period of swing. The accuracy of gravity measurements made with pendulums was limited by the difficulty of finding the location of their center of oscillation. Huygens had discovered in 1673 that a pendulum has the same period when hung from its center of oscillation as when hung from its pivot, 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 free to swing in two dimensions (later named the Foucault pendulum) from the dome of the Panthéon in Paris. The length of the cord 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. Around 1900 low thermal expansion materials began to be used for pendulum rods in the highest precision clocks and other instruments, first invar, a nickel steel alloy, and later fused quartz, which made temperature compensation trivial. The timekeeping accuracy of the pendulum was exceeded by the quartz crystal oscillator, invented in 1921, and quartz clocks, invented in 1927, replaced pendulum clocks as the world's best timekeepers, Clock pendulums ## Use for time measurementFor 300 years, from its discovery around 1602 until development of the quartz clock in the 1930s, the pendulum was the world's standard for accurate timekeeping. ## Clock pendulumsPendulums in clocks (see example at right) are usually made of a weight or bob Each time the pendulum swings through its center position, it releases one tooth of the The pendulum always has a means of adjusting the period, usually by an adjustment nut The pendulum must be suspended from a rigid support. The most common pendulum length in quality clocks, which is always used in grandfather clocks, is the seconds pendulum, about 1 meter (39 inches) long. In mantel clocks, half-second pendulums, 25 cm (10 in) long, or shorter, are used. Only a few large tower clocks use longer pendulums, the 1.5 second pendulum, 2.25 m (7 ft) long, or occasionally the two-second pendulum, 4 m (13 ft). ## Temperature compensationThe largest source of error in early pendulums was slight changes in length due to thermal expansion and contraction of the pendulum rod with changes in ambient temperature. ## Mercury pendulumThe first device to compensate for this error was the mercury pendulum, invented by George Graham ## Gridiron pendulumThe most widely used compensated pendulum was the gridiron pendulum, invented in 1726 by John Harrison. 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 in the frame caused gridiron pendulums to adjust in a series of tiny jumps. ## Invar and fused quartzAround 1900 low thermal expansion materials were developed which, when used as pendulum rods, made elaborate temperature compensation unnecessary. ## Atmospheric pressureThe presence of air around the pendulum has three effects on the period: - By Archimedes principle the effective weight of the bob is reduced by the buoyancy of the air it displaces, while the mass (inertia) remains the same, reducing the pendulum's acceleration during its swing and increasing the period. This depends on the density but not the shape of the pendulum.
- The pendulum carries an amount of air with it as it swings, and the mass of this air increases the inertia of the pendulum, again reducing the acceleration and increasing the period.
- Viscous air resistance slows the pendulum's velocity. This has a negligible effect on the period, but dissipates energy, reducing the amplitude. This reduces the pendulum's Q factor, requiring a stronger drive force from the clock's mechanism to keep it moving, which causes increased disturbance to the period.
So increases in barometric pressure slow the pendulum slightly due to the first two effects, by about 0.11 seconds per day per kilopascal (0.37 seconds per day per inch of mercury or 0.015 seconds per day per torr). ## GravityPendulums are affected by changes in gravitational acceleration, which varies by as much as 0.5% at different locations on Earth, so pendulum clocks have to be recalibrated after a move. Even moving a pendulum clock to the top of a tall building can cause it to lose measurable time from the reduction in gravity. ## Accuracy of pendulums as timekeepersThe timekeeping elements in all clocks, which include pendulums, balance wheels, the quartz crystals used in quartz watches, and even the vibrating atoms in atomic clocks, are in physics called harmonic oscillators. The reason harmonic oscillators are used in clocks is that they vibrate or oscillate at a specific resonant frequency or period and resist oscillating at other rates. However the resonant frequency is not infinitely 'sharp'. Around the resonant frequency there is a narrow natural band of frequencies (or periods), called the resonance width or bandwidth, where the harmonic oscillator will oscillate. |

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The precision of the early gravity measurements above was limited by the difficulty of measuring the length of the pendulum, *L* . *L* was the length of an idealized simple gravity pendulum (described at top), which has all its mass concentrated in a point at the end of the cord. In 1673 Huygens had shown that the period of a real pendulum (called a *compound pendulum*) was equal to the period of a simple pendulum with a length equal to the distance between the pivot point and a point called the center of oscillation, located under the center of gravity, that depends on the mass distribution along the pendulum. But there was no accurate way of determining the center of oscillation in a real pendulum.

To get around this problem, the early researchers above approximated an ideal simple pendulum as closely as possible by using a metal sphere suspended by a light wire or cord. If the wire was light enough, the center of oscillation was close to the center of gravity of the ball, at its geometric center. This "ball and wire" type of pendulum wasn't very accurate, because it didn't swing as a rigid body, and the elasticity of the wire caused its length to change slightly as the pendulum swung.

However Huygens had also proved that in any pendulum, the pivot point and the center of oscillation were interchangeable.^{[15]} That is, if a pendulum were turned upside down and hung from its center of oscillation, it would have the same period as it did in the previous position, and the old pivot point would be the new center of oscillation.

British physicist and army captain Henry Kater in 1817 realized that Huygens' principle could be used to find the length of a simple pendulum with the same period as a real pendulum.^{[48]} If a pendulum was built with a second adjustable pivot point near the bottom so it could be hung upside down, and the second pivot was adjusted until the periods when hung from both pivots were the same, the second pivot would be at the center of oscillation, and the distance between the two pivots would be the length of a simple pendulum with the same period.

Kater built a reversible pendulum (shown at right) consisting of a brass bar with two opposing pivots made of short triangular "knife" blades *(a)* near either end. It could be swung from either pivot, with the knife blades supported on agate plates. Rather than make one pivot adjustable, he attached the pivots a meter apart and instead adjusted the periods with a moveable weight on the pendulum rod *(b,c)*. In operation, the pendulum is hung in front of a precision clock, and the period timed, then turned upside down and the period timed again. The weight is adjusted with the adjustment screw until the periods are equal. Then putting this period and the distance between the pivots into equation (1) gives the gravitational acceleration *g* very accurately.

Kater timed the swing of his pendulum using the "*method of coincidences*" and measured the distance between the two pivots with a microscope. After applying corrections for the finite amplitude of swing, the buoyancy of the bob, the barometric pressure and altitude, and temperature, he obtained a value of 39.13929 inches for the seconds pendulum at London, in vacuum, at sea level, at 62 °F. The largest variation from the mean of his 12 observations was 0.00028 in.^{[96]} representing a precision of gravity measurement of 7×10^{−6} (7mGal or 70 µm/s^{2}). Kater's measurement was used as Britain's official standard of length (see below) from 1824 to 1855.

Reversible pendulums (known technically as "convertible" pendulums) employing Kater's principle were used for absolute gravity measurements into the 1930s.

The increased accuracy made possible by Kater's pendulum helped make gravimetry a standard part of geodesy. Since the exact location (latitude and longitude) of the 'station' where the gravity measurement was made was necessary, gravity measurements became part of surveying, and pendulums were taken on the great geodetic surveys of the 18th century, particularly the Great Trigonometric Survey of India.

**Invariable pendulums:**Kater introduced the idea of*relative*gravity measurements, to supplement the*absolute*measurements made by a Kater's pendulum.^{[97]}Comparing the gravity at two different points was an easier process than measuring it absolutely by the Kater method. All that was necessary was to time the period of an ordinary (single pivot) pendulum at the first point, then transport the pendulum to the other point and time its period there. Since the pendulum's length was constant, from (1) the ratio of the gravitational accelerations was equal to the square root of the ratio of the periods, and no precision length measurements were necessary. So once the gravity had been measured absolutely at some central station, by the Kater or other accurate method, the gravity at other points could be found by swinging pendulums at the central station and then taking them to the nearby point. Kater made up a set of "invariable" pendulums, with only one knife edge pivot, which were taken to many countries after first being swung at a central station at Kew Observatory, UK.

**Airy's coal pit experiments**: Starting in 1826, using methods similar to Bouguer, British astronomer George Airy attempted to determine the density of the Earth by pendulum gravity measurements at the top and bottom of a coal mine.^{[98]}^{[99]}The gravitational force below the surface of the Earth decreases rather than increasing with depth, because by Gauss's law the mass of the spherical shell of crust above the subsurface point does not contribute to the gravity. The 1826 experiment was aborted by the flooding of the mine, but in 1854 he conducted an improved experiment at the Harton coal mine, using seconds pendulums swinging on agate plates, timed by precision chronometers synchronized by an electrical circuit. He found the lower pendulum was slower by 2.24 seconds per day. This meant that the gravitational acceleration at the bottom of the mine, 1250 ft below the surface, was 1/14,000 less than it should have been from the inverse square law; that is the attraction of the spherical shell was 1/14,000 of the attraction of the Earth. From samples of surface rock he estimated the mass of the spherical shell of crust, and from this estimated that the density of the Earth was 6.565 times that of water. Von Sterneck attempted to repeat the experiment in 1882 but found inconsistent results.

**Repsold-Bessel pendulum:**It was time-consuming and error-prone to repeatedly swing the Kater's pendulum and adjust the weights until the periods were equal. Friedrich Bessel showed in 1835 that this was unnecessary.^{[100]}As long as the periods were close together, the gravity could be calculated from the two periods and the center of gravity of the pendulum.^{[101]}So the reversible pendulum didn't need to be adjustable, it could just be a bar with two pivots. Bessel also showed that if the pendulum was made symmetrical in form about its center, but was weighted internally at one end, the errors due to air drag would cancel out. Further, another error due to the finite diameter