Pendulum

A pendulum is a device made of a

, the width of the pendulum's swing.

The regular motion of pendulums was used for timekeeping and was the world's most accurate timekeeping technology until the 1930s.

Simple gravity pendulum

The simple gravity pendulum

air drag

, so the amplitude of their swings declines.

Period of oscillation

The period of swing of a

$${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}\qquad \qquad \qquad \theta _{0}\ll 1{\text{ radian}}}$$

(1)

where ${\displaystyle L}$ is the length of the pendulum and ${\displaystyle g}$ 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

Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

For larger

$${\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}\left[\sum _{n=0}^{\infty }\left({\frac {\left(2n\right)!}{2^{2n}\left(n!\right)^{2}}}\right)^{2}\sin ^{2n}\left({\frac {\theta _{0}}{2}}\right)\right]=2\pi {\sqrt {\frac {L}{g}}}\left(1+{\frac {1}{16}}\theta _{0}^{2}+{\frac {11}{3072}}\theta _{0}^{4}+\cdots \right)}$$

where ${\displaystyle \theta _{0}}$ 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:^[5]

where ${\displaystyle \varphi }$ is a constant value, dependent on .

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.^[11][13] 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.^{[citation needed]}

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 ${\displaystyle \ell ^{\mathrm {eq} }}$, called the equivalent length or radius of oscillation, equal to the distance from the pivot to a point called the center of oscillation.^[14] 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.^[15]

The radius of oscillation or equivalent length ${\displaystyle \ell ^{\mathrm {eq} }}$ of any physical pendulum can be shown to be

where ${\displaystyle I_{O}}$ is the moment of inertia of the pendulum about the pivot point ${\displaystyle O}$, ${\displaystyle m}$ is the total mass of the pendulum, and ${\displaystyle r_{\mathrm {CM} }}$ is the distance between the pivot point and the center of mass. Substituting this expression in (1) above, the period ${\displaystyle T}$ of a compound pendulum is given by

for sufficiently small oscillations.^[16]

For example, a rigid uniform rod of length ${\displaystyle \ell }$ pivoted about one end has moment of inertia ${\textstyle I_{O}={\frac {1}{3}}m\ell ^{2}}$. The center of mass is located at the center of the rod, so ${\textstyle r_{\mathrm {CM} }={\frac {1}{2}}\ell }$ Substituting these values into the above equation gives ${\textstyle T=2\pi {\sqrt {\frac {{\frac {2}{3}}\ell }{g}}}}$. This shows that a rigid rod pendulum has the same period as a simple pendulum of two-thirds its length.

, for improved measurements of the acceleration due to gravity.

History

One of the earliest known uses of a pendulum was a 1st-century

Many sources[20]^[21][22][23] 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.^{[24][25][26][27]}

During the Renaissance, large hand-pumped pendulums were used as sources of power for manual reciprocating machines such as saws, bellows, and pumps.^[28] Leonardo da Vinci made many drawings of the motion of pendulums, though without realizing its value for timekeeping.

1602: Galileo's research

Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around 1602.^[29] The first recorded interest in pendulums made by Galileo was around 1588 in his posthumously published notes titled On Motion,^{[30] [31]} 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.^[32] His biographer and student, Vincenzo Viviani, claimed his interest had been sparked around 1582 by the swinging motion of a chandelier in Pisa Cathedral.^[33][34] 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.^[35] 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.^[36] 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;^[35] Vincenzo began construction, but had not completed it when he died in 1649.^[37]

1656: The pendulum clock

In 1656 the Dutch scientist

The English scientist

During his expedition to

1673: Huygens' Horologium Oscillatorium

In 1673, 17 years after he invented the pendulum clock,

The existing clock movement, the

This became the standard escapement used in pendulum clocks.

1721: Temperature compensated pendulums

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.^[8][59] This was solved with the invention of temperature compensated pendulums, the mercury pendulum in 1721^[60] and the gridiron pendulum in 1726, reducing errors in precision pendulum clocks to a few seconds per week.^[57]

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,^[17] and the distance between the two points was equal to the length of a simple gravity pendulum of the same period.^[14] In 1818 British Captain Henry Kater invented the reversible Kater's pendulum^[61] which used this principle, making possible very accurate measurements of gravity. For the next century the reversible pendulum was the standard method of measuring absolute gravitational acceleration.

1851: Foucault pendulum

In 1851,

This was the first demonstration of the Earth's rotation that did not depend on celestial observations,[63] and a "pendulum mania" broke out, as Foucault pendulums were displayed in many cities and attracted large crowds.^[64][65]

1930: Decline in use

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.^[66] Precision pendulums were housed in low pressure tanks, which kept the air pressure constant to prevent changes in the period due to changes in buoyancy of the pendulum due to changing atmospheric pressure.^[66] The best pendulum clocks achieved accuracy of around a second per year.^[67][68]

The timekeeping accuracy of the pendulum was exceeded by the

but pendulum instruments continued to be used into the 1970s.

Use for time measurement

For 300 years, from its discovery around 1582 until development of the quartz clock in the 1930s, the pendulum was the world's standard for accurate timekeeping.^[2][71] In addition to clock pendulums, freeswinging seconds pendulums were widely used as precision timers in scientific experiments in the 17th and 18th centuries. Pendulums require great mechanical stability: a length change of only 0.02%, 0.2 mm in a grandfather clock pendulum, will cause an error of a minute per week.^[72]

Clock pendulums

Pendulums in clocks (see example at right) are usually made of a weight or

, (g,h).

Each time the pendulum swings through its centre position, it releases one tooth of the escape wheel (g). The force of the clock's

gear train

, causes the wheel to turn, and a tooth presses against one of the pallets (h), giving the pendulum a short push. The clock's wheels, geared to the escape wheel, move forward a fixed amount with each pendulum swing, advancing the clock's hands at a steady rate.

The pendulum always has a means of adjusting the period, usually by an adjustment nut (c) under the bob which moves it up or down on the rod.[8]^[76] Moving the bob up decreases the pendulum's length, causing the pendulum to swing faster and the clock to gain time. Some precision clocks have a small auxiliary adjustment weight on a threaded shaft on the bob, to allow finer adjustment. Some tower clocks and precision clocks use a tray attached near to the midpoint of the pendulum rod, to which small weights can be added or removed. This effectively shifts the centre of oscillation and allows the rate to be adjusted without stopping the clock.^[77][78]

The pendulum must be suspended from a rigid support.^[8][79] During operation, any elasticity will allow tiny imperceptible swaying motions of the support, which disturbs the clock's period, resulting in error. Pendulum clocks should be attached firmly to a sturdy wall.

The most common pendulum length in quality clocks, which is always used in grandfather clocks, is the seconds pendulum, about 1 metre (39 inches) long. In mantel clocks, half-second pendulums, 25 cm (9.8 in) long, or shorter, are used. Only a few large tower clocks use longer pendulums, the 1.5 second pendulum, 2.25 m (7.4 ft) long, or occasionally the two-second pendulum, 4 m (13 ft) ^[8][80] which is used in Big Ben.^[81]

Temperature compensation

The 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.

(ppm) with each degree Celsius increase, causing it to lose about 0.27 seconds per day for every degree Celsius increase in temperature, or 9 seconds per day for a 33 °C (59 °F) change. Wood rods expand less, losing only about 6 seconds per day for a 33 °C (59 °F) change, which is why quality clocks often had wooden pendulum rods. The wood had to be varnished to prevent water vapor from getting in, because changes in humidity also affected the length.

Mercury pendulum

The first device to compensate for this error was the mercury pendulum, invented by

Gridiron pendulum

The most widely used compensated pendulum was the

. The rods are connected by a frame, as shown in the drawing at the right, so that an increase in length of the zinc rods pushes the bob up, shortening the pendulum. With a temperature increase, the low expansion steel rods make the pendulum longer, while the high expansion zinc rods make it shorter. By making the rods of the correct lengths, the greater expansion of the zinc cancels out the expansion of the steel rods which have a greater combined length, 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 in the frame caused gridiron pendulums to adjust in a series of tiny jumps.[87] 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, but gridirons were used in quality regulator clocks.

Gridiron pendulums became so associated with good quality that, to this day, many ordinary clock pendulums have decorative 'fake' gridirons that don't actually have any temperature compensation function.

Invar and fused quartz

Around 1900, low thermal expansion materials were developed which could be used as pendulum rods in order to make elaborate temperature compensation unnecessary.

Atmospheric pressure

The effect of the surrounding air on a moving pendulum is complex and requires fluid mechanics to calculate precisely, but for most purposes its influence on the period can be accounted for by three effects:^[66][91]

• 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 air pressure and the density of the pendulum, but not its shape.
• 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. This depends on both its density and shape.
• 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.

Increases in

aneroid barometer

mechanism attached to the pendulum compensated for this effect.

Gravity

Pendulums are affected by changes in gravitational acceleration, which varies by as much as 0.5% at different locations on Earth, so precision 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 timekeepers

The timekeeping elements in all clocks, which include pendulums,

energy loss per swing of the pendulum.

Q factor

Gaithersburg, MD

, USA. It kept time with two synchronized pendulums. The master pendulum in the vacuum tank (left) swung free of virtually any disturbance, and controlled the slave pendulum in the clock case (right) which performed the impulsing and timekeeping tasks. Its accuracy was about a second per year.

The measure of a harmonic oscillator's resistance to disturbances to its oscillation period is a dimensionless parameter called the

The Q is related to how long it takes for the oscillations of an oscillator to die out. The Q of a pendulum can be measured by counting the number of oscillations it takes for the amplitude of the pendulum's swing to decay to 1/e = 36.8% of its initial swing, and multiplying by 'π.

In a clock, the pendulum must receive pushes from the clock's movement to keep it swinging, to replace the energy the pendulum loses to friction. These pushes, applied by a mechanism called the escapement, are the main source of disturbance to the pendulum's motion. The Q is equal to 2π times the energy stored in the pendulum, divided by the energy lost to friction during each oscillation period, which is the same as the energy added by the escapement each period. It can be seen that the smaller the fraction of the pendulum's energy that is lost to friction, the less energy needs to be added, the less the disturbance from the escapement, the more 'independent' the pendulum is of the clock's mechanism, and the more constant its period is. The Q of a pendulum is given by:

where M is the mass of the bob, ω = 2π/T is the pendulum's radian frequency of oscillation, and Γ is the frictional on the pendulum per unit velocity.

ω is fixed by the pendulum's period, and M is limited by the load capacity and rigidity of the suspension. So the Q of clock pendulums is increased by minimizing frictional losses (Γ). Precision pendulums are suspended on low friction pivots consisting of triangular shaped 'knife' edges resting on agate plates. Around 99% of the energy loss in a freeswinging pendulum is due to air friction, so mounting a pendulum in a vacuum tank can increase the Q, and thus the accuracy, by a factor of 100.^[99]

The Q of pendulums ranges from several thousand in an ordinary clock to several hundred thousand for precision regulator pendulums swinging in vacuum.

Their Q of 10³–10⁵ is one reason why pendulums are more accurate timekeepers than the balance wheels in watches, with Q around 100–300, but less accurate than the quartz crystals in quartz clocks, with Q of 10⁵–10⁶.^[2][100]

Escapement

Pendulums (unlike, for example, quartz crystals) have a low enough Q that the disturbance caused by the impulses to keep them moving is generally the limiting factor on their timekeeping accuracy. Therefore, the design of the escapement, the mechanism that provides these impulses, has a large effect on the accuracy of a clock pendulum. If the impulses given to the pendulum by the escapement each swing could be exactly identical, the response of the pendulum would be identical, and its period would be constant. However, this is not achievable; unavoidable random fluctuations in the force due to friction of the clock's pallets, lubrication variations, and changes in the torque provided by the clock's power source as it runs down, mean that the force of the impulse applied by the escapement varies.

If these variations in the escapement's force cause changes in the pendulum's width of swing (amplitude), this will cause corresponding slight changes in the period, since (as discussed at top) a pendulum with a finite swing is not quite isochronous. Therefore, the goal of traditional escapement design is to apply the force with the proper profile, and at the correct point in the pendulum's cycle, so force variations have no effect on the pendulum's amplitude. This is called an isochronous escapement.

The Airy condition

Clockmakers had known for centuries that the disturbing effect of the escapement's drive force on the period of a pendulum is smallest if given as a short impulse as the pendulum passes through its bottom

Gravity measurement

The presence of the

up to the 1930s.

The difference between clock pendulums and gravimeter pendulums is that to measure gravity, the pendulum's length as well as its period has to be measured. The period of freeswinging pendulums could be found to great precision by comparing their swing with a precision clock that had been adjusted to keep correct time by the passage of stars overhead. In the early measurements, a weight on a cord was suspended in front of the clock pendulum, and its length adjusted until the two pendulums swung in exact synchronism. Then the length of the cord was measured. From the length and the period, g could be calculated from equation (1).

The seconds pendulum

The

, which all had seconds pendulums. By the late 17th century, the length of the seconds pendulum became the standard measure of the strength of gravitational acceleration at a location. By 1700 its length had been measured with submillimeter accuracy at several cities in Europe. For a seconds pendulum, g is proportional to its length:

Early observations

• 1620: British scientist Francis Bacon was one of the first to propose using a pendulum to measure gravity, suggesting taking one up a mountain to see if gravity varies with altitude.^[107]
• 1644: Even before the pendulum clock, French priest Marin Mersenne first determined the length of the seconds pendulum was 39.1 inches (990 mm), by comparing the swing of a pendulum to the time it took a weight to fall a measured distance. He also was first to discover the dependence of the period on amplitude of swing.
• 1669: Jean Picard determined the length of the seconds pendulum at Paris, using a 1-inch (25 mm) copper ball suspended by an aloe fiber, obtaining 39.09 inches (993 mm).^[108] He also did the first experiments on thermal expansion and contraction of pendulum rods with temperature.
• 1672: The first observation that gravity varied at different points on Earth was made in 1672 by
  ellipticity
  of the Earth to be calculated from gravity measurements. Progressively more accurate models of the shape of the Earth followed.
• 1687: Newton experimented with pendulums (described in Principia) and found that equal length pendulums with bobs made of different materials had the same period, proving that the gravitational force on different substances was exactly proportional to their
  general theory of relativity
  .

• 1737: French mathematician
  density of the Earth
  .
• 1747: Daniel Bernoulli showed how to correct for the lengthening of the period due to a finite angle of swing θ₀ by using the first order correction θ₀²/16, giving the period of a pendulum with an extremely small swing.^[111]
• 1792: To define a pendulum standard of length for use with the new metric system, in 1792 Jean-Charles de Borda and Jean-Dominique Cassini made a precise measurement of the seconds pendulum at Paris. They used a 1+1⁄2-inch (14 mm)^{[clarification needed]} platinum ball suspended by a 12-foot (3.7 m) iron wire. Their main innovation was a technique called the "method of coincidences" which allowed the period of pendulums to be compared with great precision. (Bouguer had also used this method). The time interval Δt between the recurring instants when the two pendulums swung in synchronism was timed. From this the difference between the periods of the pendulums, T₁ and T₂, could be calculated:
  $${\displaystyle {\frac {1}{\Delta t}}={\frac {1}{T_{1}}}-{\frac {1}{T_{2}}}}$$

  
• 1821: Francesco Carlini made pendulum observations on top of Mount Cenis, Italy, from which, using methods similar to Bouguer's, he calculated the density of the Earth.^[112] He compared his measurements to an estimate of the gravity at his location assuming the mountain wasn't there, calculated from previous nearby pendulum measurements at sea level. His measurements showed 'excess' gravity, which he allocated to the effect of the mountain. Modeling the mountain as a segment of a sphere 11 miles (18 km) in diameter and 1 mile (1.6 km) high, from rock samples he calculated its gravitational field, and estimated the density of the Earth at 4.39 times that of water. Later recalculations by others gave values of 4.77 and 4.95, illustrating the uncertainties in these geographical methods.

Kater's pendulum

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 rigid bar pendulum (called a compound pendulum) was equal to the period of a simple pendulum with a length equal to the distance between the

, 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.^[17] 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.^[61] 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 L of a simple pendulum with the same period.

Kater built a reversible pendulum (see drawing) 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 micrometer. 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.

) from 1824 to 1855.

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

Later pendulum gravimeters

The increased accuracy made possible by Kater's pendulum helped make

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.
  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.^[115][116] 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.^[117] 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.^[118] 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 of the knife edges could be made to cancel out if they were interchanged between measurements. Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold, under contract by the Swiss Geodetic Commission made a pendulum along these lines. The Repsold pendulum was about 56 cm long and had a period of about 3⁄4 second. It was used extensively by European geodetic agencies, and with the Kater pendulum in the Survey of India. Similar pendulums of this type were designed by Charles Pierce and C. Defforges.

• Von Sterneck and Mendenhall gravimeters: In 1887 Austro-Hungarian scientist Robert von Sterneck developed a small gravimeter pendulum mounted in a temperature-controlled vacuum tank to eliminate the effects of temperature and air pressure. It used a "half-second pendulum," having a period close to one second, about 25 cm long. The pendulum was nonreversible, so the instrument was used for relative gravity measurements, but their small size made them small and portable. The period of the pendulum was picked off by reflecting the image of an electric spark created by a precision chronometer off a mirror mounted at the top of the pendulum rod. The Von Sterneck instrument, and a similar instrument developed by Thomas C. Mendenhall of the United States Coast and Geodetic Survey in 1890,^[119] were used extensively for surveys into the 1920s.

The Mendenhall pendulum was actually a more accurate timekeeper than the highest precision clocks of the time, and as the 'world's best clock' it was used by Albert A. Michelson in his 1924 measurements of the speed of light on Mt. Wilson, California.^[119]

• Double pendulum gravimeters: Starting in 1875, the increasing accuracy of pendulum measurements revealed another source of error in existing instruments: the swing of the pendulum caused a slight swaying of the tripod stand used to support portable pendulums, introducing error. In 1875 Charles S Peirce calculated that measurements of the length of the seconds pendulum made with the Repsold instrument required a correction of 0.2 mm due to this error.^[120] In 1880 C. Defforges used a Michelson interferometer to measure the sway of the stand dynamically, and interferometers were added to the standard Mendenhall apparatus to calculate sway corrections.^[121] A method of preventing this error was first suggested in 1877 by Hervé Faye and advocated by Peirce, Cellérier and Furtwangler: mount two identical pendulums on the same support, swinging with the same amplitude, 180° out of phase. The opposite motion of the pendulums would cancel out any sideways forces on the support. The idea was opposed due to its complexity, but by the start of the 20th century the Von Sterneck device and other instruments were modified to swing multiple pendulums simultaneously.

• Gulf gravimeter: One of the last and most accurate pendulum gravimeters was the apparatus developed in 1929 by the Gulf Research and Development Co.
  microgals or 3–5 nm/s²).^[122]
  It was used into the 1960s.

Relative pendulum gravimeters were superseded by the simpler LaCoste zero-length spring gravimeter, invented in 1934 by

Standard of length

Because the acceleration of gravity is constant at a given point on Earth, the period of a simple pendulum at a given location depends only on its length. Additionally, gravity varies only slightly at different locations. Almost from the pendulum's discovery until the early 19th century, this property led scientists to suggest using a pendulum of a given period as a standard of length.

Until the 19th century, countries based their systems of length measurement on prototypes, metal bar

, also constant. The idea was that anyone, anywhere on Earth, could recreate the standard by constructing a pendulum that swung with the defined period and measuring its length.

Virtually all proposals were based on the seconds pendulum, in which each swing (a half period) takes one second, which is about a meter (39 inches) long, because by the late 17th century it had become a standard for measuring gravity (see previous section). By the 18th century its length had been measured with sub-millimeter accuracy at a number of cities in Europe and around the world.

The initial attraction of the pendulum length standard was that it was believed (by early scientists such as Huygens and Wren) that gravity was constant over the Earth's surface, so a given pendulum had the same period at any point on Earth.

shape of the Earth, so at any given latitude (east–west line), gravity was constant enough that the length of a seconds pendulum was the same within the measurement capability of the 18th century. Thus the unit of length could be defined at a given latitude and measured at any point along that latitude. For example, a pendulum standard defined at 45° north latitude, a popular choice, could be measured in parts of France, Italy, Croatia, Serbia, Romania, Russia, Kazakhstan, China, Mongolia, the United States and Canada. In addition, it could be recreated at any location at which the gravitational acceleration had been accurately measured.

By the mid 19th century, increasingly accurate pendulum measurements by Edward Sabine and Thomas Young revealed that gravity, and thus the length of any pendulum standard, varied measurably with local geologic features such as mountains and dense subsurface rocks.^[125] So a pendulum length standard had to be defined at a single point on Earth and could only be measured there. This took much of the appeal from the concept, and efforts to adopt pendulum standards were abandoned.

Early proposals

One of the first to suggest defining length with a pendulum was Flemish scientist Isaac Beeckman^[126] who in 1631 recommended making the seconds pendulum "the invariable measure for all people at all times in all places".^[127] Marin Mersenne, who first measured the seconds pendulum in 1644, also suggested it. The first official proposal for a pendulum standard was made by the British Royal Society in 1660, advocated by Christiaan Huygens and Ole Rømer, basing it on Mersenne's work,^[128] and Huygens in Horologium Oscillatorium proposed a "horary foot" defined as 1/3 of the seconds pendulum. Christopher Wren was another early supporter. The idea of a pendulum standard of length must have been familiar to people as early as 1663, because Samuel Butler satirizes it in Hudibras:^[129]

Upon the bench I will so handle ‘em

That the vibration of this pendulum

Shall make all taylors’ yards of one

Unanimous opinion

In 1671

were also supporters.

By the end of the 18th century, when many nations were reforming their

When his efforts to promote a joint British–French–American metric system fell through in 1790, he proposed a British system based on the length of the seconds pendulum at London. This standard was adopted in 1824 (below).

The metre

In the discussions leading up to the French adoption of the

ancien regime

.

Although not defined by the pendulum, the final length chosen for the metre, 10⁻⁷ of the pole-to-equator meridian arc, was very close to the length of the seconds pendulum (0.9937 m), within 0.63%. Although no reason for this particular choice was given at the time, it was probably to facilitate the use of the seconds pendulum as a secondary standard, as was proposed in the official document. So the modern world's standard unit of length is certainly closely linked historically with the seconds pendulum.

Britain and Denmark

Britain and Denmark appear to be the only nations that (for a short time) based their units of length on the pendulum. In 1821 the Danish inch was defined as 1/38 of the length of the mean solar seconds pendulum at 45° latitude at the meridian of Skagen, at sea level, in vacuum.^[133][134] The British parliament passed the Imperial Weights and Measures Act in 1824, a reform of the British standard system which declared that if the prototype standard yard was destroyed, it would be recovered by defining the inch so that the length of the solar seconds pendulum at London, at sea level, in a vacuum, at 62 °F was 39.1393 inches.^[135] This also became the US standard, since at the time the US used British measures. However, when the prototype yard was lost in the 1834 Houses of Parliament fire, it proved impossible to recreate it accurately from the pendulum definition, and in 1855 Britain repealed the pendulum standard and returned to prototype standards.

Other uses

Seismometers

A pendulum in which the rod is not vertical but almost horizontal was used in early

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.

Schuler tuning

As first explained by

stable is modified so the device acts as though it is attached to such a pendulum, keeping the platform always facing down as the vehicle moves on the curved surface of the Earth.

Coupled pendulums

In 1665 Huygens made a curious observation about pendulum clocks. Two clocks had been placed on his

The cause of this behavior was that the two pendulums were affecting each other through slight motions of the supporting mantlepiece. This process is called

Religious practice

Pendulum motion appears in religious ceremonies as well. The swinging

.

Education

Pendulums are widely used in

or the pendulum is initially released with a push (so that when it returns it surpasses the release position).

Torture device

It is claimed that the pendulum was used as an instrument of

but there is considerable skepticism that it actually was used.

Most knowledgeable sources are skeptical that this torture was ever actually used.[148]^[149][150] The only evidence of its use is one paragraph in the preface to Llorente's 1826 History,^[145] relating a second-hand account by a single prisoner released from the Inquisition's Madrid dungeon in 1820, who purportedly described the pendulum torture method. Modern sources point out that due to Jesus' admonition against bloodshed, Inquisitors were only allowed to use torture methods which did not spill blood, and the pendulum method would have violated this stricture. One theory is that Llorente misunderstood the account he heard; the prisoner was actually referring to another common Inquisition torture, the strappado (garrucha), in which the prisoner has his hands tied behind his back and is hoisted off the floor by a rope tied to his hands.^[150] This method was also known as the "pendulum". Poe's popular horror tale, and public awareness of the Inquisition's other brutal methods, has kept the myth of this elaborate torture method alive.

Pendulum wave

A pendulum wave is a physics demonstration and kinetic art comprising several uncoupled pendulums with different lengths. As the pendulums oscillate, they appear to produce travelling and standing waves, beating, and random motion.^[151]

See also

Notes

The value of g reflected by the period of a pendulum varies from place to place. The gravitational force varies with distance from the center of the Earth, i.e. with altitude – or because the Earth's shape is oblate, g varies with latitude. A more important cause of this reduction in g at the equator is because the equator is spinning at one revolution per day, so the acceleration by the gravitational force is partially canceled there by the centrifugal force.

References

Further reading

• G. L. Baker and J. A. Blackburn (2009). The Pendulum: A Case Study in Physics (Oxford University Press).
• M. Gitterman (2010). The Chaotic Pendulum (World Scientific).
• Michael R. Matthews, Arthur Stinner, Colin F. Gauld (2005) The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives, Springer
• Matthews, Michael R.; Gauld, Colin; Stinner, Arthur (2005). "The Pendulum: Its Place in Science, Culture and Pedagogy". Science & Education. 13 (4/5): 261–277.
  S2CID 195221704
  .
• Schlomo Silbermann,(2014) "Pendulum Fundamental; The Path Of Nowhere" (Book)
• Matthys, Robert J. (2004). Accurate Pendulum Clocks. UK: Oxford Univ. Press.
  ISBN 978-0-19-852971-2
  .
• Nelson, Robert; M. G. Olsson (February 1986). "The pendulum – Rich physics from a simple system". American Journal of Physics. 54 (2): 112–121.
  S2CID 121907349
  .
• L. P. Pook (2011). Understanding Pendulums: A Brief Introduction (Springer).

External links

Media related to Pendulums at Wikimedia Commons