Editing Pendulum (section)

== 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.<ref name="Marrison" /><ref>Milham 1945, p.334</ref>  In addition to clock pendulums, freeswinging [[seconds pendulum]]s 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&nbsp;mm in a grandfather clock pendulum, will cause an error of a minute per week.<ref>calculated from equation (1)</ref>

{{multiple image
| align    = center
| header   = Clock pendulums
| header_align = center
| image1   = Grandfather clock pendulum.png
| caption1 = [[Longcase clock]]  (Grandfather clock) pendulum
| width1   = 111
| image2   = Gaine Comtoise.jpg
| caption2 = Ornamented pendulum in a French Comtoise clock 
| width2   = 89
| image3   = Mercury pendulum.png
| caption3 = Mercury pendulum
| width3   = 122
| image4   = Tidens naturlære fig22.png
| caption4 = [[Gridiron pendulum]]
| width4   = 112
| image5   = Ellicott pendulum.png
| caption5 = Ellicott pendulum, another temperature compensated type
| width5   = 129
| image6   = Riefler clock NIST.jpg
| caption6 = [[Invar]] pendulum in low pressure tank in [[Riefler escapement|Riefler regulator clock]], used as the US time standard from 1909 to 1929
| width6   = 106
}}

=== Clock pendulums ===
{{main|Pendulum clock}}
{{multiple image
| align    = right
| image1   = Pendulum-with-Escapement.png
| caption1 = Pendulum and [[anchor escapement]] from a [[grandfather clock]]
| width1   = 100
| image2   = Anchor escapement animation 217x328px.gif
| caption2 = Animation of [[anchor escapement]], one of the most widely used escapements in pendulum clocks
| width2   = 140
}}

Pendulums in clocks (see example at right) are usually made of a weight or [[bob (physics)|bob]] ''<span style="color:red;">(b)</span>''  suspended by a rod of wood or metal ''<span style="color:red;">(a)</span>''.<ref name="Milham1945" /><ref>{{cite book
| last=Glasgow
| first=David
| title=Watch and Clock Making
| year=1885
| publisher=Cassel & Co.
| location=London
| pages = [https://archive.org/details/watchandclockma00glasgoog/page/n265 279]–284
| url=https://archive.org/details/watchandclockma00glasgoog}}</ref>  To reduce [[air resistance]] (which accounts for most of the energy loss in precision clocks)<ref>{{cite book
  | last = Matthys
  | first = Robert J.
  | title = Accurate Pendulum Clocks
  | publisher = Oxford Univ. Press
  | year = 2004
  | location = UK
  | page = 4
  | url = https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA4
  | isbn = 978-0-19-852971-2}}</ref>  the bob is traditionally a smooth disk with a lens-shaped cross section, although in antique clocks it often had carvings or decorations specific to the type of clock. In quality clocks the bob is made as heavy as the suspension can support and the movement can drive, since this improves the regulation of the clock (see [[#Accuracy of pendulums as timekeepers|Accuracy]] below). A common weight for [[seconds pendulum]] bobs is {{convert|15|lb|kg}}.<ref>[https://books.google.com/books?id=_78S_w3EBmAC&dq=matthys+%22common+bob+size&pg=PA13 Mattheys, 2004, p. 13]</ref>  Instead of hanging from a [[wikt:pivot|pivot]], clock pendulums are usually supported by a short straight [[spring (device)|spring]] ''<span style="color:red;">(d)</span>'' of flexible metal ribbon. This avoids the friction and 'play' caused by a pivot, and the slight bending force of the spring merely adds to the pendulum's [[restoring force]]. The highest precision clocks have pivots of 'knife' blades resting on agate plates. The impulses to keep the pendulum swinging are provided by an arm hanging behind the pendulum called the ''crutch'', ''<span style="color:red;">(e)</span>'', which ends in a ''fork'', ''<span style="color:red;">(f)</span>'' whose prongs embrace the pendulum rod. The crutch is pushed back and forth by the clock's [[escapement]], ''<span style="color:red;">(g,h)</span>''.

Each time the pendulum swings through its centre position, it releases one tooth of the ''escape wheel'' ''<span style="color:red;">(g)</span>''. The force of the clock's [[mainspring]] or a driving weight hanging from a pulley, transmitted through the clock's [[Wheel train (horology)|gear train]], causes the wheel to turn, and a tooth presses against one of the pallets ''<span style="color:red;">(h)</span>'', 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 ''<span style="color:red;">(c)</span>'' under the bob which moves it up or down on the rod.<ref name="Milham1945" /><ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA91 Matthys 2004], p.91-92</ref>  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 [[Turret clock|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.<ref>[https://books.google.com/books?id=OvQ3AAAAMAAJ&pg=PA48 Beckett 1874], p.48</ref><ref>{{cite web
  | title = Regulation
  | website = Encyclopedia of Clocks and Watches
  | publisher = Old and Sold antiques marketplace
  | year = 2006
  | url = http://www.oldandsold.com/articles02/clocks-r.shtml
  | access-date = 2009-03-09}}</ref>

The pendulum must be suspended from a rigid support.<ref name="Milham1945" /><ref>[https://books.google.com/books?id=OvQ3AAAAMAAJ&pg=PA43 Beckett 1874], p.43</ref>  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 clock]]s, is the [[seconds pendulum]], about {{convert|1|m|in|abbr=off}} long.  In [[mantel clock]]s, half-second pendulums, {{convert|25|cm|in|abbr=on}} long, or shorter, are used. Only a few large [[turret clock|tower clocks]] use longer pendulums, the 1.5 second pendulum, {{convert|2.25|m|ft|abbr=on}} long, or occasionally the two-second pendulum, {{convert|4|m|ft|abbr=on}} <ref name="Milham1945" /><ref>[https://archive.org/details/watchandclockma00glasgoog/page/n268 <!-- pg=282 --> Glasgow 1885], p.282</ref> which is used in [[Big Ben]].<ref>{{cite web
 |url=http://www.parliament.uk/about/livingheritage/building/big_ben/facts_figures/great_clock_facts.cfm 
 |title=Great Clock facts 
 |date=13 November 2009 
 |website=Big Ben 
 |publisher=UK Parliament 
 |access-date=31 October 2012 
 |location=London 
 |url-status=dead 
 |archive-url=https://web.archive.org/web/20091007101459/http://www.parliament.uk/about/livingheritage/building/big_ben/facts_figures/great_clock_facts.cfm 
 |archive-date=7 October 2009 
}}</ref>

=== Temperature compensation ===
[[Image:Observatório Astronômico da Universidade Federal do Rio Grande do Sul 04 clock closeup.jpg|thumb|upright=0.6|Mercury pendulum in  astronomical regulator clock by Adolf Opperman, late 1800s]]
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.<ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA3 Matthys 2004], p.3</ref>  This was discovered when people noticed that pendulum clocks ran slower in summer, by as much as a minute per week<ref name="Beckett1874" /><ref name="BritannicaCompensation">{{cite EB1911|wstitle= Clock |volume= 06 |last= Penderel-Brodhurst |first= James George Joseph |author-link= James George Joseph Penderel-Brodhurst | pages = 536&ndash;553; see pages 539 and 540 |quote= }}</ref> (one of the first was [[Godefroy Wendelin]], as reported by Huygens in 1658).<ref>{{cite book
  | last =Huygens
  | first = Christiaan
  | title = Horologium
  | publisher = Adrian Vlaqc
  | year = 1658
  | location = The Hague
  | url = http://www.antique-horology.org/_Editorial/Horologium/Horologium.pdf
  }}, translation by Ernest L. Edwardes (December 1970) ''Antiquarian Horology'', Vol.7, No.1</ref>  Thermal expansion of pendulum rods was first studied by [[Jean Picard]] in 1669.<ref name="Zupko">{{cite book
  | last = Zupko
  | first = Ronald Edward
  | author-link = Ronald Edward Zupko
  | title = Revolution in Measurement: Western European Weights and Measures since the Age of Science
  | publisher = Diane Publishing
  | year = 1990
  | page = 131
  | isbn = 978-0-87169-186-6}}</ref><ref>{{cite book |last= Picard |first= Jean |date= 1671 |title= La Mesure de la Terre |trans-title= The Measurement of the Earth |language= fr |location= Paris, France |publisher= Imprimerie Royale |page= 4 |url= https://gallica.bnf.fr/ark:/12148/btv1b7300361b/f14.item }} Picard described a pendulum consisting of a copper ball which was an inch (2.54&nbsp;mm) in diameter and was suspended by a strand of {{lang|fr|pite}}, a fiber from the aloe plant. Picard then mentions that temperature slightly effects the length of this pendulum: {{lang|fr|"Il est vray que cette longueur ne s'est pas toûjours trouvées si précise, & qu'il a semblé qu'elle devoit estre toûjours un peu accourcie en Hyver, & allongée en esté; mais c'est seulement de la dixieme partie d'une ligne&nbsp;..."}} ("It is true that this length [of the pendulum] is not always found [to be] so precise, and that it seemed that it should be always a bit shortened in winter, and lengthened in summer; but it is only by a tenth part of a line&nbsp;...") [1 {{lang|fr|[[ligne]]}} (line)&nbsp;= 2.2558&nbsp;mm].</ref> A pendulum with a steel rod will expand by about 11.3 [[parts per million]] (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 {{convert|33|C-change|abbr=on}} change. Wood rods expand less, losing only about 6 seconds per day for a {{convert|33|C-change|abbr=on}} 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 [[George Graham (clockmaker)|George Graham]]<ref name="Graham1726" /> in 1721.<ref name="Milham1945" /><ref name="BritannicaCompensation" />  The liquid metal [[mercury (element)|mercury]] expands in volume with temperature. In a mercury pendulum, the pendulum's weight (bob) is a container of mercury. With a temperature rise, the pendulum rod gets longer, but the mercury also expands and its surface level rises slightly in the container, moving its [[centre 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 centre 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 a day or two to reach the new temperature, causing the rate to deviate during that time.<ref name="MatthysCompensation">[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA7 Matthys 2004], p.7-12</ref>  To improve thermal accommodation several thin containers were often used, made of metal. Mercury pendulums were the standard used in precision regulator clocks into the 20th century.<ref>Milham 1945, p.335</ref>

==== Gridiron pendulum ====
[[Image:BanjoPendulum.svg|thumb|Diagram of a gridiron pendulum{{ordered list
 | list_style=margin-left:0;
 | item_style=list-style-position:inside;
 | list-style-type=upper-alpha
 | exterior schematic
 | normal temperature
 | higher temperature
}}]]
{{Main|Gridiron pendulum}}
The most widely used compensated pendulum was the [[gridiron pendulum]], invented in 1726 by [[John Harrison]].<ref name="Milham1945" /><ref name="BritannicaCompensation" /><ref name="MatthysCompensation" />  This consists of alternating rods of two different metals, one with lower thermal expansion ([[Coefficient of thermal expansion|CTE]]), [[steel]], and one with higher thermal expansion, [[zinc]] or [[brass]]. 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.<ref name="MatthysCompensation" /> 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 (deformation)|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.<ref name="Milham1945" /><ref name="BritannicaCompensation" /> These were only used in a few of the highest precision clocks before the pendulum became obsolete as a time standard. In 1896 [[Charles Édouard Guillaume]] invented the [[nickel]] [[steel]] [[alloy]] [[Invar]]. This has a [[coefficient of thermal expansion|CTE]] of around {{val|0.9|ul=ppm|up=°C}} ({{val|0.5|u=ppm|up=°F}}), resulting in pendulum temperature errors over {{convert|71|°F|°C °F|order=out}} of only 1.3&nbsp;seconds per day, and this residual error could be compensated to zero with a few centimeters of aluminium under the pendulum bob<ref name="Marrison" /><ref name="MatthysCompensation" /> (this can be seen in the Riefler clock image above). Invar pendulums were first used in 1898 in the [[Riefler escapement|Riefler regulator clock]]<ref>Milham 1945, p.331-332</ref>  which achieved accuracy of 15&nbsp;milliseconds per day. Suspension springs of [[Elinvar]] were used to eliminate temperature variation of the spring's [[restoring force]] on the pendulum. Later [[fused quartz]] was used which had even lower CTE. These materials are the choice for modern high accuracy pendulums.<ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA153 Matthys 2004], Part 3, p.153-179</ref>

=== 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:<ref name="BritannicaP540" /><ref>[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA13 Poynting & Thompson, 1907, p.13-14]</ref>
* By [[Archimedes' principle]] the effective [[weight]] of the [[bob (physics)|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 [[barometric pressure]] increase a pendulum's period slightly due to the first two effects, by about {{convert|0.11|s/day/kPa|s/day/inHg s/day/torr|lk=out|abbr=none}}.<ref name="BritannicaP540" /> Researchers using pendulums to measure the [[Gravity of Earth|acceleration of gravity]] had to correct the period for the air pressure at the altitude of measurement, computing the equivalent period of a pendulum swinging in vacuum. A pendulum clock was first operated in a constant-pressure tank by Friedrich Tiede in 1865 at the [[Berlin Observatory]],<ref>{{cite journal
  | last = Updegraff
  | first = Milton
  | title = On the measurement of time
  | journal = Science
  | volume = 15
  | issue = 371
  | pages = 218–219
  | date = February 7, 1902
  | url = https://books.google.com/books?id=O44CAAAAYAAJ&q=tiede+clock+observatory&pg=RA1-PA219
  | pmid = 17793345| doi = 10.1126/science.ns-15.374.218-a
  | s2cid = 21030470
  | access-date = 2009-07-13}}</ref><ref>{{cite book
  | last = Dunwoody
  | first = Halsey
  | title = Notes, Problems, and Laboratory Exercises in Mechanics, Sound, Light, Thermo-Mechanics and Hydraulics, 1st Ed
  | publisher = John Wiley & Sons
  | year = 1917
  | location = New York
  | page = 87
  | url = https://books.google.com/books?id=ZDe5XCIug_0C&pg=PA87
  }}</ref> and by 1900 the highest precision clocks were mounted in tanks that were kept at a constant pressure to eliminate changes in atmospheric pressure. Alternatively, in some a small [[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.