Editing Pendulum clock (section)

==Gravity-swing pendulum==
{{multiple image
| align    = right
| image1   = Pendulum-with-Escapement.png
| caption1 = Pendulum and [[anchor escapement]] from a [[grandfather clock]]
| width1   = 102
| image2   = Grandfather clock pendulum.png
| caption2 = Grandfather clock pendulum
| width2   = 82
| image3   = Mercury pendulum.png
| caption3 = Mercury pendulum
| width3   = 91
| image4   = Tidens naturlære fig22.png
| caption4 = [[Gridiron pendulum]]
| width4   = 84
}}

The pendulum in most clocks ''(see diagram)'' consists of a wood or metal rod ''<span style="color:red;">(a)</span>'' with a metal weight called the [[bob (physics)|bob]] ''<span style="color:red;">(b)</span>'' on the end. The bob is traditionally lens-shaped to reduce air drag. Wooden rods were often used in quality clocks because wood had a lower [[coefficient of thermal expansion]] than metal. The rod is usually suspended from the clock frame with a short straight spring of metal ribbon ''<span style="color:red;">(d)</span>''; this avoids instabilities that were introduced by a conventional pivot. In the most accurate regulator clocks the pendulum is suspended by metal knife edges resting on flat [[agate]] (a hard mineral that will retain a highly polished surface).

The pendulum is driven by an arm hanging behind it attached to the anchor piece ''<span style="color:red;">(h)</span>'' of the [[escapement]], called the "crutch" ''<span style="color:red;">(e)</span>'', ending in a "fork" ''<span style="color:red;">(f)</span>'' which embraces the pendulum rod. Each swing of the pendulum releases the escape wheel, and a tooth of the wheel presses against one of the [[pallet]]s, exerting a brief push through the crutch and fork on the pendulum rod to keep it swinging.

Most quality clocks, including all grandfather clocks, have a "seconds pendulum", in which each swing of the pendulum takes one [[second]] (a complete cycle takes two seconds), which is approximately {{convert|1|m|in|abbr=off|spell=in}} long from pivot to center of bob. [[Mantel clock]]s often have a half-second pendulum, which is approximately {{convert|25|cm|in}} long. Only a few [[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}} which is used in the Great Clock of Westminster which houses [[Big Ben]].

The pendulum swings with a period that varies with the square root of its effective length. For small swings the period ''T'', the time for one complete cycle (two swings), is

:<math>T = 2 \pi \sqrt{\frac{L}{g}} \,</math>

where ''L'' is the length of the pendulum and ''g'' is the local [[gravitational acceleration|acceleration of gravity]]. All pendulum clocks have a means of adjusting the rate. This is usually an adjustment nut ''<span style="color:red;">(c)</span>'' under the pendulum bob which moves the bob up or down on its rod. Moving the bob up reduces the length of the pendulum, reducing the pendulum's period so the clock gains time. In some pendulum clocks, fine adjustment is done with an auxiliary adjustment, which may be a small weight that is moved up or down the pendulum rod. In some master clocks and tower clocks, adjustment is accomplished by a small tray mounted on the rod where small weights are placed or removed to change the effective length, so the rate can be adjusted without stopping the clock.

The period of a pendulum increases slightly with the width (amplitude) of its swing; this is called ''circular error''. The ''rate'' of error increases with amplitude, so when limited to small swings of a few degrees the pendulum is nearly ''isochronous''; its period is independent of changes in amplitude. Therefore, the swing of the pendulum in clocks is limited to 2° to 4°.

Small swing angles tend toward isochronous behavior due to the mathematical fact that the approximation <math>\sin (x) = x</math> becomes valid as the angle approaches zero. With that substitution made, the pendulum equation becomes the equation of a harmonic oscillator, which has a fixed period in all cases. As the swing angle becomes larger, the approximation gradually fails and the period is no longer fixed.

=== Temperature compensation ===
A major source of error in pendulum clocks is thermal expansion; the pendulum rod changes in length slightly with changes in temperature, causing changes in the rate of the clock. An increase in temperature causes the rod to expand, making the pendulum longer, so its period increases and the clock loses time. Many older quality clocks used wooden pendulum rods to reduce this error, as wood expands less than metal.

The first pendulum to correct for this error was the ''mercury pendulum'' invented by Graham in 1721, which was used in precision regulator clocks into the 20th century. These had a bob consisting of a container of the liquid metal [[Mercury (element)|mercury]]. An increase in temperature would cause the pendulum rod to expand, but the mercury in the container would also expand and its level would rise slightly in the container, moving the center of gravity of the pendulum up toward the pivot. By using the correct amount of mercury, the centre of gravity of the pendulum remained at a constant height, and thus its period remained constant, despite changes in temperature.

The most widely used temperature-compensated pendulum was the [[gridiron pendulum]] invented by [[John Harrison]] around 1726. This consisted of a "grid" of parallel rods of high-thermal-expansion metal such as [[zinc]] or [[brass]] and low-thermal-expansion metal such as [[steel]]. If properly combined, the length change of the high-expansion rods compensated for the length change of the low-expansion rods, again achieving a constant period of the pendulum with temperature changes.
This type of pendulum became so associated with quality that decorative "fake" gridirons are often seen on pendulum clocks, that have no actual temperature compensation function.

Beginning around 1900, some of the highest precision scientific clocks had pendulums made of ultra-low-expansion materials such as the nickel steel alloy [[Invar]] or [[fused silica]], which required very little compensation for the effects of temperature.

===Atmospheric drag===
The viscosity of the air through which the pendulum swings will vary with atmospheric pressure, humidity, and temperature. This drag also requires power that could otherwise be applied to extending the time between windings. Traditionally the pendulum bob is made with a narrow streamlined lens shape to reduce air drag, which is where most of the driving power goes in a quality clock. In the late 19th century and early 20th century, pendulums for precision [[regulator clock]]s in astronomical observatories were often operated in a chamber that had been pumped to a low pressure to reduce drag and make the pendulum's operation even more accurate by avoiding changes in atmospheric pressure. Fine adjustment of the rate of the clock could be made by slight changes to the internal pressure in the sealed housing.

===Leveling and "beat"===
To keep time accurately, pendulum clocks must be level. If they are not, the pendulum swings more to one side than the other, upsetting the symmetrical operation of the escapement. This condition can often be heard audibly in the ticking sound of the clock. The ticks or "beats" should be at precisely equally spaced intervals to give a sound of, "tick...tock...tick...tock"; if they are not, and have the sound  "tick-tock...tick-tock..." the clock is ''out of beat'' and needs to be leveled. This problem can easily cause the clock to stop working, and is one of the most common reasons for service calls. A [[spirit level]] or [[watch timing machine]] can achieve a higher accuracy than relying on the sound of the beat; precision regulators often have a built-in spirit level for the task. Older freestanding clocks often have feet with adjustable screws to level them, more recent ones have a leveling adjustment in the movement. Some modern pendulum clocks have 'auto-beat' or 'self-regulating beat adjustment' devices, and do not need this adjustment.

===Local gravity===
[[File:Reloj de pendulo Ansonia C-1904.jpg|thumb|upright|Pendulum clock Ansonia. C. 1904, SANTIAGO, hanging oak gingerbread clock, eight-day time and strike.]]
Since the pendulum rate will increase with an increase in gravity, and local [[gravitational acceleration]] <math>g</math> varies with latitude and elevation on Earth, the highest precision pendulum clocks must be readjusted to keep time after a move. For example, a pendulum clock moved from sea level to {{convert|4000|ft}} will lose 16 seconds per day.<ref>{{cite web
 |last        = Arnstein
 |first       = Walt
 |title       = The Gravity Pendulum and its Horological Quirks
 |work        = Community Articles
 |publisher   = Timezone.com website
 |url         = http://www.timezone.com/library/comarticles/comarticles0013
 |access-date  = 2011-04-01
 |url-status     = dead
 |archive-url  = https://archive.today/20130204110512/http://www.timezone.com/library/comarticles/comarticles0013
 |archive-date = 2013-02-04
}}</ref>   With the most accurate pendulum clocks, even moving the clock to the top of a tall building would cause it to lose measurable time due to lower gravity.<ref>{{cite journal
 | last = Gore
 | first = Jeff
 | author2=Alexander van Oudenaarden
 | title = The Yin and Yang of Nature
 | journal = Nature
 | volume = 457
 | issue = 7227
 | pages = 271–2
 | publisher = MacMillan
 | date = January 15, 2009
 | url = http://web.mit.edu/biophysics/papers/NATURENV2009.pdf
 | doi = 10.1038/457271a
 | access-date = 2009-07-22
 | pmid = 19148089|bibcode = 2009Natur.457..271G | s2cid = 205043569
 }}</ref> The local gravity also varies by about 0.5% with [[latitude]] between the [[equator]] and the poles, with gravity increasing at higher latitudes due to the [[oblate]] shape of the Earth. Thus precision regulator clocks used for [[celestial navigation]] in the early 20th century had to be recalibrated when moved to a different latitude.