Editing Gridiron pendulum (section)

===Temperature error===
All substances expand with an increase in temperature <math>\theta</math>, so uncompensated pendulum rods get longer with a temperature increase, causing the clock to slow down, and get shorter with a temperature decrease, causing the clock to speed up.  The amount depends on the [[coefficient of thermal expansion|linear coefficient of thermal expansion]] (CTE) <math>\alpha</math> of the material they are composed of. CTE is usually given in parts per million (ppm) per degree Celsius. If a rod has a length <math>L</math> at some standard temperature <math>\theta_\text{0}</math>, the length of the rod as a function of temperature is 
:<math>L(\theta) = L + \alpha L(\theta - \theta_\text{0}) = L[1 + \alpha(\theta - \theta_\text{0})]</math>
If  <math>\Delta L = L(\theta) - L</math> and <math>\Delta\theta = \theta - \theta_\text{0}</math>, the expansion or contraction of a rod of length <math>L</math> with a coefficient of expansion <math>\alpha</math> caused by a temperature change <math>\Delta\theta</math> is<ref name="Baker2005">{{cite book
  | last1  = Baker
  | first1 = Gregory L.
  | last2  = Blackburn
  | first2 = James A. 
  | title  = The Pendulum: A Case Study in Physics
  | publisher = Oxford University Press
  | date = 2005
  | location =
  | pages = 
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  | url = https://books.google.com/books?id=t4ISDAAAQBAJ&pg=PA250
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 9780198567547
  | mr = 
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  | jfm =}}</ref>{{rp|p.250,eq.10.19}}
:<math>\Delta L = \alpha L \Delta\theta</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

The [[Frequency|period of oscillation]] <math>T</math> of the pendulum (the time interval for a right swing and a left swing) is<ref name="Baker2005" />{{rp|p.239,eq.10.2}}
:<math>T = 2\pi\sqrt{L \over g}</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)
A change in length <math>\Delta L</math> due to a temperature change <math>\Delta\theta</math> will cause a change in the period <math>\Delta T</math>.  Since the expansion coefficient is so small, the length changes due to temperature are very small, parts per million, so <math>\Delta T << T</math> and the change in period can be approximated to first order as a linear function<ref name="Baker2005" />{{rp|p.250}}
:<math>\Delta T = {dT \over dL}\Delta L</math>
:<math>\qquad =  {d \over dL}\Big( 2\pi\sqrt{L \over g}\Big)\Delta L = \pi{\Delta L \over \sqrt{gL}}</math> 
Substituting equation (1), the change in the pendulum's period caused by a change in temperature <math>\Delta\theta</math> is
:<math>\qquad = \pi{\alpha L \Delta\theta \over \sqrt{gL}} = \alpha\pi\sqrt{L \over g}\Delta\theta</math>
:<math>\Delta T = {\alpha T\Delta\theta \over 2}</math>
{{Equation box 1 |indent = |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent
|equation = <math>\quad{\Delta T \over T} = {1 \over 2}\alpha\Delta\theta\quad</math>
}}
So the fractional change in an uncompensated pendulum's period is equal to one-half the coefficient of expansion times the change in temperature.

Steel has a CTE of 11.5 parts per million per °C so a pendulum with a steel rod will have a thermal error rate of 
5.7 parts per million or 0.5 seconds per day per degree Celsius (0.9 seconds per day per degree Fahrenheit).  Before 1900 most buildings were unheated, so clocks in temperate climates like Europe and North America would experience a summer/winter temperature variation of around {{convert|25|F-change|C-change|order=flip}} resulting in an error rate of 6.8 seconds per day.<ref name="Kater">{{cite book
  | last1  = Kater
  | first1 = Henry
  | last2  = Lardner
  | first2 = Dionysus
  | title = A Treatise on Mechanics
  | publisher = Carey and Lea
  | date = 1831
  | location = Philadelphia
  | pages = 
  | url = https://archive.org/details/atreatiseonmech00lardgoog/page/n276/mode/2up
  | archive-url= 
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  | doi = 
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  | jfm =}}</ref>{{rp|p.259}}  

Although this seems like a small error, it should be kept in mind that in the 1700s and 1800s pendulum clocks were primary standards used for exacting tasks like keeping trains on schedule, and that outside big cities there were no time standards, so it was a difficult process to set a clock accurately to the correct time.  A [[transit telescope]] instrument was required to observe the exact moment when the sun or a star passed overhead, then [[almanac]] tables were consulted to determine the time the clock should be set to.   So clocks in rural areas typically had to run for long periods between being set.  A 6.8 second per day temperature error accumulates a 21 minute error over 6 months.   

Wood has a smaller CTE of 4.9 ppm per °C thus a pendulum with a wood rod will have a smaller thermal error of 0.21 sec per day per °C or 2.9 seconds per day for a 14°C seasonal change, so wood pendulum rods were often used in quality domestic clocks.  The wood had to be varnished to protect it from the atmosphere as [[humidity]] could also cause changes in length.