Editing Pendulum (section)

=== ''Q'' factor ===
[[File:Shortt Synchronome free pendulum clock.jpg|thumb|250px|A [[Shortt-synchronome clock|Shortt-Synchronome free pendulum clock]], the most accurate pendulum clock ever made, at the [[NIST]] museum, [[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 [[Q factor|''Q'' factor]] equal to the resonant frequency divided by the [[resonance width]].<ref name="Jespersen" /><ref>{{cite book
  | last = Matthys
  | first = Robert J.
  | title = Accurate Pendulum Clocks
  | publisher = Oxford Univ. Press
  | year = 2004
  | location = UK
  | pages = 27–36
  | url = https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA27
  | isbn = 978-0-19-852971-2}} has an excellent comprehensive discussion of the controversy over the applicability of ''Q'' to the accuracy of pendulums.</ref>  The higher the ''Q'', the smaller the resonance width, and the more constant the frequency or period of the oscillator for a given disturbance.<ref>{{cite web
  | title = Quality Factor, Q
  | website = Glossary
  | publisher = Time and Frequency Division, US National Institute of Standards and Technology
  | year = 2009
  | url = http://tf.nist.gov/general/enc-q.htm
  | access-date = 2009-02-21
  | archive-url = https://web.archive.org/web/20080504160852/http://tf.nist.gov/general/enc-q.htm
  | archive-date = 2008-05-04
  | url-status = dead
  }}</ref>  The reciprocal of the Q is roughly proportional to the limiting accuracy achievable by a harmonic oscillator as a time standard.<ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA32 Matthys, 2004, p.32, fig. 7.2 and text]</ref>

The ''Q'' is related to how long it takes for the oscillations of an oscillator to die out. The [[Q factor|''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 (clockwork)|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 2''π'' 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:
<math display="block">Q = \frac{M \omega}{\Gamma} </math>
where ''M'' is the mass of the bob,  {{math|1=''ω'' = 2''π''/''T''}} is the pendulum's radian frequency of oscillation, and Γ is the frictional [[Damping ratio|damping force]] 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.<ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA81 Matthys, 2004, p.81]</ref>

The ''Q'' of pendulums ranges from several thousand in an ordinary clock to several hundred thousand for precision regulator pendulums swinging in vacuum.<ref name="Orologeria">{{cite web
  | title = Q, Quality Factor
  | website = Watch and clock magazine
  | publisher = Orologeria Lamberlin website
  | url = http://www.orologeria.com/english/magazine/magazine4.htm
  | access-date = 2009-02-21}}</ref>  A quality home pendulum clock might have a ''Q'' of 10,000 and an accuracy of 10 seconds per month. The most accurate commercially produced pendulum clock was the [[Shortt-synchronome clock|Shortt-Synchronome free pendulum clock]], invented in 1921.<ref name="Marrison" /><ref name="Jones2000" /><ref>Milham 1945, p.615</ref><ref>{{cite web
  | title = The Reifler and Shortt clocks
  | publisher = JagAir Institute of Time and Technology
  | url = http://www.clockvault.com/heritage/index.htm
  | access-date = 2009-12-29}}</ref><ref name="Betts">{{cite web
  | last = Betts
  | first = Jonathan
  | title = Expert's Statement, Case 6 (2008-09) William Hamilton Shortt regulator
  | website = Export licensing hearing, Reviewing Committee on the Export of Works of Art and Objects of Cultural Interest
  | publisher = UK Museums, Libraries, and Archives Council
  | date = May 22, 2008
  | url = http://www.mla.gov.uk/what/cultural/export/reviewing_cttee/~/media/Files/word/2009/RCEWA/Cases%202008-09/Case%206%202008-09%20Regulator/internet%20experts%20statement%20shortt.ashx
  | format = DOC
  | access-date = 2009-12-29
  | url-status = dead
  | archive-url = https://web.archive.org/web/20091025180404/http://www.mla.gov.uk/what/cultural/export/reviewing_cttee/~/media/Files/word/2009/RCEWA/Cases%202008-09/Case%206%202008-09%20Regulator/internet%20experts%20statement%20shortt.ashx
  | archive-date = October 25, 2009
  }}</ref>  Its [[Invar]] master pendulum swinging in a vacuum tank had a ''Q'' of 110,000<ref name="Orologeria" /> and an error rate of around a second per year.<ref name="Jones2000" />

Their Q of 10<sup>3</sup>–10<sup>5</sup>  is one reason why pendulums are more accurate timekeepers than the [[balance wheel]]s in watches, with ''Q'' around 100–300, but less accurate than the [[Crystal oscillator|quartz crystals]] in [[quartz clock]]s, with ''Q'' of 10<sup>5</sup>–10<sup>6</sup>.<ref name="Marrison" /><ref name="Orologeria" />