Editing Pendulum (section)

=== Period of oscillation ===
{{multiple image
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| image1            = Pendulum 30deg.gif
| image2            = Pendulum 60deg.gif
| image3            = Pendulum 120deg.gif
| image4            = Pendulum 170deg.gif
| footer            = The period of a pendulum gets longer as the amplitude ''θ''<sub>0</sub> (width of swing) increases.
}}

The period of swing of a [[pendulum (mathematics)#Simple gravity pendulum|simple gravity pendulum]] depends on its [[length]], the local [[Gravitational acceleration|strength of gravity]], and to a small extent on the maximum [[angle]] that the pendulum swings away from vertical, ''θ''<sub>0</sub>, called the [[amplitude]].<ref name="Milham1945">{{cite book
 | last=Milham | first=Willis I.
 | title=Time and Timekeepers
 | date=1945
 | publisher=MacMillan
}}, p.188-194</ref>  It is independent of the [[mass]] of the bob.  If the amplitude is limited to small swings,<ref group = Note>A "small" swing is one in which the angle {{mvar|θ}} is small enough that {{math|sin(''θ'')}} can be approximated by {{mvar|θ}} when {{mvar|θ}} is measured in radians</ref>  the [[Frequency|period]] {{mvar|T}} of a simple pendulum, the time taken for a complete cycle, is:<ref>{{cite book
 |last       = Halliday |first = David
 |author2    = Robert Resnick
 |author3    = Jearl Walker
 |title      = Fundamentals of Physics, 5th Ed
 |publisher  = John Wiley & Sons.
 |year       = 1997
 |location   = New York
 |page       = [https://archive.org/details/fundamentalsofp000davi/page/381 381]
 |url        = https://archive.org/details/fundamentalsofp000davi/page/381
 |url-access = registration
 |isbn       = 978-0-471-14854-8
}}</ref>
{{NumBlk||<math display="block">
  T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad
  \theta_0 \ll 1\text{ radian}
</math>|{{EquationRef|1}}}}
where <math>L</math> is the length of the pendulum and <math>g</math> is the local [[Gravitational acceleration|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 [[isochronism]], is the reason pendulums are so useful for timekeeping.<ref>{{cite book
 | last = Cooper | first = Herbert J.
 | title = Scientific Instruments
 | publisher = Hutchinson's
 | year = 2007
 | location = New York
 | page = 162
 | url = https://books.google.com/books?id=t7OoPLzXwiQC&pg=PA162
 | isbn = 978-1-4067-6879-4
}}</ref>  Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

For larger [[amplitude]]s, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of ''θ''<sub>0</sub> = 0.4 radians (23°) it is 1% larger than given by (1). The period increases asymptotically (to infinity) as ''θ''<sub>0</sub> approaches π radians (180°), because the value ''θ''<sub>0</sub> = π is an [[mechanical equilibrium|unstable equilibrium point]] for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see [[pendulum (mechanics)]]), one example being the [[infinite series]]:<ref name="Nelson">{{cite journal
 | last = Nelson | first = Robert
 |author2=M. G. Olsson
 | title = The pendulum – Rich physics from a simple system
 | journal = American Journal of Physics
 | volume = 54 | issue = 2 | pages = 112–121
 | date = February 1987
 | url = http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf
 | doi = 10.1119/1.14703
 | access-date = 2008-10-29
 | bibcode = 1986AmJPh..54..112N
| s2cid = 121907349
 }}</ref><ref>{{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 page 538 |quote= Pendulum.—Suppose that we have a body...}} includes a derivation</ref>
<math display="block">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)
</math>
where <math>\theta_0</math> 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&nbsp;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]]:<ref name="Hyperphysics" />
<math display="block">\theta (t) = \theta_0 \cos \left( \frac{2\pi}{T}\, t +\varphi \right)</math>
where <math>\varphi</math> is a constant value, dependent on [[initial conditions]].

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.<ref name="Nelson" /><ref>{{cite journal | last1 = Deschaine | first1 = J. S. | last2 = Suits | first2 = B. H. | year = 2008 | title = The hanging cord with a real tip mass | journal = European Journal of Physics | volume = 29 | issue = 6| pages = 1211–1222 | doi=10.1088/0143-0807/29/6/010| bibcode = 2008EJPh...29.1211D | s2cid = 122637957 }}</ref>  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 [[Chaos theory|chaotic]] system.<ref>{{cite web
 | last       = Bevivino
 | first      = Josh
 | title      = The Path From the Simple Pendulum to Chaos
 | publisher  = Department of Physics, Colorado State University
 | date       = 2009
 | url        = https://www.math.colostate.edu/~shipman/47/volume12009/bevivino.pdf
 | accessdate = 4 May 2025
}}</ref><ref>{{cite web |last=Fowler |first=Michael |title=22a: Driven Damped Pendulum: Period Doubling, Chaos, Strange Attractors |url=https://galileoandeinstein.phys.virginia.edu/7010/CM_22a_Period_Doubling_Chaos.html |accessdate=4 May 2025 |work=Graduate Classical Mechanics |publisher=University of Virginia Department of Physics}}</ref>