Editing Pendulum (section)

== Mechanics ==

{{Main|Pendulum (mechanics)}}

=== Simple gravity pendulum {{anchor|Simple}} ===

The ''simple gravity pendulum''<ref>defined by Christiaan Huygens: {{cite web
 | last = Huygens | first = Christian
 | title = Horologium Oscillatorium
 | website = 17centurymaths
 | publisher = 17thcenturymaths.com
 | year = 1673
 | url = http://www.17centurymaths.com/contents/huygens/horologiumpart4a.pdf
 | access-date = 2009-03-01
}}, Part 4, Definition 3, translated July 2007 by Ian Bruce
</ref> is an idealized mathematical model of a pendulum.<ref name="Hyperphysics">{{cite web
 | last = Nave | first = Carl R.
 | title = Simple pendulum
 | website = Hyperphysics
 | publisher = Georgia State Univ.
 | year = 2006
 | url = http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html
 | access-date = 2008-12-10
}}</ref><ref>{{cite web
 | last = Xue | first = Linwei
 | title = Pendulum Systems
 | website = Seeing and Touching Structural Concepts
 | publisher = Civil Engineering Dept., Univ. of Manchester, UK
 | year = 2007
 | url = http://www.mace.manchester.ac.uk/project/teaching/civil/structuralconcepts/Dynamics/pendulum/pendulum_con.php
 | access-date = 2008-12-10
}}</ref><ref>{{cite web
 | last = Weisstein | first = Eric W.
 | title = Simple Pendulum
 | website = Eric Weisstein's world of science
 | publisher = Wolfram Research
 | year = 2007
 | url = http://scienceworld.wolfram.com/physics/SimplePendulum.html
 | access-date = 2009-03-09
}}</ref> This is a weight (or [[Bob (physics)|bob]]) on the end of a massless cord suspended from a [[wikt:pivot|pivot]], without [[friction]]. When given an initial push, it will swing back and forth at a constant [[amplitude]]. Real pendulums are subject to friction and [[air drag]], so the amplitude of their swings declines.

{{multiple image
<!-- Essential parameters -->| align             = right
| direction         = vertical
<!-- Header -->| header            = Pendulum
<!-- Images -->| width             = 200
| image1            = PenduloTmg.gif
| caption1          = Animation of a pendulum showing forces acting on the bob: the tension ''T'' in the rod and the gravitational force ''mg''.
| image2            = Oscillating pendulum.gif
| caption2          = Animation of a pendulum showing the [[Equations of motion|velocity]] and acceleration vectors
}}

=== Period of oscillation ===
{{multiple image
| direction         = horizontal
| width             = 80
| 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>

=== Compound pendulum ===

Any swinging [[rigid body]] free to rotate about a fixed horizontal axis is called a '''compound pendulum''' or '''physical pendulum'''. A compound pendulum has the same period as a simple gravity pendulum of length <math>\ell^\mathrm{eq}</math>, called the ''equivalent length'' or ''radius of oscillation'', equal to the distance from the pivot to a point called the ''[[center of percussion|center of oscillation]]''.<ref name="HuygensCompound">{{cite web
 | first = Christian | last = Huygens
 | translator-first=Ian | translator-last=Bruce
 | title = Horologium Oscillatorium
 | website = 17centurymaths
 | publisher = 17thcenturymaths.com
 | year = 1673
 | url = http://www.17centurymaths.com/contents/huygenscontents.html
 | access-date = 2009-03-01
}}, Part 4, Proposition 5</ref> This point is located under the [[center of mass]] of the pendulum, at a distance which depends on the mass distribution of the pendulum. If most of the mass is concentrated in a relatively small bob compared to the pendulum length, the center of oscillation is close to the center of mass.<ref>{{cite book
 | last=Glasgow | first=David
 | title=Watch and Clock Making
 | year=1885
 | publisher=Cassel & Co.
 | location=London
 | page = [https://archive.org/details/watchandclockma00glasgoog/page/n264 278]
 | url=https://archive.org/details/watchandclockma00glasgoog
}}</ref>

The radius of oscillation or equivalent length <math>\ell^\mathrm{eq}</math> of any physical pendulum can be shown to be
<math display="block">\ell^\mathrm{eq} = \frac{I_O}{mr_\mathrm{CM}}</math> 

where <math>I_O</math> is the [[moment of inertia]] of the pendulum about the pivot point <math>O</math>, <math>m</math> is the total mass of the pendulum, and <math>r_\mathrm{CM}</math> is the distance between the pivot point and the [[center of mass]].
Substituting this expression in (1) above, the period <math>T</math> of a compound pendulum is given by
<math display="block">T = 2\pi \sqrt\frac{I_O}{mgr_\mathrm{CM}}</math>
for sufficiently small oscillations.<ref>{{cite book
 | last=Fowles
 | first=Grant R
 | title=Analytical Mechanics, 4th Ed
 | year=1986
 | publisher=Saunders
 | location=NY, NY
 | pages = 202 ff
}}</ref>

For example, a rigid uniform rod of length <math>\ell</math> pivoted about one end has moment of inertia <math display="inline">I_O = \frac{1}{3}m\ell^2</math>. The center of mass is located at the center of the rod, so <math display="inline">r_\mathrm{CM} = \frac{1}{2}\ell</math> Substituting these values into the above equation gives <math display="inline">T = 2\pi\sqrt{\frac{\frac{2}{3}\ell}{g}}</math>. This shows that a rigid rod pendulum has the same period as a simple pendulum of two-thirds its length.

[[Christiaan Huygens]] proved in 1673 that the pivot point and the center of oscillation are interchangeable.<ref name="HuygensReciprocity">[http://www.17centurymaths.com/contents/huygenscontents.html Huygens (1673) Horologium Oscillatorium], Part 4, Proposition 20</ref>  This means if any pendulum is turned upside down and swung from a pivot located at its previous center of oscillation, it will have the same period as before and the new center of oscillation will be at the old pivot point. In 1817 [[Henry Kater]] used this idea to produce a type of reversible pendulum, now known as a [[Kater pendulum]], for improved measurements of the acceleration due to gravity.

=== Double pendulum ===
[[File:Double-compound-pendulum.gif|thumb|Animation of a double compound pendulum showing chaotic behaviour. The two sections have the same length and mass, with the mass being distributed evenly along the length of each section, and the pivots being at the very ends. Motion computed by fourth-order Runge–Kutta method.]]
{{Main|Double pendulum}}

In [[physics]] and [[mathematics]], in the area of [[dynamical systems]], a [[double pendulum]], also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple [[physical system]] that exhibits rich [[dynamical systems|dynamic behavior]] with a [[butterfly effect|strong sensitivity to initial conditions]].<ref>{{cite journal |last1=Levien |first1=R. B. |last2=Tan |first2=S. M. |title=Double Pendulum: An experiment in chaos |journal=[[American Journal of Physics]] |year=1993 |volume=61 |issue=11 |page=1038 |doi=10.1119/1.17335 |bibcode=1993AmJPh..61.1038L }}</ref> The motion of a double pendulum is governed by a set of coupled [[ordinary differential equation]]s and is [[chaos theory|chaotic]].