Difference between revisions of "Pendulum (mathematics)"

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{{Dynamics}}
 
{{Dynamics}}
The mathematics of [[pendulum]]s are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple '''pendulum''' allows the equations of motion to be solved analytically for small-angle oscillations.
+
{{seeintro}}
 +
The mathematics of [[pendulum]]s are in general quite complicated. Simplifying assumptions can be made, which in the case of a [[#Simple gravity pendulum|simple pendulum]] allows the equations of motion to be solved analytically for small-angle oscillations.
  
 
== Simple gravity pendulum ==
 
== Simple gravity pendulum ==
A simple pendulum is an idealization of a real pendulum using the following assumptions:
+
[[File:Oscillating pendulum.gif|right|300px|thumb|Animation of a pendulum showing the [[Equations of motion|velocity and acceleration vectors]].]]
 +
 
 +
A so-called "simple pendulum" is an idealization of a "real pendulum" but in an [[isolated system]] using the following assumptions:
 
* The rod or cord on which the bob swings is massless, inextensible and always remains taut;
 
* The rod or cord on which the bob swings is massless, inextensible and always remains taut;
 +
* The bob is a point mass;
 
* Motion occurs only in [[two dimensions]], i.e. the bob does not trace an [[ellipse]] but an [[Arc (geometry)|arc]].
 
* Motion occurs only in [[two dimensions]], i.e. the bob does not trace an [[ellipse]] but an [[Arc (geometry)|arc]].
 
* The motion does not lose energy to [[friction]] or [[air resistance]].
 
* The motion does not lose energy to [[friction]] or [[air resistance]].
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The [[differential equation]] which represents the motion of a simple pendulum is
 
The [[differential equation]] which represents the motion of a simple pendulum is
  
:<math>{d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0 \quad\quad\quad\quad\quad(1)</math>
+
{{NumBlk|:|<math>{d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0</math>|{{EquationRef|Eq. 1}}|LnSty=1px dashed black}}
where g is acceleration due to gravity, <math>\ell</math> is the length of the pendulum, and θ is the angular displacement.  
+
where <math>g</math> is acceleration due to gravity, <math>\ell</math> is the length of the pendulum, and <math>\theta</math> is the angular displacement.
  
 
{| class="wikitable" width=100%;
 
{| class="wikitable" width=100%;
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|
 
|
 
{{show
 
{{show
|"Force" derivation of (1)
+
|"Force" derivation of ({{EquationNote|Eq. 1}})
 
|
 
|
[[File:Pendulum 2.png|thumb|'''Figure 1.''' Force diagram of a simple gravity pendulum.|right|300px]]
+
[[File:Pendulum gravity.svg|thumb|'''Figure 1.''' Force diagram of a simple gravity pendulum.|right|300px]]
Consider Figure 1, showing the forces acting on a simple pendulum. Note that the path of the pendulum sweeps out an arc of a circle. The angle <math>\theta</math> is measured in [[radians]], and this is crucial for this formula. The blue arrow is the [[gravitational force]] acting on the bob, and the violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion. The direction of the bob's instantaneous [[velocity]] always points along the red axis, which is considered the tangential axis because its direction is always tangent to the circle. Consider Newton's second law,
+
Please take the time to consider Figure 1 on the right, showing the forces acting on a simple pendulum. Note that the path of the pendulum sweeps out an arc of a circle. The angle <math>\theta</math> is measured in [[radian]]s, and this is crucial for this formula. The blue arrow is the [[gravitational force]] acting on the bob, and the violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion. The direction of the bob's instantaneous [[velocity]] always points along the red axis, which is considered the tangential axis because its direction is always tangent to the circle. Consider Newton's second law,
  
 
:<math>F=ma\,</math>
 
:<math>F=ma\,</math>
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:<math>a =  -g \sin\theta\,</math>
 
:<math>a =  -g \sin\theta\,</math>
 
where
 
where
:''g'' is the acceleration due to gravity near the surface of the earth. The negative sign on the right hand side implies that <math>\theta</math> and ''a'' always point in opposite directions. This makes sense because when a pendulum swings further to the left, we would expect it to accelerate back toward the right.
+
:<math>g</math> is the acceleration due to gravity near the surface of the earth. The negative sign on the right hand side implies that <math>\theta</math> and <math>a</math> always point in opposite directions. This makes sense because when a pendulum swings further to the left, we would expect it to accelerate back toward the right.
  
This linear acceleration <math>a</math> along the red axis can be related to the change in angle <math>\theta</math> by the arc length formulas; ''s'' is arc length:
+
This linear acceleration <math>a</math> along the red axis can be related to the change in angle <math>\theta</math> by the arc length formulas; <math>s</math> is arc length:
 
:<math> s = \ell\theta\,</math>
 
:<math> s = \ell\theta\,</math>
 
:<math> v = {ds\over dt} = \ell{d\theta\over dt}</math>
 
:<math> v = {ds\over dt} = \ell{d\theta\over dt}</math>
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thus:
 
thus:
 +
 +
:<math>\ell{d^2\theta\over dt^2} = -g \sin\theta</math>
 
:<math>{d^2\theta\over dt^2} + {g\over\ell}\sin\theta = 0</math>
 
:<math>{d^2\theta\over dt^2} + {g\over\ell}\sin\theta = 0</math>
 
}}
 
}}
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|
 
|
 
{{show
 
{{show
|"Torque" derivation of (1)
+
|"Torque" derivation of ({{EquationNote|Eq. 1}})
 
|
 
|
 +
Equation (1) can be obtained using two definitions for torque.
 +
:<math>\mathbf{\tau} = \mathbf{r} \times \mathbf{F} = {{d\mathbf{ L}} \over {dt}}</math>
 +
 +
First start by defining the torque on the pendulum bob using the force due to gravity.
 +
:<math>\mathbf{ \tau } = \mathbf{ l \times F_g },</math>
 +
where <math>\mathbf{l}</math> is the length vector of the pendulum and <math>\mathbf{F_g}</math> is the force due to gravity.
  
Consider the torque and angular momentum of this system,
+
For now just consider the magnitude of the torque on the pendulum.
 +
:<math>\mathbf{ |\tau| } = -mg l \sin\theta,</math>
 +
where <math>m</math> is the mass of the pendulum, <math>g</math> is the acceleration due to gravity, <math>l</math> is the length of the pendulum
 +
and <math>\theta</math> is the angle between the length vector and the force due to gravity.
  
:<math>\mathbf{ \tau } = \mathbf{ r \times f }= -mg\ell \sin\theta </math>,
+
Next rewrite the angular momentum.
 +
:<math>\mathbf{ L } =  \mathbf{ r \times p } = m\mathbf{ r \times (\omega \times r) }</math>.
 +
Again just consider the magnitude of the angular momentum.
 +
:<math> \mathbf{ |L| } = mr^2 \omega = m l^2 {d\theta \over dt} </math>.
 +
and its time derivative
  
:<math>\mathbf{ L } =  \mathbf{ r \times p } = m\mathbf{ r \times (\omega \times r) } = mr^2 \omega = m\ell^2 {d\theta \over dt} </math>.
+
:<math> {d \over dt}\mathbf{|L|} = m l^2 {d^2\theta \over dt^2} </math>,
  
According to <math>{ \mathbf{ \tau } = d \mathbf{ L} /dt }</math>, we can get
+
According to <math>{ \mathbf{ \tau } = {d \mathbf{ L} \over dt} }</math>, we can get by comparing the magnitudes
  
:<math>-mg\ell \sin\theta = m\ell^2 {d^2\theta \over dt^2} </math>,
+
:<math>-mgl \sin\theta = m l^2 {d^2\theta \over dt^2} </math>,
  
 
thus:
 
thus:
  
:<math>{d^2\theta\over dt^2} + {g\over \ell}\sin\theta = 0,</math>
+
:<math>{d^2\theta\over dt^2} + {g\over l}\sin\theta = 0,</math>
  
 
which is the same result as obtained through force analysis.
 
which is the same result as obtained through force analysis.
 
 
}}
 
}}
 
|}
 
|}
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|
 
|
 
{{show
 
{{show
|"Energy" derivation of (1)
+
|"Energy" derivation of ({{EquationNote|Eq. 1}})
 
|
 
|
 
[[Image:simple pendulum height.svg|thumb|'''Figure 2.''' Trigonometry of a simple gravity pendulum.|right|300px]]
 
[[Image:simple pendulum height.svg|thumb|'''Figure 2.''' Trigonometry of a simple gravity pendulum.|right|300px]]
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in terms of <math>{d\theta\over dt}</math> gives
 
in terms of <math>{d\theta\over dt}</math> gives
  
:<math>{d\theta\over dt} = \sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)}\quad\quad\quad (2)</math>
+
{{NumBlk|:|<math>{d\theta\over dt} = \sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)}</math>|{{EquationRef|Eq. 2}}|LnSty=1px dashed black}}
  
 
This equation is known as the ''first integral of motion'', it gives the velocity in terms of the location and includes an integration constant related to the initial displacement (<math>\theta_0</math>).  We can differentiate, by applying the [[chain rule]], with respect to time to get the acceleration
 
This equation is known as the ''first integral of motion'', it gives the velocity in terms of the location and includes an integration constant related to the initial displacement (<math>\theta_0</math>).  We can differentiate, by applying the [[chain rule]], with respect to time to get the acceleration
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==Small-angle approximation==
 
==Small-angle approximation==
The differential equation given above is not easily solved. However adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian, or
+
The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian, or
  
 
:<math>\theta \ll 1\,</math>,
 
:<math>\theta \ll 1\,</math>,
  
then substituting for sin θ into (1) using the [[small-angle approximation]],
+
then substituting for sin θ into {{EquationNote|Eq. 1}} using the [[small-angle approximation]],
  
 
:<math>\sin\theta\approx\theta\,</math>,
 
:<math>\sin\theta\approx\theta\,</math>,
  
yields the equation for a [[harmonic oscillator]]
+
yields the equation for a [[harmonic oscillator]],
  
 
:<math>{d^2\theta\over dt^2}+{g\over \ell}\theta=0.</math>
 
:<math>{d^2\theta\over dt^2}+{g\over \ell}\theta=0.</math>
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:<math>T_0 = 2\pi\sqrt{\frac{\ell}{g}}</math> can be expressed as <math>\ell = {\frac{g}{\pi^2}}\times{\frac{T_0^2}{4}}.</math>
 
:<math>T_0 = 2\pi\sqrt{\frac{\ell}{g}}</math> can be expressed as <math>\ell = {\frac{g}{\pi^2}}\times{\frac{T_0^2}{4}}.</math>
  
If [[International System of Units|SI units]] are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then <math>\scriptstyle g\approx9.81</math>  m/s<sup>2</sup>, and <math>\scriptstyle g/\pi^2\approx{1}</math> (the exact figure is 0.994 to 3 decimal places).
+
If [[International System of Units|SI units]] are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then <math>g\approx9.81</math>  m/s<sup>2</sup>, and <math>g/\pi^2\approx{1}</math> (0.994 is the approximation to 3 decimal places).
  
 
Therefore a relatively reasonable approximation for the length and period are,
 
Therefore a relatively reasonable approximation for the length and period are,
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== Arbitrary-amplitude period ==
 
== Arbitrary-amplitude period ==
 +
For amplitudes beyond the [[small angle approximation]], one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method ({{EquationNote|Eq. 2}}),
  
For amplitudes beyond the [[small angle approximation]], one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method,
+
[[Image:Pendulum period.svg|thumb|'''Figure 3.''' Deviation of the "true" period of a pendulum from the small-angle approximation of the period. "True" value was obtained using Matlab to numerically evaluate the elliptic integral. |right|300px]]
 
 
[[Image:Pendulum period.svg|thumb|'''Figure 3.''' Deviation of the period from small-angle approximation.|right|300px]]
 
  
 
[[Image:Pendulum Rel Error90a.png|thumb|'''Figure 4.''' Relative errors using the power series.|right|300px]]
 
[[Image:Pendulum Rel Error90a.png|thumb|'''Figure 4.''' Relative errors using the power series.|right|300px]]
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:<math>{dt\over d\theta} = \sqrt{\ell\over 2g}{1\over\sqrt{\cos\theta-\cos\theta_0}}</math>
 
:<math>{dt\over d\theta} = \sqrt{\ell\over 2g}{1\over\sqrt{\cos\theta-\cos\theta_0}}</math>
  
and then integrating over one complete cycle,  
+
and then integrating over one complete cycle,
  
 
:<math>T = t(\theta_0\rightarrow0\rightarrow-\theta_0\rightarrow0\rightarrow\theta_0),</math>
 
:<math>T = t(\theta_0\rightarrow0\rightarrow-\theta_0\rightarrow0\rightarrow\theta_0),</math>
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:<math>T = 4\sqrt{\ell\over 2g}\int^{\theta_0}_0 {1\over\sqrt{\cos\theta-\cos\theta_0}}\,d\theta.</math>
 
:<math>T = 4\sqrt{\ell\over 2g}\int^{\theta_0}_0 {1\over\sqrt{\cos\theta-\cos\theta_0}}\,d\theta.</math>
  
This integral can be re-written in terms of [[elliptic integral|elliptic functions]] as
+
Note that this integral diverges as <math> \theta_0 </math> approaches the vertical
 +
 
 +
:<math> \lim_{\theta_0 \rightarrow \pi} T = \infty </math>,
  
:<math>T = 4\sqrt{\ell\over g}F\left( {\theta_0\over 2}, \csc{\theta_0\over2}\right)\csc {\theta_0\over 2}</math>
+
so that a pendulum with just the right energy to go vertical will never actually get there.  (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.)
  
where <math>F</math> is the [[elliptic integral|incomplete elliptic function of the first kind]] defined by
+
This integral can be re-written in terms of [[elliptic integral]]s as
  
: <math>F(\varphi , k) = \int_0^\varphi {1\over\sqrt{1-k^2\sin^2{u}}}\,du\,,</math>
+
:<math>T = 4\sqrt{\ell\over g}F\left( {\theta_0}, \csc{\theta_0\over2}\right)\csc {\theta_0\over 2}</math>
  
or more concisely by the [[integration by substitution|substitution]]  <math>\sin{u} = \frac{\sin{\theta\over 2}}{\sin{\theta_0\over 2}}</math> expressing <math>\theta</math> in terms of <math>u</math>,
+
where <math>F</math> is the [[elliptic integral#Incomplete elliptic integral of the first kind|incomplete elliptic integral of the first kind]] defined by
  
:<math>T = 4\sqrt{\ell\over g}\,K\left( \sin{\theta_0\over 2} \right)</math>
+
: <math>F(\varphi , k) = \int_0^\varphi {1\over\sqrt{1-k^2\sin^2{u}}}\,du\,.</math>
  
where <math>K</math> is the [[elliptic integral|complete elliptic function of the first kind]] defined by
+
Or more concisely by the [[integration by substitution|substitution]] <math>\sin{u} = \frac{\sin{\theta\over 2}}{\sin{\theta_0\over 2}}</math> expressing <math>\theta</math> in terms of <math>u</math>,
  
: <math>K(k) = F \left( {\pi\over 2}, k \right) = \int_0^{\pi/2} {1\over\sqrt{1-k^2\sin^2{u}}}\,du\,.</math>  
+
{{NumBlk|:|<math>T = 4\sqrt{\ell\over g}\,K\left( \sin{\theta_0\over 2} \right)</math>|{{EquationRef|Eq. 3}}|LnSty=1px dashed black}}
 +
where  <math>K</math> is the [[elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] defined by
  
Figure 3 shows the deviation of ''T'' from ''T''<sub>0</sub>, the period obtained from small-angle approximation.
+
: <math>K(k) = F \left( {\pi\over 2}, k \right) = \int_0^{\pi/2} {1\over\sqrt{1-k^2\sin^2{u}}}\,du\,.</math>
  
Value for the complete elliptic function can be computed using the following series:
+
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (''g'' = 9.80665&nbsp;m/s<sup>2</sup>) at initial angle 10 degrees is <math>4\sqrt{1\ \mathrm{m}\over g}K\left( {\sin {10^\circ\over 2}} \right) \approx 2.0102\ \mathrm{s}</math>. The linear approximation gives <math>2\pi \sqrt{1\ \mathrm{m}\over g} \approx 2.0064\ \mathrm{s}</math>. The difference between the two values, less than 0.2%, is much less than that caused by the variation of g with geographical location.
  
 +
From here there are many ways to proceed to calculate the elliptic integral:
 +
 +
===Legendre polynomial solution for the elliptic integral===
 +
Given {{EquationNote|Eq. 3}} and the [[Legendre polynomial]] solution for the elliptic integral:
 +
:<math>K(k) = \frac{\pi}{2}\left\{1 + \left(\frac{1}{2}\right)^2 k^{2} + \left(\frac{1 \cdot 3}{2 \cdot 4}\right)^2 k^{4} + \cdots + \left[\frac{\left(2n - 1\right)!!}{\left(2n\right)!!}\right]^2 k^{2n} + \cdots \right\},</math>
 +
where {{math|''n''!!}} denotes the [[double factorial]], an exact solution to the period of a pendulum is:
 
:<math>\begin{alignat}{2}
 
:<math>\begin{alignat}{2}
 
T & = 2\pi \sqrt{\ell\over g} \left( 1+ \left( \frac{1}{2} \right)^2 \sin^2\left(\frac{\theta_0}{2}\right) + \left( \frac{1 \cdot 3}{2 \cdot 4} \right)^2 \sin^4\left(\frac{\theta_0}{2}\right) + \left( \frac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \right)^2 \sin^6\left(\frac{\theta_0}{2}\right) + \cdots \right) \\
 
T & = 2\pi \sqrt{\ell\over g} \left( 1+ \left( \frac{1}{2} \right)^2 \sin^2\left(\frac{\theta_0}{2}\right) + \left( \frac{1 \cdot 3}{2 \cdot 4} \right)^2 \sin^4\left(\frac{\theta_0}{2}\right) + \left( \frac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \right)^2 \sin^6\left(\frac{\theta_0}{2}\right) + \cdots \right) \\
Line 220: Line 246:
  
 
Figure 4 shows the relative errors using the power series.  ''T''<sub>0</sub> is the linear approximation, and ''T''<sub>2</sub> to ''T''<sub>10</sub> include respectively the terms up to the 2nd to the 10th powers.
 
Figure 4 shows the relative errors using the power series.  ''T''<sub>0</sub> is the linear approximation, and ''T''<sub>2</sub> to ''T''<sub>10</sub> include respectively the terms up to the 2nd to the 10th powers.
 
+
[[File:Pendulum phase portrait.svg|thumb|312x312px|'''Figure 5.''' Potential energy and phase portrait of a simple pendulum. Note that the ''x''-axis, being angle, wraps onto itself after every 2π radians.]]
For a swing of exactly 180° the bob is balanced over its pivot point and so ''T''&nbsp;=&nbsp;∞.
+
=== Power series solution for the elliptic integral ===
 
+
Another formulation of the above solution can be found if the following Maclaurin series:
[[Image:Pendulumphase.png|right|300px|thumb| '''Figure 5.''' Potential energy and phase portrait of a simple pendulum. Note that the ''x''-axis, being angle, wraps onto itself after every 2π radians.]]
+
:<math>\sin {\theta_0 \over 2}=\frac{1}{2}\theta_0 - \frac{1}{48}\theta_0^3 + \frac{1}{3840}\theta_0^5 - \frac{1}{645120}\theta_0^7 + \cdots.</math>
 
+
is used in the Legendre polynomial solution above.
For example, the period of a pendulum of length 1 m on Earth (''g'' = 9.80665&nbsp;m/s<sup>2</sup>) at initial angle 10 degrees is <math>4\sqrt{1\ \mathrm{m}\over g}K\left( {\sin {10^\circ\over 2}} \right) \approx 2.0102\ \mathrm{s}</math>, where the linear approximation gives <math>2\pi \sqrt{1\ \mathrm{m}\over g} \approx 2.0064\ \mathrm{s}</math>.<br>
+
The resulting power series is:<ref>{{cite journal
The difference (less than 0.2%) is much less than that caused by the variation of g with geographical location.
 
 
 
By using the following [[Maclaurin series]]:
 
 
 
:<math>
 
K \left(x \right)={\pi \over 2} \left( 1 + \frac{1}{4}x^2 + \frac{9}{64}x^4 + \frac{25}{256}x^6 + \frac{1225}{16384}x^8 + \cdots\right)
 
</math>
 
 
 
:<math>
 
\sin \left({\theta_0 \over 2}\right)=\left(\frac{1}{2}\theta_0 - \frac{1}{48}\theta_0^3 + \frac{1}{3840}\theta_0^5 - \frac{1}{645120}\theta_0^7 + \cdots\right)
 
</math>
 
 
 
The equivalent power series is:<ref>{{cite journal
 
 
   | last = Nelson
 
   | last = Nelson
 
   | first = Robert
 
   | first = Robert
 
   | authorlink =  
 
   | authorlink =  
   | coauthors = M. G. Olsson
+
   |author2=M. G. Olsson
 
   | title = The pendulum &mdash; Rich physics from a simple system
 
   | title = The pendulum &mdash; Rich physics from a simple system
 
   | journal = American Journal of Physics
 
   | journal = American Journal of Physics
Line 254: Line 267:
 
   | doi =  
 
   | doi =  
 
   10.1119/1.14703| id =  
 
   10.1119/1.14703| id =  
   | accessdate = 2012-4-30}}</ref>
+
   | accessdate = 2012-04-30}}</ref>
  
 
:<math>\begin{alignat}{2}
 
:<math>\begin{alignat}{2}
 
T & = 2\pi \sqrt{\ell\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \frac{173}{737280}\theta_0^6 + \frac{22931}{1321205760}\theta_0^8 + \frac{1319183}{951268147200}\theta_0^{10} + \frac{233526463}{2009078326886400}\theta_0^{12} + . . . \right)  
 
T & = 2\pi \sqrt{\ell\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \frac{173}{737280}\theta_0^6 + \frac{22931}{1321205760}\theta_0^8 + \frac{1319183}{951268147200}\theta_0^{10} + \frac{233526463}{2009078326886400}\theta_0^{12} + . . . \right)  
\end{alignat}</math>
+
\end{alignat}.</math>
 
 
A much simpler and better converging formula for the period is discussed on pp. 1096-1097 of the September 2012 issue of the Notices of the AMS<ref>Adlaj, S. ''An eloquent formula for the perimeter of an ellipse'', Notices of the AMS 59(8), pp. 1096-1097.</ref>:
 
  
:<math>T = \frac{2\pi}{M(\cos(\theta_0/2))} \sqrt\frac{L}{g},</math>
+
===Arithmetic-geometric mean solution for elliptic integral===
 +
Given {{EquationNote|Eq. 3}} and the [[Arithmetic-geometric mean]] solution of the elliptic integral:
 +
:<math>K(k) = \frac {\pi /2}{M(1-k,1+k)},</math>
 +
where <math>M(x,y)</math> is the arithmetic-geometric mean of <math>x</math> and <math>y</math>.
  
where <math>M(x)</math> is the [[Arithmetic-geometric_mean|arithmetic-geometric mean]] of 1 and <math>x</math>.
+
This yields an alternative and faster-converging formula for the period:<ref>{{Citation |title=Approximations for the period of the simple pendulum based on the arithmetic-geometric mean |url=http://suppes-corpus.stanford.edu/articles/physics/431.pdf |first1=Claudio G. |last1=Carvalhaes |first2=Patrick |last2=Suppes |journal=[[American Journal of Physics|Am. J. Phys.]] |issn=0002-9505 |volume=76 |issue=12͒ |date=December 2008 |pages=1150–1154 |doi=10.1119/1.2968864͔ |accessdate=2013-12-14}}</ref><ref>{{Citation |title=An eloquent formula for the perimeter of an ellipse | url=http://www.ams.org/notices/201208/rtx120801094p.pdf |first=Semjon | last=Adlaj |journal=[[Notices of the American Mathematial Society |Notices of the AMS]] |issn=1088-9477 |volume=76 |issue=8 |date=September 2012 |pages=1094・099 | doi=}}</ref>
 +
 +
:<math>T = \frac{2\pi}{M(1, \cos(\theta_0/2))} \sqrt\frac{\ell}{g}.</math>
  
 
== Examples ==
 
== Examples ==
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File:Pendulum_135deg.gif|Initial angle of 135°
 
File:Pendulum_135deg.gif|Initial angle of 135°
 
File:Pendulum_170deg.gif|Initial angle of 170°
 
File:Pendulum_170deg.gif|Initial angle of 170°
 +
File:Pendulum_180deg.gif|Initial angle of 180°, unstable equilibrium.
 
File:Pendulum_190deg.gif|Pendulum with just barely enough energy for a full swing
 
File:Pendulum_190deg.gif|Pendulum with just barely enough energy for a full swing
 
File:Pendulum_220deg.gif|Pendulum with enough energy for a full swing
 
File:Pendulum_220deg.gif|Pendulum with enough energy for a full swing
File:Pendulum_180deg.gif|Initial angle of 180°, unstable equilibrium.
 
 
</gallery>
 
</gallery>
  
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== Physical interpretation of the imaginary period ==
 
== Physical interpretation of the imaginary period ==
  
The [[Jacobi's elliptic functions|Jacobian elliptic function]] that expresses the position of a pendulum as a function of time is a [[doubly periodic function]] with a [[real number|real]] period and an [[imaginary number|imaginary]] period.  The real period is of course the time it takes the pendulum to go through one full cycle.  [[Paul Appell]] pointed out a physical interpretation of the imaginary period:<ref>Paul Appell, "Sur une interprétation des valeurs imaginaires du temps en Mécanique", ''[[Comptes Rendus]] Hebdomadaires des Scéances de l'Académie des Sciences'', volume 87, number 1, July, 1878</ref> if ''θ''<sub>0</sub> is the maximum angle of one pendulum and 180°&nbsp;&minus;&nbsp;''θ''<sub>0</sub> is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.
+
The [[Jacobi's elliptic functions|Jacobian elliptic function]] that expresses the position of a pendulum as a function of time is a [[doubly periodic function]] with a [[real number|real]] period and an [[imaginary number|imaginary]] period.  The real period is of course the time it takes the pendulum to go through one full cycle.  [[Paul Appell]] pointed out a physical interpretation of the imaginary period:<ref>Paul Appell, "Sur une interprétation des valeurs imaginaires du temps en Mécanique", ''[[Comptes Rendus]] Hebdomadaires des Scéances de l'Académie des Sciences'', volume 87, number 1, July, 1878</ref> if ''θ''<sub>0</sub> is the maximum angle of one pendulum and 180°&nbsp;&minus;&nbsp;''θ''<sub>0</sub> is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other. This interpretation, involving dual forces in opposite directions, might be further clarified and generalized to other classical problems in mechanics with dual solutions.<ref>Adlaj, S. ''Mechanical interpretation of negative and imaginary tension of a tether in a linear parallel force field '', Selected papers of the International Scientific Conference on Mechanics "SIXTH POLYAKHOV READINGS", January 31 - February 3, 2012, Saint-Petersburg, Russia, pp. 13-18.</ref>
  
 
==See also==
 
==See also==
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*[[Double pendulum]]
 
*[[Double pendulum]]
 
*[[Inverted pendulum]]
 
*[[Inverted pendulum]]
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*[[Kapitza's pendulum]]
 
*[[Spring pendulum]]
 
*[[Spring pendulum]]
 
*[[Mathieu function]]
 
*[[Mathieu function]]
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*[http://mathworld.wolfram.com/MathieuFunction.html Mathworld article on Mathieu Function]
 
*[http://mathworld.wolfram.com/MathieuFunction.html Mathworld article on Mathieu Function]
  
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[[Category:Horology]]
 
[[Category:Mathematical physics]]
 
[[Category:Mathematical physics]]
 
[[Category:Pendulums]]
 
[[Category:Pendulums]]
 
[[ar:رقاص (رياضيات)]]
 
[[ca:Pèndol matemàtic]]
 
[[cs:Matematické kyvadlo]]
 
[[da:Matematisk pendul]]
 
[[de:Mathematisches Pendel]]
 
[[et:Matemaatiline pendel]]
 
[[es:Péndulo simple]]
 
[[eo:Matematika pendolo]]
 
[[he:מטוטלת מתמטית]]
 
[[kk:Математикалық және серіппелі маятниктердің тербелістері]]
 
[[lt:Matematinė svyruoklė]]
 
[[hu:Matematikai inga]]
 
[[pt:Equação do pêndulo]]
 
[[ru:Математический маятник]]
 
[[sl:Matematično nihalo]]
 
[[sr:Математичко клатно]]
 
[[uk:Математичний маятник]]
 

Latest revision as of 20:37, 29 November 2014

Template:Dynamics Template:Seeintro The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations.

Simple gravity pendulum

Animation of a pendulum showing the velocity and acceleration vectors.

A so-called "simple pendulum" is an idealization of a "real pendulum" but in an isolated system using the following assumptions:

  • The rod or cord on which the bob swings is massless, inextensible and always remains taut;
  • The bob is a point mass;
  • Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
  • The motion does not lose energy to friction or air resistance.

The differential equation which represents the motion of a simple pendulum is

Template:NumBlk where is acceleration due to gravity, is the length of the pendulum, and is the angular displacement.

Template:Show

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Small-angle approximation

The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian, or

,

then substituting for sin θ into Template:EquationNote using the small-angle approximation,

,

yields the equation for a harmonic oscillator,

The error due to the approximation is of order θ 3 (from the Maclaurin series for sin θ).

Given the initial conditions θ(0) = θ0 and /dt(0) = 0, the solution becomes,

The motion is simple harmonic motion where θ0 is the semi-amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The period of the motion, the time for a complete oscillation (outward and return) is

which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ0; this is the property of isochronism that Galileo discovered.

Rule of thumb for pendulum length

can be expressed as

If SI units are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then m/s2, and (0.994 is the approximation to 3 decimal places).

Therefore a relatively reasonable approximation for the length and period are,

Arbitrary-amplitude period

For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (Template:EquationNote),

Figure 3. Deviation of the "true" period of a pendulum from the small-angle approximation of the period. "True" value was obtained using Matlab to numerically evaluate the elliptic integral.

and then integrating over one complete cycle,

or twice the half-cycle

or 4 times the quarter-cycle

which leads to

Note that this integral diverges as approaches the vertical

,

so that a pendulum with just the right energy to go vertical will never actually get there. (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.)

This integral can be re-written in terms of elliptic integrals as

where is the incomplete elliptic integral of the first kind defined by

Or more concisely by the substitution expressing in terms of ,

Template:NumBlk where is the complete elliptic integral of the first kind defined by

For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (g = 9.80665 m/s2) at initial angle 10 degrees is . The linear approximation gives . The difference between the two values, less than 0.2%, is much less than that caused by the variation of g with geographical location.

From here there are many ways to proceed to calculate the elliptic integral:

Legendre polynomial solution for the elliptic integral

Given Template:EquationNote and the Legendre polynomial solution for the elliptic integral:

where n!! denotes the double factorial, an exact solution to the period of a pendulum is:

Figure 4 shows the relative errors using the power series. T0 is the linear approximation, and T2 to T10 include respectively the terms up to the 2nd to the 10th powers.

Figure 5. Potential energy and phase portrait of a simple pendulum. Note that the x-axis, being angle, wraps onto itself after every 2π radians.

Power series solution for the elliptic integral

Another formulation of the above solution can be found if the following Maclaurin series:

is used in the Legendre polynomial solution above. The resulting power series is:[1]

Arithmetic-geometric mean solution for elliptic integral

Given Template:EquationNote and the Arithmetic-geometric mean solution of the elliptic integral:

where is the arithmetic-geometric mean of and .

This yields an alternative and faster-converging formula for the period:[2][3]

Examples

The animations below depict several different modes of oscillation given different initial conditions. The small graph above the pendulums are their phase portraits.

Compound pendulum

A compound pendulum (or physical pendulum) is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the pendulum's period depends on its moment of inertia I around the pivot point.

The equation of torque gives:

where:

is the angular acceleration.
is the torque

The torque is generated by gravity so:

where:

L is the distance from the pivot to the center of mass of the pendulum
θ is the angle from the vertical

Hence, under the small-angle approximation ,

This is of the same form as the conventional simple pendulum and this gives a period of:

[4]

And a frequency of:

Physical interpretation of the imaginary period

The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. The real period is of course the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period:[5] if θ0 is the maximum angle of one pendulum and 180° − θ0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other. This interpretation, involving dual forces in opposite directions, might be further clarified and generalized to other classical problems in mechanics with dual solutions.[6]

See also

References

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  4. Physical Pendulum
  5. Paul Appell, "Sur une interprétation des valeurs imaginaires du temps en Mécanique", Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences, volume 87, number 1, July, 1878
  6. Adlaj, S. Mechanical interpretation of negative and imaginary tension of a tether in a linear parallel force field , Selected papers of the International Scientific Conference on Mechanics "SIXTH POLYAKHOV READINGS", January 31 - February 3, 2012, Saint-Petersburg, Russia, pp. 13-18.

Further reading

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External links