File:Pendulum period.svg

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Pendulum_period.svg‎ (SVG file, nominally 730 × 591 pixels, file size: 28 KB)

[edit] Summary

Description A plot of the ratio between the actual period of a pendulum and the approximate value obtained for small angles, as a function of the amplitude. According to Wikipedia:Pendulum (mathematics), the oscillation period for small angles is given by:
T_0 = 2 \pi \sqrt{\ell\over g}

while the actual period for any angle is given by:

T = 4\sqrt{\ell\over g}F\left({\sin \frac{\theta_0}{2}}, {\pi \over 2} \right)

where:

F(k,\phi) = \int^{\phi}_0 {1\over\sqrt{1-k^2\sin^2{\theta}}}\,d\theta.

so the ratio is given by:

\frac{T}{T_0} = \frac{2}{\pi} F \left({\sin \frac{\theta_0}{2}}, {\pi \over 2} \right)

and this is the function plotted in the graph. First, with the following Matlab code I created a file called pendulum_period.dat:


res=2000; % resolution
sup=pi/2; % max angle to be used in the plot (in radians, less than pi)

phi=pi/2; % integration upper limit

% inizialization
T=zeros(1,res);

for i=1:res
    theta0=i*sup/res; % theta0 is the pendulum amplitude
    k = sin(theta0/2);
    F = @(t) 1./sqrt(1-(k*sin(t)).^2);
    T(i)=quad(F,0,phi); % numerical integration 
end

T = 2./pi.*T; % normalization
deg = 180/pi*sup*(1:res)./res; % conversion to degrees

% saving in the external file
temp = [deg; T];
temp = temp';
save -ascii 'pendulum_period.dat' temp;

then, in order to plot it, I used the following Gnuplot code:

# set the output
set terminal svg
set output "pendulum_period.svg"

# axis properties
set yrange [0.99:1.08]
set xzeroaxis linetype -1 linewidth 0.5
set yzeroaxis linetype -1 linewidth 0.5
set xtics axis
set ytics axis
set key off

plot "pendulum_period.dat" using 1:2 with lines linewidth 2
This code creates a file called pendulum_period.svg. I heavily post-processed it with Inkscape.
Date 29 November 2006(2006-11-29)
Source I created it by myself using Matlab, Gnuplot and Inkscape
Inkscape Logo.svg
 This vector image was created with Inkscape.
Lissajous small.svg
 This vector image was created with gnuplot.
Author Alessio Damato; thanks to John wayman, he let me notice a mistake in the code.


[edit] Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.

w:en:Creative Commons
attribution share alike		 This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
  • share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
This licensing tag was added to this file as part of the GFDL licensing update.

w:en:Creative Commons
attribution share alike		 This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic, 2.0 Generic and 1.0 Generic license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
  • share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
You may select the license of your choice.

File history

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Date/TimeThumbnailDimensionsUserComment
current19:54, 16 April 2007Thumbnail for version as of 19:54, 16 April 2007730 × 591 (28 KB)Alejo2083fixed mistake
13:48, 29 November 2006Thumbnail for version as of 13:48, 29 November 2006648 × 480 (24 KB)Alejo2083 {{Information |Description=The plot represents the ration between the period an oscillator and the approximated value obtained for small angles. According to the relative [http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29 article on wikipedia], the
The following pages on the English Wikipedia link to this file (pages on other projects are not listed):

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This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.
Width 730
Height 591
Retrieved from "http://en.wikipedia.org/wiki/File:Pendulum_period.svg"
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