Image:Pendulum period.svg

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The graph as well as other important points are simply wrong: The time at 90° amplitude is 18% larger than T0 and not 7.3% as shown in the plot. Then, the author plots the integrand instead of the integral. Furthermore, he calculates sin(th0)/2 instead of sin(th0/2). Means, this page is seriously flawed!

[edit] Summary

Description

The plot represents the ration between the period an oscillator and the approximated value obtained for small angles. According to the relative article on wikipedia, 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:

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

where:

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

so the ratio is given by:

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

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

res=1000; % resolution
phi=pi/2;

T=zeros(1,res); % initialization

for i=1:res
    theta=i*pi/(2*res);
    k = sin(theta)./2;
    F = @(t) 1./sqrt(1-(k*sin(t)).^2);
    T(i)=quad(F,0,phi);
end

T = 2.*T./pi; % normalization
angle=90*(1:res)./res; % to plot agains degrees

% saving in the external file
temp = [angle; 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

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.

Source

I created it by myself using Matlab, Gnuplot and Inkscape

This vector image was created with Inkscape.

This vector image was created with Gnuplot.

Date

29/11/2006

Author

Alessio Damato

Permission

see below

[edit] Licensing

I, the author of this work, hereby publish it under the following licenses:

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".