2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The period \(T\) of a moving pendulum without friction around the fixed point \(O\), whose mass is concentrated at the center of gravity \(G\) (\(\mathrm{OG}=l\)) and whose maximum angular amplitude is \({\theta }_{0}\) is given by the series [bib1]:
\(T=2\pi \sqrt{\frac{l}{g}}\left[1+\underset{n=1}{\overset{\infty }{\Sigma }}{a}_{n}^{2}{(\mathrm{sin}\frac{{\theta }_{0}}{2})}^{\mathrm{2n}}\right]\)
with
\({a}_{n}=\frac{\mathrm{2n}-1}{\mathrm{2n}}\)
2.2. Benchmark results#
For \(l=0.5m\), \(g=9.81m/{s}^{2}\), and \({\theta }_{0}=\pi /2\), we get: \(T=1.6744s\)
2.3. Uncertainty about the solution#
The terms of the series were summed up to and including \(n=12\), with the last term taken into account being less than \({10}^{-5}\) times the calculated sum.
2.4. Bibliographical references#
HAAG, « Vibratory movements », P.U.F. (1952).