How is the period of a simple pendulum growing with increasing amplitude? (English) Zbl 1484.70008
Summary: For the period \(T(\alpha)\) of a simple pendulum with the length \(L\) and the amplitude (the initial elongation) \(\alpha\in (0,\pi)\), a strictly increasing sequence \(T_n(\alpha)\) is constructed such that the relations
\begin{align*}
& T_1(\alpha)=2\sqrt{\frac{L}{g}}\left[\pi-2+\frac{1}{\epsilon} \ln\left(\frac{1+\varepsilon}{1-\varepsilon}\right)+\left(\frac{\pi}{4}-\frac{2}{3}\right)\varepsilon^2\right],\\
& T_{n+1}(\alpha)=T_n(\alpha)+2\sqrt{\frac{L}{g}}\left(\pi w_{n+1}^2 - \frac{2}{2n+3}\right)\varepsilon^{2n+2},
\end{align*}
and
\[
0<\frac{T(\alpha)-T_n(\alpha)}{T(\alpha)}<\frac{2\varepsilon^{2n+2}}{\pi(2n+1)},
\]
holds true, for \(\alpha\in (0,\pi)\), \(n\in\mathbb{N}\), \(w_n:=\prod_{k=1}^n\frac{2k-1}{2k}\) (the \(n\)-th Wallis’ ratio) and \(\varepsilon=\sin(\alpha/2)\).
MSC:
| 70E17 | Motion of a rigid body with a fixed point |
| 70F20 | Holonomic systems related to the dynamics of a system of particles |
| 40A25 | Approximation to limiting values (summation of series, etc.) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 65B10 | Numerical summation of series |
Software:
MathematicaReferences:
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