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How is the period of a simple pendulum growing with increasing amplitude?

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Veröffentlicht/Copyright: 14. April 2021
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Abstract

For the period T(α) of a simple pendulum with the length L and the amplitude (the initial elongation) α ∈ (0, π), a strictly increasing sequence Tn(α) is constructed such that the relations

T1(α)=2Lgπ2+1ϵln1+ϵ1ϵ+π423ϵ2,Tn+1(α)=Tn(α)+2Lgπwn+1222n+3ϵ2n+2,

and

0<T(α)Tn(α)T(α)<2ϵ2n+2π(2n+1),

holds true, for α ∈ (0, π), n ∈ ℕ, wn:=k=1n2k12k (the nth Wallis’ ratio) and ϵ = sin(α/2).

  1. Communicated by Marek Balcerzak

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Received: 2020-03-02
Accepted: 2020-04-21
Published Online: 2021-04-14
Published in Print: 2021-04-27

© 2021 Mathematical Institute Slovak Academy of Sciences

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