A review of lienar approximations for large oscillation amplitudes of a simple pendulum

Lopes, Flávio Silva; Suave, Rogério Netto; Nogueira, Jose Alexandre

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Revista Brasileira de Ensino de Física

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Artigos Gerais • Rev. Bras. Ensino Fís. 40 (3) • 2018 • https://doi.org/10.1590/1806-9126-RBEF-2017-0393 copy

A review of lienar approximations for large oscillation amplitudes of a simple pendulum

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Although the $sin θ \approx θ$ approximation to calculate the period of a simple pendulum is well-known and studied from the first year of high school, it only has good results for small oscillation amplitudes. Now, the approximations for large oscillation amplitudes are almost completely unknown by the vast majority of theachers and students of high school and of undergraduate. Hence, in this work, we review the main approximation suggestions for large oscillation amplitudes to calculate the period of a simple pendulum and we make a comparison between their results, too.

Keywords:
simple pendulum; linear approximations; large amplitudes

$θ_{M}$	Erro %
$10^{0}$	$0, 1904$
$30^{0}$	$1, 711$
$60^{0}$	$6, 819$
$90^{0}$	$15, 27$
$114, 6^{0}$	$24, 75$

$θ_{M}$	Erro %
$10^{0}$	$4, 240 \times 10^{- 5}$
$30^{0}$	$3, 510 \times 10^{- 3}$
$60^{0}$	$6, 066 \times 10^{- 2}$
$90^{0}$	$3, 547 \times 10^{- 1}$
$114, 6^{0}$	$1, 129$

$θ_{M}$	Erro %
$10^{0}$	$9, 085 \times 10^{- 5}$
$30^{0}$	$7, 512 \times 10^{- 3}$
$60^{0}$	$1, 293 \times 10^{- 1}$
$90^{0}$	$7, 512 \times 10^{- 1}$
$114, 6^{0}$	$2, 3736$

$θ_{M}$	Erro %
$10^{0}$	$6, 092 \times 10^{- 6}$
$30^{0}$	$5, 274 \times 10^{- 4}$
$60^{0}$	$1, 055 \times 10^{- 2}$
$90^{0}$	$7, 772 \times 10^{- 2}$
$114, 6^{0}$	$3, 168 \times 10^{- 1}$

	Erro %
$θ_{M}$	$sin θ \approx θ$	Molina	Kidd-Fogg	Ganley-Parwani
$10^{0}$	$0, 1904$	$4, 240 \times 10^{- 5}$	$9, 085 \times 10^{- 5}$	$6, 092 \times 10^{- 6}$
$30^{0}$	$1, 711$	$3, 510 \times 10^{- 3}$	$7, 512 \times 10^{- 3}$	$5, 274 \times 10^{- 4}$
$60^{0}$	$6, 819$	$6, 066 \times 10^{- 2}$	$1, 293 \times 10^{- 1}$	$1, 055 \times 10^{- 2}$
$90^{0}$	$15, 27$	$3, 547 \times 10^{- 1}$	$7, 512 \times 10^{- 1}$	$7, 772 \times 10^{- 2}$
$114, 6^{0}$	$24, 75$	$1, 129$	$2, 3736$	$3, 168 \times 10^{- 1}$

Sociedade Brasileira de Física Caixa Postal 66328, 05389-970 São Paulo SP - Brazil - São Paulo - SP - Brazil
E-mail: marcio@sbfisica.org.br

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[1] Endereço atual: Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)