We show that there are an infinite number of tautochrone curves in addition to the cycloid solution first obtained by Christiaan Huygens in 1658. We begin by reviewing the inverse problem of finding the possible potential energy functions that lead to periodic motions of a particle whose period is a given function of its mechanical energy. There are infinitely many such solutions, called “sheared” potentials. As an interesting example, we show that a Pöschl-Teller potential and the one-dimensional Morse potentials are sheared relative to one another for negative energies, clarifying why they share the same oscillation periods for their bounded solutions. We then consider periodic motions of a particle sliding without friction over a track around its minimum under the influence of a constant gravitational field. After a brief historical survey of the tautochrone problem we show that, given the oscillation period, there is an infinity of tracks that lead to the same period. As a bonus, we show that there are infinitely many tautochrones.
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December 2016
PAPERS|
December 01 2016
Is the tautochrone curve unique?
Pedro Terra;
Pedro Terra
a)
Instituto de Física,
Universidade Federal do Rio de Janeiro
, Rio de Janeiro RJ 21945-970, Brazil
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Reinaldo de Melo e Souza;
Reinaldo de Melo e Souza
b)
Instituto de Física,
Universidade Federal do Rio de Janeiro
, Rio de Janeiro RJ 21945-970, Brazil
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a)
Electronic mail: terra@if.ufrj.br
b)
Electronic mail: reinaldo@if.ufrj.br
c)
Electronic mail: farina@if.ufrj.br
Am. J. Phys. 84, 917–923 (2016)
Article history
Received:
February 22 2016
Accepted:
September 09 2016
Citation
Pedro Terra, Reinaldo de Melo e Souza, C. Farina; Is the tautochrone curve unique?. Am. J. Phys. 1 December 2016; 84 (12): 917–923. https://doi.org/10.1119/1.4963770
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