A256514 -id:A256514

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A376643

Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

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4, 0, 3, 7, 8, 1, 1, 6, 3, 9, 9, 5, 6, 8, 4, 6, 4, 3, 1, 1, 6, 8, 0, 2, 8, 8, 7, 9, 9, 9, 7, 8, 6, 4, 9, 3, 0, 1, 3, 6, 0, 8, 3, 9, 9, 3, 4, 0, 8, 8, 0, 6, 8, 5, 7, 8, 6, 3, 4, 9, 6, 1, 5, 9, 8, 9, 7, 7, 7, 3, 8, 3, 7, 8, 6, 5, 3, 1, 9, 4, 7, 4, 4, 4, 0, 7, 7, 0, 1, 5, 0, 7, 0, 3, 3, 7, 9, 1, 9, 6, 9, 1, 0, 5, 7

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OFFSET

1,1

COMMENTS

A point mass is attached to a frictionless pivot by a massless string of length L and revolves in a vertical circle about the pivot in a uniform gravitational field with an acceleration g. The slowest possible motion occurs when the tension in the string is momentarily zero at the top of the route, and the longest-possible period is then c * sqrt(L/g), where c is this constant.

LINKS

Table of n, a(n) for n=1..105.

Physics StackExchange, Maximum period of a vertically spinning ball, 2016.

Physics StackExchange, Maximal time for an object vertically swinging on a rope to complete a full circle, 2023.

Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.

Wikipedia, Elliptic integral: Complete elliptic integral of the first kind.

FORMULA

Equals 2 * Integral_{0..Pi} (1/sqrt(3 + 2*cos(x))) dx.

EXAMPLE

4.03781163995684643116802887999786493013608399340880...

MATHEMATICA

RealDigits[4 * EllipticK[4/5] / Sqrt[5], 10, 120][[1]]