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A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

%I A376643 #6 Oct 01 2024 08:51:19
%S A376643 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,
%T A376643 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,
%U A376643 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
%N A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.
%C A376643 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.
%H A376643 Physics StackExchange, <a href="https://physics.stackexchange.com/questions/239661/maximum-period-of-a-vertically-spinning-ball">Maximum period of a vertically spinning ball</a>, 2016.
%H A376643 Physics StackExchange, <a href="https://physics.stackexchange.com/questions/751432/maximal-time-for-an-object-vertically-swinging-on-a-rope-to-complete-a-full-circ">Maximal time for an object vertically swinging on a rope to complete a full circle</a>, 2023.
%H A376643 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>.
%H A376643 Wikipedia, <a href="https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind">Elliptic integral: Complete elliptic integral of the first kind</a>.
%F A376643 Equals 2 * Integral_{0..Pi} (1/sqrt(3 + 2*cos(x))) dx.
%e A376643 4.03781163995684643116802887999786493013608399340880...
%t A376643 RealDigits[4 * EllipticK[4/5] / Sqrt[5], 10, 120][[1]]
%o A376643 (PARI) 4*ellK(sqrt(4/5))/sqrt(5)
%Y A376643 Constants related to similar physical problems: A019692, A038533, A038534, A175574, A256514, A309893, A310000.
%K A376643 nonn,cons
%O A376643 1,1
%A A376643 _Amiram Eldar_, Oct 01 2024