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
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.
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]]
PROG
(PARI) 4*ellK(sqrt(4/5))/sqrt(5)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 01 2024
STATUS
approved