OFFSET
0,1
COMMENTS
The absolute value of the total gravitational attraction force between two identical homogeneous cubes, each with mass M and edge length s, whose centers are at distance s is c*G*M^2/s^2, where G is the gravitational constant (A070058) and c is this constant.
The calculation of the closed-form formula for this constant was done by Prof. Bengt Fornberg of the University of Colorado (Trefethen, 2011).
LINKS
Folkmar Bornemann, The Challenge of Sixfold Integrals: The Closed-Form Evaluation of Newton Potentials between Two Cubes, arXiv:2204.02793 [math.CA], 2022.
Jeff Sanny and David M. Smith, How Spherical Is a Cube (Gravitationally)?, The Physics Teacher, Vol. 53 (2015), pp. 111-113; alternative link.
Lloyd N. Trefethen, Ten digit problems, in: D. Schleicher and M. Lackmann (eds.), An Invitation to Mathematics, Springer, Berlin, Heidelberg, 2011, pp. 119-136; alternative link.
Lloyd N. Trefethen, Two Cubes, LMS Newsletter, Issue 491 (November 2020), p. 17.
Michael Trott, Calculating the energy between two cubes, News, Views and Insights from Wolfram, Wolfram Blog, October 23, 2012.
FORMULA
Equals (26*Pi/3 - 14 + 2*sqrt(2) - 4*sqrt(3) + 10*sqrt(5) - 2*sqrt(6) + 26*log(2) - 2*log(5) + 10*log(sqrt(2) + 1) + 20*log(sqrt(3) + 1) - 35*log(sqrt(5) + 1) + 6*log(sqrt(6) + 1) - 2*log(sqrt(6) + 4) - 22*arctan(2*sqrt(6)))/3.
EXAMPLE
0.92598126055729142809343668703833155990642541428277...
MATHEMATICA
RealDigits[(26*Pi/3 - 14 + 2*Sqrt[2] - 4*Sqrt[3] + 10*Sqrt[5] - 2*Sqrt[6] + 26*Log[2] - 2*Log[5] + 10*Log[Sqrt[2] + 1] + 20*Log[Sqrt[3] + 1] - 35*Log[Sqrt[5] + 1] + 6*Log[Sqrt[6] + 1] - 2*Log[Sqrt[6] + 4] - 22*ArcTan[2*Sqrt[6]])/3, 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2022
STATUS
approved