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Search: a353770 -id:a353770
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A353769
Decimal expansion of the gravitational acceleration generated at the center of a face by a unit-mass cube with edge length 2 in units where the gravitational constant is G = 1.
+10
5
6, 4, 9, 2, 2, 4, 1, 4, 4, 5, 6, 4, 5, 9, 1, 2, 6, 4, 7, 1, 2, 4, 7, 4, 7, 4, 2, 4, 4, 6, 6, 8, 2, 0, 3, 1, 5, 3, 5, 9, 5, 0, 1, 6, 4, 6, 9, 1, 0, 4, 1, 9, 3, 1, 3, 4, 8, 7, 8, 0, 0, 3, 3, 4, 0, 3, 3, 2, 2, 1, 2, 8, 6, 1, 7, 1, 1, 1, 5, 9, 9, 4, 3, 1, 3, 1, 4, 4, 2, 9, 8, 3, 8, 6, 5, 2, 6, 4, 0, 8, 2, 9, 9, 0, 0
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
The absolute value of the gravitational attraction force between a homogeneous cube with mass M and edge length 2*s and a test particle with mass m located at the cube's center of face is c*G*M*m/s^2, where G is the gravitational constant (A070058) and c is this constant.
The centers of the faces are the positions where the gravitational field that is generated by the cube attains its maximum absolute value.
LINKS
Table of n, a(n) for n=0..104.
Murray S. Klamkin, Extreme Gravitational Attraction, Problem 92-5, SIAM Review, Vol. 34, No. 1 (1992), pp. 120-121; Solution, by Carl C. Grosjean, ibid., Vol. 38, No. 3 (1996), pp. 515-520.
J. A. Lira, If the Earth were a cube, what would be the value of the acceleration of gravity at the center of each face?, Physics Education, Vol. 53, No. 6 (2018), 065013.
Eric Weisstein's World of Physics, Cube Gravitational Force.
Eric Weisstein's World of Physics, Polyhedron Gravitational Force.
FORMULA
Equals Pi/2 + log((sqrt(2) + 1)*(sqrt(6) - 1)/sqrt(5)) - 2*arcsin(sqrt(2/5)).
EXAMPLE
0.64922414456459126471247474244668203153595016469104...
MATHEMATICA
RealDigits[Pi/2 + Log[(Sqrt[2] + 1)*(Sqrt[6] - 1)/Sqrt[5]] - 2*ArcSin[Sqrt[2/5]], 10, 100][[1]]
CROSSREFS
Cf. A353770, A353771, A353772, A353773.
Cf. A070058, A336274, A353407.
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2022
STATUS
approved
A353771
Decimal expansion of the gravitational acceleration generated at the center of a face by unit-mass regular tetrahedron with edge length 2 in units where the gravitational constant is G = 1.
+10
4
2, 5, 6, 3, 3, 1, 1, 8, 1, 6, 1, 4, 3, 6, 4, 9, 4, 6, 6, 8, 8, 2, 2, 9, 3, 9, 5, 7, 5, 4, 8, 4, 0, 7, 9, 5, 1, 8, 3, 4, 5, 8, 5, 1, 1, 7, 5, 9, 1, 1, 8, 4, 4, 9, 6, 7, 7, 0, 3, 9, 4, 4, 9, 2, 4, 6, 4, 9, 0, 1, 6, 3, 8, 2, 5, 4, 0, 1, 8, 9, 5, 0, 9, 0, 7, 3, 0, 4, 6, 7, 2, 2, 8, 6, 8, 0, 9, 4, 5, 2, 9, 5, 2, 0, 7
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The absolute value of the gravitational attraction force between a homogeneous regular tetrahedron with mass M and edge length 2*s and a test particle with mass m located at the tetrahedron's center of face is c*G*M*m/s^2, where G is the gravitational constant (A070058) and c is this constant.
The centers of the faces are the positions where the gravitational field that is generated by the tetrahedron attains its maximum absolute value.
LINKS
Table of n, a(n) for n=1..105.
Murray S. Klamkin, Extreme Gravitational Attraction, Problem 92-5, SIAM Review, Vol. 34, No. 1 (1992), pp. 120-121; Solution, by Carl C. Grosjean, ibid., Vol. 38, No. 3 (1996), pp. 515-520.
Eric Weisstein's World of Physics, Polyhedron Gravitational Force.
Eric Weisstein's World of Physics, Tetrahedron Gravitational Force.
FORMULA
Equals 2*Pi/(3*sqrt(3)) + sqrt(6)*log(sqrt(3) + 2) - 2*sqrt(6)*log(sqrt(3) + sqrt(2))/3.
EXAMPLE
2.56331181614364946688229395754840795183458511759118...
MATHEMATICA
RealDigits[2*Pi/(3*Sqrt[3]) + Sqrt[6]*Log[Sqrt[3] + 2] - 2*Sqrt[6]*Log[Sqrt[3] + Sqrt[2]]/3, 10, 100][[1]]
CROSSREFS
Cf. A070058, A353769, A353770, A353772, A353773.
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2022
STATUS
approved
A353772
Decimal expansion of the gravitational acceleration generated at a vertex by a unit-mass regular tetrahedron with edge length 2 in units where the gravitational constant is G = 1.
+10
4
9, 5, 4, 8, 5, 4, 6, 6, 5, 9, 6, 6, 1, 5, 6, 7, 8, 0, 1, 4, 5, 5, 0, 9, 5, 2, 8, 0, 3, 3, 6, 9, 0, 5, 8, 9, 6, 0, 2, 4, 7, 1, 4, 7, 0, 9, 8, 7, 5, 7, 2, 3, 4, 0, 9, 8, 0, 2, 0, 0, 8, 3, 5, 1, 3, 3, 4, 2, 7, 0, 0, 4, 5, 7, 9, 9, 0, 5, 9, 5, 5, 1, 3, 2, 1, 0, 3, 7, 3, 5, 2, 7, 7, 0, 0, 1, 0, 4, 7, 9, 0, 6, 2, 6, 2
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
The absolute value of the gravitational attraction force between a homogeneous regular tetrahedron with mass M and edge length 2*s and a test particle with mass m located at the tetrahedron's vertex is c*G*M*m/s^2, where G is the gravitational constant (A070058) and c is this constant.
The vertices are the positions where the gravitational field that is generated by the tetrahedron on its surface attains its minimum absolute value.
LINKS
Table of n, a(n) for n=0..104.
Murray S. Klamkin, Extreme Gravitational Attraction, Problem 92-5, SIAM Review, Vol. 34, No. 1 (1992), pp. 120-121; Solution, by Carl C. Grosjean, ibid., Vol. 38, No. 3 (1996), pp. 515-520.
Eric Weisstein's World of Physics, Polyhedron Gravitational Force.
Eric Weisstein's World of Physics, Tetrahedron Gravitational Force.
FORMULA
Equals 6*sqrt(3)*(Pi/3 - arctan(sqrt(2))).
Equals 3*sqrt(3)*(Pi/6 - arctan(sqrt(2)/4)).
EXAMPLE
0.95485466596615678014550952803369058960247147098757...
MATHEMATICA
RealDigits[6*Sqrt[3]*(Pi/3 - ArcTan[Sqrt[2]]), 10, 100][[1]]
CROSSREFS
Cf. A070058, A353769, A353770, A353771, A353773.
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2022
STATUS
approved
A353773
Decimal expansion of the gravitational acceleration generated at a vertex by a unit-mass regular octahedron with edge length 2 in units where the gravitational constant is G = 1.
+10
4
6, 4, 7, 0, 5, 8, 8, 2, 7, 5, 9, 7, 8, 4, 7, 3, 5, 8, 2, 3, 7, 9, 0, 6, 1, 9, 4, 7, 4, 6, 1, 7, 4, 6, 6, 8, 5, 4, 7, 7, 4, 2, 9, 8, 0, 4, 6, 7, 8, 6, 6, 8, 1, 4, 6, 7, 8, 0, 1, 1, 8, 3, 1, 5, 5, 4, 1, 6, 2, 6, 2, 7, 5, 5, 0, 0, 0, 9, 4, 1, 3, 7, 3, 1, 6, 0, 7, 7, 7, 9, 2, 6, 2, 3, 0, 1, 8, 8, 4, 7, 6, 7, 8, 3, 1
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
The absolute value of the gravitational attraction force between a homogeneous regular octahedron with mass M and edge length 2*s and a test particle with mass m located at the octahedron's vertex is c*G*M*m/s^2, where G is the gravitational constant (A070058) and c is this constant.
The vertices are the positions where the gravitational field that is generated by the octahedron on its surface attains its minimum absolute value.
LINKS
Table of n, a(n) for n=0..104.
Murray S. Klamkin, Extreme Gravitational Attraction, Problem 92-5, SIAM Review, Vol. 34, No. 1 (1992), pp. 120-121; Solution, by Carl C. Grosjean, ibid., Vol. 38, No. 3 (1996), pp. 515-520.
Eric Weisstein's World of Physics, Octahedron Gravitational Force.
Eric Weisstein's World of Physics, Polyhedron Gravitational Force.
FORMULA
Equals sqrt(2)*log(3*(sqrt(2)-1)) + arctan(sqrt(2)/4).
EXAMPLE
0.64705882759784735823790619474617466854774298046786...
MATHEMATICA
RealDigits[Sqrt[2]*Log[3*(Sqrt[2]-1)] + ArcTan[Sqrt[2]/4], 10, 100][[1]]
CROSSREFS
Cf. A070058, A353769, A353770, A353771, A353772.
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2022
STATUS
approved
A353407
Decimal expansion of the gravitational force between two unit-edge-length unit-mass cubes whose centers are a unit distance apart, so they are in contact along one face, in units where the gravitational constant is G = 1.
+10
2
9, 2, 5, 9, 8, 1, 2, 6, 0, 5, 5, 7, 2, 9, 1, 4, 2, 8, 0, 9, 3, 4, 3, 6, 6, 8, 7, 0, 3, 8, 3, 3, 1, 5, 5, 9, 9, 0, 6, 4, 2, 5, 4, 1, 4, 2, 8, 2, 7, 7, 7, 8, 6, 5, 5, 9, 8, 7, 3, 4, 3, 4, 5, 4, 0, 9, 5, 9, 8, 4, 2, 2, 4, 9, 8, 6, 3, 2, 8, 6, 2, 2, 1, 4, 8, 5, 4, 1, 6, 8, 0, 8, 2, 6, 5, 1, 3, 3, 4, 0, 8, 5, 4, 0, 1
(list; constant; graph; refs; listen; history; text; internal format)
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
Table of n, a(n) for n=0..104.
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
Cf. A070058, A336274, A353769, A353770.
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2022
STATUS
approved
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