A336274 -id:A336274 - cp's OEIS frontend

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.

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

Offset: 0

Amiram Eldar, May 07 2022

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.

			0.64922414456459126471247474244668203153595016469104...

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.

Cf. A353770, A353771, A353772, A353773.

Cf. A070058, A336274, A353407.

RealDigits[Pi/2 + Log[(Sqrt[2] + 1)*(Sqrt[6] - 1)/Sqrt[5]] - 2*ArcSin[Sqrt[2/5]], 10, 100][[1]]

Equals Pi/2 + log((sqrt(2) + 1)*(sqrt(6) - 1)/sqrt(5)) - 2*arcsin(sqrt(2/5)).

A353770 Decimal expansion of the gravitational acceleration generated at a vertex by a unit-mass cube with edge length 2 in units where the gravitational constant is G = 1.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 07 2022

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 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 cube on its surface attains its minimum absolute value.

			0.41975733988710629187374768720081390960585610276177...

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.

Cf. A353769, A353771, A353772, A353773.

Cf. A070058, A336274, A353407.

RealDigits[(Sqrt[3]/2)*(Pi/12 + Log[Sqrt[2] + 1] - Log[Sqrt[3] + 2]/2), 10, 100][[1]]

Equals (sqrt(3)/2)*(Pi/12 + log(sqrt(2) + 1) - log(sqrt(3) + 2)/2).

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.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 07 2022

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).

			0.92598126055729142809343668703833155990642541428277...

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.

Cf. A070058, A336274, A353769, A353770.

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]]

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.

A336275 Decimal expansion of the dimensionless coefficient of the Coulomb self-energy of a uniformly charged two-dimensional square.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 15 2020

Coulomb self-energy of a system of electric charges is the total electrostatic potential energy of interaction between charge elements.

For a uniformly charged two-dimensional square with a total charge Q and a side length L it is equal to c * k*Q^2/L, where k is the Coulomb constant (A182999) and c is this constant.

			1.486604799123689351264092833819785899009872739605305...

Orion Ciftja, Coulomb self-energy and electrostatic potential of a uniformly charged square in two dimensions, Physics Letters A, Vol. 374, No. 7 (2010), pp. 981-983, alternative link.
Wikipedia, Electric potential energy.

Cf. A085469, A182999, A336274.

RealDigits[2*(1 - Sqrt[2])/3 + 2*Log[1 + Sqrt[2]], 10, 100][[1]]

2*(1 - sqrt(2))/3 + 2 * log(1 + sqrt(2)) \\ Michel Marcus, Jul 15 2020

Equals 2*(1 - sqrt(2))/3 + 2 * log(1 + sqrt(2)).

cp's OEIS Frontend

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.

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A353770 Decimal expansion of the gravitational acceleration generated at a vertex by a unit-mass cube with edge length 2 in units where the gravitational constant is G = 1.

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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.

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A336275 Decimal expansion of the dimensionless coefficient of the Coulomb self-energy of a uniformly charged two-dimensional square.

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