inverse-square - 2024 ARCADIA

Question

Answer the following about things that make 3D special in physics, for 10 points each.

[10m] Gravitational orbits and atoms can only exist in 3D since solving the Poisson equation in 3D gives force laws with this property. Only force laws with this form and linear force laws give rise to stable orbits by Bertrand’s theorem.

ANSWER: inverse-square [accept one over r-squared or one over d-squared or one over distance-squared or one over radius-squared; prompt on central forces]

[10h] Tensor quantities like angular velocity and magnetic field can only be represented as vectors in 3D, because 2-forms and 1-forms are Hodge dual in 3D. The representations can be mapped to each other using this operator, symbolized epsilon-sub-ijk in 3D.

ANSWER: Levi-Civita symbol

[10e] The duality of vectors and rank-2 tensors explains why this operation is defined in 3D. Taking this operation of two vectors outputs a vector that is perpendicular to both.

ANSWER: cross product [prompt on vector product; reject “dot product”]

<VD, Physics>

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Summary


2024 ARCADIA at BU	2024-12-06	Y	7	8.57	57%	29%	0%
2024 ARCADIA at Duke	2024-12-06	Y	2	15.00	100%	0%	50%
2024 ARCADIA at Missouri S&T	2024-12-06	Y	2	25.00	100%	100%	50%
2024 ARCADIA at Penn State	2024-12-06	Y	2	25.00	100%	100%	50%
2024 ARCADIA at Florida Tech	2024-12-06	Y	4	10.00	75%	25%	0%
2024 ARCADIA at UC Berkeley	2024-12-06	Y	3	23.33	100%	67%	67%
2024 ARCADIA at Waterloo	2024-11-09	Y	6	16.67	100%	33%	33%
2024 ARCADIA at GT	2024-12-06	Y	5	10.00	100%	0%	0%
2024 ARCADIA at Illinois	2024-11-09	Y	4	20.00	100%	50%	50%
2024 ARCADIA at Warwick	2024-12-06	Y	5	24.00	100%	60%	80%
2024 ARCADIA Online	2025-05-17	N	5	16.00	100%	40%	20%
2024 ARCADIA at Claremont	2024-11-02	N	3	23.33	100%	67%	67%

Data


Tufts B	Boston College B	0	0	10	10
Brandeis A	Brandeis B	0	0	10	10
Clark House	Bowdoin B	0	0	0	0
Clark Wilson	Bowdoin C	0	0	0	0
Boston College A	Tufts A	0	0	0	0
Harvard	Yale	10	0	10	20
Virginia Tech A	Virginia Tech C	0	0	10	10
North Carolina B	Virginia Tech B	0	10	10	20
Truman	Missouri	10	0	10	20
WashU	Squidward Community College	10	10	10	30
Pitt	Penn State	10	0	10	20
RIT	Maryland	10	10	10	30
UCF B	USF	0	0	0	0
UF A	UF B	10	0	10	20
UCF A	UF C	0	0	10	10
Florida Tech	Valencia	0	0	10	10
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Stanford	Berkeley A	10	10	10	30
Berkeley B	Berkeley Past	0	0	10	10
MIT	Amherst	10	0	10	20
McMaster ApocolocyntosisBidii	Toronto Chestnut Rice and Kamehameha	0	0	10	10
Toronto A	Ottawa Absolomabsolomabsolom	10	0	10	20
Toronto B	Waterloo Aspidistra	10	10	10	30
McGill A	Toronto Metropolitan A	0	0	10	10
Toronto Disband the Club 2k24	Waterloo Basic	0	0	10	10
Carleton A	Waterloo Cloze	0	10	10	20
Georgia Tech C	Milton	0	0	10	10
Georgia Tech A	Georgia Tech D	0	0	10	10
Innovation Academy A	Mississippi State	0	0	10	10
Innovation Academy B	Innovation Academy C	0	0	10	10
Johns Creek	Georgia Tech B	0	0	10	10
Illinois Orange	Illinois Blue	10	0	10	20
Chicago B	Purdue B	0	10	10	20
Purdue A	SIUE	10	0	10	20
Cambridge A	Warwick A	10	10	10	30
Imperial B	LSE A	0	10	10	20
Chicago A	Notre Dame	0	10	10	20
Imperial A	LSE B	10	10	10	30
Bristol	Warwick B	10	10	10	30
Warwick C	Southampton	0	0	10	10