parallel axis theorem - 2024 ARCADIA

Question

Consider a stick of mass M and length L, which is constrained to move such that it is always a chord of a fixed vertical, circular ring of radius R. For 10 points each:

[10m] The moment of inertia of the stick relative to the ring’s center can be calculated using this theorem, coming out to be M times the quantity R-squared minus one-sixth L squared.

ANSWER: parallel axis theorem [or Huygens-Steiner theorem; reject “perpendicular axis theorem”]

[10h] This quantity for the system equals “one-half I theta-dot squared, minus Mgd times the quantity one minus cosine theta,” where d is the distance from the center of the ring to the stick.

ANSWER: Lagrangian [prompt on L]

[10e] Solving the equation of motion for small perturbations about the equilibrium position shows the system exhibits this behavior. Systems obeying Hooke’s law display the “simple harmonic” form of this motion.

ANSWER: oscillation [accept simple harmonic oscillation or SHO]

<BW, Physics>

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Summary


2024 ARCADIA at BU	2024-12-06	Y	8	16.25	100%	63%	0%
2024 ARCADIA at Duke	2024-12-06	Y	3	13.33	100%	33%	0%
2024 ARCADIA at Missouri S&T	2024-12-06	Y	2	15.00	100%	50%	0%
2024 ARCADIA Online	2025-05-17	Y	5	18.00	100%	80%	0%
2024 ARCADIA at Penn State	2024-12-06	Y	2	20.00	100%	50%	50%
2024 ARCADIA at Florida Tech	2024-12-06	Y	4	12.50	75%	50%	0%
2024 ARCADIA at UC Berkeley	2024-12-06	Y	2	20.00	100%	100%	0%
2024 ARCADIA at Waterloo	2024-11-09	Y	5	16.00	100%	40%	20%
2024 ARCADIA at GT	2024-12-06	Y	5	16.00	80%	80%	0%
2024 ARCADIA at Illinois	2024-11-09	Y	4	22.50	100%	100%	25%
2024 ARCADIA at Warwick	2024-12-06	Y	5	16.00	100%	40%	20%
2024 ARCADIA at Claremont	2024-11-02	Y	3	26.67	100%	100%	67%

Data


BU	Brandeis A	0	0	10	10
Boston College B	Bowdoin B	0	0	10	10
Bowdoin A	Brandeis B	10	0	10	20
Harvard	Tufts A	10	0	10	20
MIT	Clark Wilson	10	0	10	20
Boston College A	Tufts B	0	0	10	10
Yale	Bowdoin C	10	0	10	20
North Carolina A	Virginia Tech B	0	0	10	10
Amherst	Clark House	10	0	10	20
Virginia Tech A	South Carolina	10	0	10	20
Missouri	Squidward Community College	0	0	10	10
North Carolina B	Virginia Tech C	0	0	10	10
Social credit go vroom	ASU->Stanford->ASU pipeline	10	0	10	20
bingy academy	In the Mood for Buzz	10	0	10	20
she limp on my waltz till i feel pathetique	Madmen and Specialists	10	0	10	20
Paddington in Peru (2024)	The cult of SGA	0	0	10	10
Toyota Tundra Turbos — Twisting Truths - Tackling Trivia - Taming Titans	Old and Young	10	0	10	20
Maryland	Penn State	0	0	10	10
RIT	Pitt	10	10	10	30
UCF B	Florida Tech	0	0	10	10
UCF A	UF A	10	0	10	20
UF B	UF C	10	0	10	20
USF	Valencia	0	0	0	0
Berkeley Past	Stanford	10	0	10	20
Berkeley A	Berkeley B	10	0	10	20
Carleton A	McGill A	0	10	10	20
Toronto Metropolitan A	McMaster ApocolocyntosisBidii	0	0	10	10
Toronto A	Toronto B	10	0	10	20
Toronto Chestnut Rice and Kamehameha	Waterloo Cloze	0	0	10	10
Waterloo Basic	Waterloo Aspidistra	10	0	10	20
Georgia Tech A	Mississippi State	10	0	10	20
Innovation Academy C	Lambert	10	0	10	20
Milton	Innovation Academy B	0	0	0	0
Georgia Tech D	Johns Creek	10	0	10	20
Georgia Tech C	Innovation Academy A	10	0	10	20
Chicago A	Purdue B	10	10	10	30
Illinois Blue	Notre Dame	10	0	10	20
Chicago B	Purdue A	10	0	10	20
SIUE	Illinois Orange	10	0	10	20
Cambridge B	Imperial B	10	0	10	20
LSE A	Warwick C	0	0	10	10
Cambridge A	Southampton	0	0	10	10
Imperial A	Warwick A	10	10	10	30
Warwick B	LSE B	0	0	10	10
Claremont	Anderson et al.	10	10	10	30
SGV Ventures	Jason et al.	10	0	10	20
ALCU	U[C]SD	10	10	10	30
WashU	Truman	10	0	10	20