Citation
Math: Student Research Spotlight

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Title:
Math: Student Research Spotlight
Series Title:
Quest
Creator:
SUNY Oswego
Gelnett, Ryan ( Speaker )
Dawson, Brian ( Speaker )
Hanusch, Sarah ( Speaker )
Publication Date:
Copyright Date:
2021

Notes

Abstract:
The Lagrangian and The Problem of N Connected Pendulums by Ryan Gelnett. Abstract: When addressing the problem of $n$ connected pendulums the traditional way is to consider Newton's second law, $F=m\ddot{\bar{x}}$. The problem that arises is the requirement to know all of the forces, gravity and tension, acting in the system to map out the motion of the n-pendulum; this tends to become very convoluted especially when non-conservative forces are taken into effect. Today we analyze this problem and others alike is through use of the Lagrangian which requires scalar quantities of energy; this is possible in a closed system where energy is conserved and air resistance is negligible. The Lagrangian's perspective of handling this problem involves less work and more mathematical elegance when compared to the Newtonian Method. ( , )
Abstract:
Math Literacy: A Primer to Understanding Word Problems by Brian Dawson. Abstract With the implementation of the Common Core Curriculum, math in secondary school has changed from focusing on procedural skills to more abstract conceptual understandings. Students have entered a new realm where being able to justify concepts, make predictions, and reason mathematically is the norm while solving commuting problems with a simple algorithm is no longer a top priority. Past research has indicated that there are four critical components within every mathematical word problem. These four critical components are Functional Vocabulary, Mathematical Vocabulary, Syntax, and the Actual Mathematics Involved. The researcher looks to find trends and frequencies within these four critical components that can lead to a more challenging word problem. This has been accomplished through analyzing word problems present on the New York State Regents examination from the years 2010 - 2019. In this talk, the researcher will present findings pertaining to the Functional Vocabulary words and Syntax of a math word problem and why this is prevalent to mathematics teachers.
Summary:
Session Chair: Sarah Hanusch
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Collected for SUNY Oswego Institutional Repository by the online self-submittal tool. Submitted by Zach Vickery.

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography TheLagrangianandTheProblemof N ConnectedPendulums By:RyanCharlesGelnett At:SUNYOswego April9,2021 By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Derivation:FunctionforMotion Newton'sMethodonthe DoublePendulum TheLagrangianMethod DoublePendulum The N -LinkPendulum AnAnalysis ChaosTheory Fractals Questions&Bibliography By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod Newton'sMethodontheDoublePendulum Newton'sSecondLawin2-D: ~ F = m ~ a = m x i ^ x + m y i ^ y By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod m 1 x 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 1 sin 1 + T 2 sin 2 m 1 y 1 = T 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 1 g m 2 x 2 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 2 sin 2 m 2 y 2 = T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 g x 1 = L 1 sin 1 y 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(L 1 cos 1 x 2 = x 1 + L 2 sin 2 y 2 = y 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(L 2 cos 2 By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod m 1 x 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 1 sin 1 + T 2 sin 2 m 1 y 1 = T 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 1 g m 2 x 2 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 2 sin 2 m 2 y 2 = T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 g x 1 = L 1 sin 1 y 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(L 1 cos 1 x 2 = x 1 + L 2 sin 2 y 2 = y 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(L 2 cos 2 By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod m 1 x 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 1 sin 1 + T 2 sin 2 m 1 y 1 = T 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 1 g m 2 x 2 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 2 sin 2 m 2 y 2 = T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 g x 1 = L 1 sin 1 y 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(L 1 cos 1 x 2 = x 1 + L 2 sin 2 y 2 = y 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(L 2 cos 2 By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod m 1 x 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 1 sin 1 + T 2 sin 2 m 1 y 1 = T 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 1 g m 2 x 2 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(T 2 sin 2 m 2 y 2 = T 2 cos 2 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 g x 1 = L 1 sin 1 y 1 = )]TJ/F66 10.9091 Tf 8.485 0 Td [(L 1 cos 1 x 2 = x 1 + L 2 sin 2 y 2 = y 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(L 2 cos 2 By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod sin 1 m 1 y 1 "+ m 2 y 2 "+ m 2 g + m 1 g = )]TJ/F16 10.9091 Tf 10.303 0 Td [(cos 2 m 1 x 1 "+ m 2 x 2 " sin 2 m 2 y 2 "+ m 2 g = )]TJ/F16 10.9091 Tf 10.303 0 Td [(cos 2 m 2 x 2 " By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod sin 1 m 1 y 1 "+ m 2 y 2 "+ m 2 g + m 1 g = )]TJ/F16 10.9091 Tf 10.303 0 Td [(cos 2 m 1 x 1 "+ m 2 x 2 " sin 2 m 2 y 2 "+ m 2 g = )]TJ/F16 10.9091 Tf 10.303 0 Td [(cos 2 m 2 x 2 " By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod TheLagrangianMethodontheDoublePendulum U = )]TJ/F66 10.9091 Tf 8.484 0 Td [(mghT = 1 2 mv 2 v 2 = L _ 2 PE = )]TJ/F16 10.9091 Tf 8.485 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F56 10.9091 Tf 10.395 0 Td [( 2 ] : L = KE )]TJ/F66 10.9091 Tf 10.909 0 Td [(PE @ L @ )]TJ/F66 10.9091 Tf 13.951 7.38 Td [(d dt @ L @ _ =0 : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod TheLagrangianMethodontheDoublePendulum U = )]TJ/F66 10.9091 Tf 8.484 0 Td [(mghT = 1 2 mv 2 v 2 = L _ 2 PE = )]TJ/F16 10.9091 Tf 8.485 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F56 10.9091 Tf 10.395 0 Td [( 2 ] : L = KE )]TJ/F66 10.9091 Tf 10.909 0 Td [(PE @ L @ )]TJ/F66 10.9091 Tf 13.951 7.38 Td [(d dt @ L @ _ =0 : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod TheLagrangianMethodontheDoublePendulum U = )]TJ/F66 10.9091 Tf 8.484 0 Td [(mghT = 1 2 mv 2 v 2 = L _ 2 PE = )]TJ/F16 10.9091 Tf 8.485 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F66 10.9091 Tf 10.909 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F56 10.9091 Tf 10.395 0 Td [( 2 ] : L = KE )]TJ/F66 10.9091 Tf 10.909 0 Td [(PE @ L @ )]TJ/F66 10.9091 Tf 13.951 7.38 Td [(d dt @ L @ _ =0 : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

PAGE 15

Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod @ L @ _ 1 = m 1 l 2 1 _ 1 + m 2 l 2 1 _ 1 + m 2 l 1 l 2 _ 2 cos 1 )]TJ/F50 8.9664 Tf 9.215 0 Td [( 2 ; d dt @ L @ _ 1 = m 1 + m 2 l 2 1 1 + m 2 l 1 l 2 2 cos 1 )]TJ/F50 8.9664 Tf 9.215 0 Td [( 2 )]TJ/F66 9.4645 Tf 7.168 0 Td [(m 2 l 1 l 2 _ 2 sin 1 )]TJ/F50 8.9664 Tf 9.216 0 Td [( 2 )]TJ/F49 8.9664 Tf 7.316 -5.3 Td [(_ 1 )]TJ/F49 8.9664 Tf 12.193 2.242 Td [(_ 2 ; @ L @ 1 = )]TJ/F66 9.4645 Tf 7.168 0 Td [(l 1 g m 1 + m 2 sin 1 )]TJ/F66 9.4645 Tf 9.216 0 Td [(m 2 l 1 l 2 _ 1 _ 2 sin 1 )]TJ/F50 8.9664 Tf 9.215 0 Td [( 2 : @ L @ _ 2 = m 2 l 2 2 _ 2 + m 2 l 1 l 2 _ 1 cos 1 )]TJ/F50 8.9664 Tf 9.215 0 Td [( 2 ; d dt @ L @ _ 2 = m 2 l 2 2 2 + m 2 l 1 l 2 1 cos 1 )]TJ/F50 8.9664 Tf 9.215 0 Td [( 2 )]TJ/F66 9.4645 Tf 7.176 0 Td [(m 2 l 1 l 2 _ 1 sin 1 )]TJ/F50 8.9664 Tf 9.216 0 Td [( 2 )]TJ/F49 8.9664 Tf 7.317 -5.3 Td [(_ 1 )]TJ/F49 8.9664 Tf 12.194 2.242 Td [(_ 2 ; @ L @ 2 = m 2 l 1 l 2 _ 1 _ 2 sin 1 )]TJ/F50 8.9664 Tf 9.216 0 Td [( 2 )]TJ/F66 9.4645 Tf 9.216 0 Td [(l 2 m 2 g sin 2 : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod Domyowntypeofpendulum???multiplemassesxedtoamass orinitialstationarypositionwithxedanglebetweenthem By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod The N -LinkPendulum N-Link Pendulum N-LinkKEN-LinkPE PE = )]TJ/F46 6.9738 Tf 6.227 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F28 6.9738 Tf 8.041 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F10 6.9738 Tf 8.041 0 Td [( 2 ] : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod The N -LinkPendulum N-Link Pendulum N-LinkKEN-LinkPE PE = )]TJ/F46 6.9738 Tf 6.227 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F28 6.9738 Tf 8.041 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F10 6.9738 Tf 8.041 0 Td [( 2 ] : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

PAGE 19

Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod The N -LinkPendulum N-Link Pendulum N-LinkKEN-LinkPE PE = )]TJ/F46 6.9738 Tf 6.227 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F28 6.9738 Tf 8.041 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F10 6.9738 Tf 8.041 0 Td [( 2 ] : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

PAGE 20

Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod The N -LinkPendulum N-Link Pendulum N-LinkKEN-LinkPE PE = )]TJ/F46 6.9738 Tf 6.227 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F28 6.9738 Tf 8.041 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F10 6.9738 Tf 8.041 0 Td [( 2 ] : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

PAGE 21

Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod The N -LinkPendulum N-Link Pendulum N-LinkKEN-LinkPE PE = )]TJ/F46 6.9738 Tf 6.227 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F28 6.9738 Tf 8.041 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F10 6.9738 Tf 8.041 0 Td [( 2 ] : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

PAGE 22

Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Newton'sMethodontheDoublePendulum TheLagrangianMethod The N -LinkPendulum N-Link Pendulum N-LinkKEN-LinkPE PE = )]TJ/F46 6.9738 Tf 6.227 0 Td [( m 1 + m 2 gL 1 cos 1 )]TJ/F28 6.9738 Tf 8.041 0 Td [(m 2 gL 2 cos 2 ; KE = 1 2 m 1 L 2 1 _ 2 1 + 1 2 m 2 [ L 2 1 _ 2 1 + L 2 2 _ 2 2 +2 L 1 L 2 _ 1 _ 2 cos 1 )]TJ/F10 6.9738 Tf 8.041 0 Td [( 2 ] : By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography ChaosTheory Fractals ChaosTheory By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography ChaosTheory Fractals InitialConditionsforwhentheDuoblePendulumFlips By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

PAGE 25

Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Questions?I EricW.Weisstein. Doublependulum,1996-2007. EricW.Weisstein. Euler-lagrangedierentialequation,1996-2007. EarthquakeProtectionSystems. Triplependulumisolator,2020. ErikNeumann. Doublependulum,2004-2016. GrantSanderson. Thelagrangian,2021. By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Questions?II DavidMoran. `TheLagrangeMethod',IntroductiontoClassicalMechanicswith ProblemsandSolutions . Harvard,2007. AntonioM.Lopes&J.A.TenreiroMachado. Dynamicsofthen-linkpendulum:afractionalperspective. InternationalJournalofControl ,1-9.,90:6,2017. MattGuthrie. Theoriginofthelagrangian,2013. AntonioM.Lopes&J.A.TenreiroMachado. TheN-linkpendulum:Embeddingnonlineardynamicsintothe multidimensionalscalingmethod . Harvard,2016. By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums

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Derivation:FunctionforMotion AnAnalysis Questions&Bibliography Questions?III JeremyS.Heyl. TheDoublePendulumFractal . UniversityofBritishColumbia,2008. Thanksforcoming! By:RyanCharlesGelnett TheLagrangianandTheProblemof N ConnectedPendulums