Three-body inertia tensor.
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| Title: | Three-body inertia tensor. |
|---|---|
| Authors: | Ee, June-Haak (AUTHOR) chodigi@gmail.com, Jung, Dong-Won (AUTHOR) jungil@korea.ac.kr, Kim, U-Rae (AUTHOR), Kim, Dohyun (AUTHOR), Lee, Jungil (AUTHOR) |
| Source: | European Journal of Physics. Sep2021, Vol. 42 Issue 5, p1-18. 18p. |
| Subjects: | Center of mass, Gauge field theory, Classical mechanics, Calculus of tensors, Moments of inertia |
| Abstract: | We derive a general formula for the inertia tensor of a rigid body consisting of three particles with which students can learn basic properties of the inertia tensor without calculus. The inertia-tensor operator is constructed by employing the Dirac's bra-ket notation to obtain the inertia tensor in an arbitrary frame of reference covariantly. The principal axes and moments of inertia are computed when the axis of rotation passes the center of mass. The formulas are expressed in terms of the relative displacements of particles that are determined by introducing Lagrange's undetermined multipliers. This is a heuristic example analogous to the addition of a gauge-fixing term to the Lagrangian density in gauge field theories. We confirm that the principal moments satisfy the perpendicular-axis theorem of planar lamina. Two special cases are considered as pedagogical examples. One is a water-molecule-like system in which a particle is placed on the vertical bisector of two identical particles. The other is the case in which the center of mass coincides with the incenter of the triangle whose vertices are placed at the particles. The principal moment of the latter example about the normal axis is remarkably simple and proportional to the product 'abc' of the three relative distances. We expect that this new formula can be used in actual laboratory classes for general physics or undergraduate classical mechanics. [ABSTRACT FROM AUTHOR] |
| Copyright of European Journal of Physics is the property of IOP Publishing and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 152289743 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Three-body inertia tensor. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Ee%2C+June-Haak%22">Ee, June-Haak</searchLink> (AUTHOR)<i> chodigi@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Jung%2C+Dong-Won%22">Jung, Dong-Won</searchLink> (AUTHOR)<i> jungil@korea.ac.kr</i><br /><searchLink fieldCode="AR" term="%22Kim%2C+U-Rae%22">Kim, U-Rae</searchLink> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kim%2C+Dohyun%22">Kim, Dohyun</searchLink> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Lee%2C+Jungil%22">Lee, Jungil</searchLink> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22European+Journal+of+Physics%22">European Journal of Physics</searchLink>. Sep2021, Vol. 42 Issue 5, p1-18. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Center+of+mass%22">Center of mass</searchLink><br /><searchLink fieldCode="DE" term="%22Gauge+field+theory%22">Gauge field theory</searchLink><br /><searchLink fieldCode="DE" term="%22Classical+mechanics%22">Classical mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Calculus+of+tensors%22">Calculus of tensors</searchLink><br /><searchLink fieldCode="DE" term="%22Moments+of+inertia%22">Moments of inertia</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We derive a general formula for the inertia tensor of a rigid body consisting of three particles with which students can learn basic properties of the inertia tensor without calculus. The inertia-tensor operator is constructed by employing the Dirac's bra-ket notation to obtain the inertia tensor in an arbitrary frame of reference covariantly. The principal axes and moments of inertia are computed when the axis of rotation passes the center of mass. The formulas are expressed in terms of the relative displacements of particles that are determined by introducing Lagrange's undetermined multipliers. This is a heuristic example analogous to the addition of a gauge-fixing term to the Lagrangian density in gauge field theories. We confirm that the principal moments satisfy the perpendicular-axis theorem of planar lamina. Two special cases are considered as pedagogical examples. One is a water-molecule-like system in which a particle is placed on the vertical bisector of two identical particles. The other is the case in which the center of mass coincides with the incenter of the triangle whose vertices are placed at the particles. The principal moment of the latter example about the normal axis is remarkably simple and proportional to the product 'abc' of the three relative distances. We expect that this new formula can be used in actual laboratory classes for general physics or undergraduate classical mechanics. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of European Journal of Physics is the property of IOP Publishing and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1088/1361-6404/abf8c6 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 1 Subjects: – SubjectFull: Center of mass Type: general – SubjectFull: Gauge field theory Type: general – SubjectFull: Classical mechanics Type: general – SubjectFull: Calculus of tensors Type: general – SubjectFull: Moments of inertia Type: general Titles: – TitleFull: Three-body inertia tensor. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Ee, June-Haak – PersonEntity: Name: NameFull: Jung, Dong-Won – PersonEntity: Name: NameFull: Kim, U-Rae – PersonEntity: Name: NameFull: Kim, Dohyun – PersonEntity: Name: NameFull: Lee, Jungil IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2021 Type: published Y: 2021 Identifiers: – Type: issn-print Value: 01430807 Numbering: – Type: volume Value: 42 – Type: issue Value: 5 Titles: – TitleFull: European Journal of Physics Type: main |
| ResultId | 1 |
