We employ the barycentric coordinate system to evaluate the inertia tensor of an arbitrary triangular plate of uniform mass distribution. We find that the physical quantities involving the computation are expressed in terms of a single master integral over barycentric coordinates. To expedite the computation in the barycentric coordinates, we employ Lagrange undetermined multipliers. The moment of inertia is expressed in terms of mass, barycentric coordinates of the pivot, and side lengths. The expression is unique and the most compact in comparison with popular expressions that are commonly used in the field of mechanical engineering. A master integral that is necessary to compute the integral over the triangle in the barycentric coordinate system and derivations of the barycentric coordinates of common triangle centers are provided in appendices. We expect that the barycentric coordinates are particularly efficient in computing physical quantities like the electrostatic potential of a triangular charge distribution. We also illustrate a practical experimental design that can be immediately applied to general-physics experiments.
Bibliographical noteFunding Information:
As members of the Korea Pragmatist Organization for Physics Education (KPOP), the authors thank the remaining members of KPOP for useful discussions. The work is supported in part by the National Research Foundation of Korea (NRF) under the BK21 FOUR program at Korea University, Initiative for science frontiers on upcoming challenges. This work is also supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) under Contract Nos. NRF-2020R1A2C3009918 (U-R.K. and J.L.), NRF-2017R1E1A1A01074699 (D.W.J. and J.L.), NRF-2018R1D1A1B07047812 (D.W.J.). The work of C.Y. is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R1I1A1A01073770).
© 2021, The Korean Physical Society.
- Barycentric Coordinates
- Classical Mechanics
- Inertia Tensor
- Lagrange Multipliers
ASJC Scopus subject areas
- Physics and Astronomy(all)