Inertia tensor of a triangle in barycentric coordinates

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.

Automated summary

This study evaluates the inertia tensor of a triangular plate of uniform mass distribution using the barycentric coordinate system, expressing physical quantities in terms of a single master integral and employing Lagrange undetermined multipliers for faster computation. The moment of inertia is uniquely expressed in terms of mass, barycentric coordinates of the pivot, and side lengths, with the most compact expression in comparison to commonly used ones in mechanical engineering. Appendices provide necessary master integrals for computing integrals over triangles in the barycentric coordinate system and derivations of barycentric coordinates of common triangle centers. The barycentric coordinates are expected to be efficient in computing physical quantities like the electrostatic potential of a triangular charge distribution and can be applied to general-physics experiments through practical experimental designs.