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.

我们推导出了一个由三个粒子组成的刚体的惯性张量的一般公式，学生可以用它来学习惯性张量的基本性质，无需微积分。惯性张量算子是通过使用狄拉克的括号符号构造的，以协变地获得任意参考系中的惯性张量。当旋转轴通过质心时，计算主轴和转动惯量。这些公式用粒子的相对位移表示，这些位移是通过引入拉格朗日未定乘子确定的。这是一个启发式的例子，类似于规范场理论中向拉格朗日密度添加规范固定项。我们确认主矩满足平面薄片的垂直轴定理。两个特殊案例被视为教学示例。一种是类似水分子的系统，其中一个粒子被放置在两个相同粒子的垂直平分线上。另一种情况是质心与三角形的中心重合，三角形的顶点位于粒子处。后一个例子关于法线轴的主力矩非常简单，并且与乘积成正比 'abc ' 的三个相对距离。我们希望这个新公式可以用于普通物理或本科经典力学的实际实验室课程。