Pääinertiamomentit

Pääinertiamomentit, or principal moments of inertia, are a fundamental concept in rigid body dynamics. They represent the eigenvalues of the inertia tensor, which is a symmetric, 3x3 matrix that describes how an object's mass is distributed relative to its axis of rotation. For a given rigid body, there are always three principal moments of inertia, denoted as I₁, I₂, and I₃. These moments are associated with three mutually perpendicular axes passing through the body's center of mass, known as the principal axes.

The inertia tensor depends on the object's shape and mass distribution. When expressed in the coordinate system aligned with the principal axes, the inertia tensor becomes a diagonal matrix, with the principal moments of inertia as its diagonal elements. This simplifies many calculations in rotational motion. The principal moments of inertia are invariant under rotation of the coordinate system as long as the new system remains aligned with the principal axes.

The concept of principal moments of inertia is crucial for understanding the rotational behavior of rigid bodies. For example, an object will rotate stably around an axis corresponding to its largest or smallest principal moment of inertia. If the object is spinning around an axis that is not a principal axis, it will experience torques that tend to change its orientation, leading to more complex motion. Calculating these moments is often done through integration over the body's volume, considering the density and position of each infinitesimal mass element.