Massenträgheitsmoment

Massenträgheitsmoment, often translated as moment of inertia, is a fundamental concept in physics describing an object's resistance to rotational motion. It is the rotational analog of mass, which measures resistance to linear acceleration. Just as a more massive object requires a greater force to change its linear velocity, a larger mass moment of inertia requires a greater torque to change its angular velocity. The mass moment of inertia depends not only on the total mass of an object but also on how that mass is distributed relative to the axis of rotation. Mass located farther from the axis contributes more significantly to the moment of inertia than mass located closer to it.

Mathematically, the mass moment of inertia is defined as the integral of the square of the distance of each infinitesimal mass element from the axis of rotation, multiplied by that mass element. For a collection of discrete point masses, it is the sum of the product of each mass and the square of its distance from the axis. Different shapes and mass distributions have distinct formulas for their mass moments of inertia about specific axes. For instance, a thin rod rotated about its center has a different moment of inertia than the same rod rotated about one of its ends. This property is crucial in the analysis of rotating systems, from simple pendulums to complex machinery and celestial bodies.