ParallelachsenTheorem

ParallelachsenTheorem, also known as Steiner's theorem, is a principle in classical mechanics that relates the moment of inertia of a rigid body about different parallel axes. The theorem states that the moment of inertia I about any axis parallel to an axis through the body's center of mass equals the sum of the moment of inertia about the parallel axis through the center of mass, I_cm, and the product of the body's mass M and the square of the perpendicular distance d between the two axes: I = I_cm + M d^2. The distance d is measured perpendicularly between the axes.

The relation holds for rigid bodies with a fixed axis of rotation, and for axes that are parallel. It is particularly useful because I_cm depends only on the body's shape and mass distribution, while d accounts for the shift of the axis.

Derivation idea: by expressing the position of mass elements relative to the center of mass and to the shift vector, one finds that the squared distance to the shifted axis equals the sum of the squared distance to the center-of-mass axis plus the square of the shift, leading to the additive term M d^2 when integrating over the body.

Common applications include calculating moments of inertia for rods, disks, and plates about axes other than those through the center of mass, such as an axis through an end or a fixed pivot. For example, a thin rod of length L has I_cm = (1/12) M L^2 about an axis perpendicular to the rod through its center; about a parallel axis through one end, I = I_cm + M (L/2)^2 = (1/3) M L^2.

Limitations: the theorem assumes a rigid body and fixed axes. In three dimensions, the complete description of inertia is given by the inertia tensor, of which the ParallelachsenTheorem is a scalar special case.