parallelaxes

Parallel axes refers to a relation in rotational dynamics between moments of inertia about different, parallel axes. It is most commonly encapsulated in the parallel axis theorem, also known as Steiner’s theorem. The theorem states that the moment of inertia I of a rigid body about any axis parallel to an axis through the body's center of mass is equal to the moment of inertia I_cm about the latter axis plus 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 perpendicular to both axes.

Key elements of the theorem include that the two axes are parallel, the distance between them is orthogonal to the axes, and I_cm is the moment of inertia about the axis through the center of mass with the same orientation as the parallel axis. The theorem applies to rigid bodies in rotation and is widely used in engineering, physics, and biomechanics to compute moments of inertia for complex mounting or support configurations.

Examples often cited include a uniform disk or wheel rotated about an axis parallel to the central axis but offset by a distance d. For a uniform disk, the central axis moment I_cm = (1/2) m R^2; about a parallel axis at distance d, I = (1/2) m R^2 + m d^2.

Beyond the parallel axis theorem, the concept is related to the inertia tensor, which generalizes moments of inertia for arbitrary orientations. The theorem is a fundamental tool for analyzing rotational dynamics, stability, and energy in systems with offset axes. See also Steiner’s theorem, moment of inertia, and inertia tensor.