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Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis; it is the rotational analogue to mass (which determines an object's resistance to linear acceleration). The moments of inertia of a mass have units of dimension ML 2 ([mass] × [length] 2).
In geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with a ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular , and gives a constructive procedure for finding them.
A body's moment of inertia about a particular axis depends both on the mass and its distribution relative to the axis, increasing with mass & distance from the axis. It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation.
In classical mechanics, the stretch rule (sometimes referred to as Routh's rule) states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution of mass remains unchanged except in the direction parallel to the axis. [1]
where M is the applied torques and I is the inertia matrix. The vector ˙ is the angular acceleration. Again, note that all quantities are defined in the rotating reference frame. In orthogonal principal axes of inertia coordinates the equations become
The tennis racket theorem or intermediate axis theorem, is a kinetic phenomenon of classical mechanics which describes the movement of a rigid body with three distinct principal moments of inertia. It has also been dubbed the Dzhanibekov effect , after Soviet cosmonaut Vladimir Dzhanibekov , who noticed one of the theorem's logical consequences ...
In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial distribution of mass inside a planet or satellite. Since a moment of inertia has dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.
The parallel axis theorem can be used to determine the second moment of area of a rigid body about any axis, given the body's second moment of area about a parallel axis through the body's centroid, the area of the cross section, and the perpendicular distance (d) between the axes. ′ = +