For a planar distribution of mass rotating about some axis in the plane of the mass, the radius of gyration is the distance from the axis that all mass can be concentrated to obtain the same mass moment of inertia. Thus, the radius of gyration is the ``equivalent distance'' of the mass from the axis of rotation. In this context, gyration can be defined as rotation of a planar region about some axis lying in the plane.
For a bar cross-section with area , the radius of gyration is given by
where is the area moment of inertia (§B.4.8) of the cross-section about a given axis of rotation lying in the plane of the cross-section (usually passing through its centroid):
For a rectangular cross-section of height and width , area , the area moment of inertia about the horizontal midline is given by
The radius of gyration can be thought of as the ``effective radius'' of the mass distribution with respect to its inertial response to rotation (``gyration'') about the chosen axis.
Most cross-sectional shapes (e.g., rectangular), have at least two radii of gyration. A circular cross-section has only one, and its radius of gyration is equal to half its radius, as shown in the next section.
Using the elementrary trig identity , we readily derive
For a circular tube in which the mass of the cross-section lies within a circular annulus having inner radius and outer radius , the radius of gyration is given by
Two Masses Connected by a Rod
Area Moment of Inertia