### Radius of Gyration

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

 (B.11)

where is the area moment of inertiaB.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):

where denotes the distance of the differential area element from the axis of gyration.

#### Rectangular Cross-Section

For a rectangular cross-section of height and width , area , the area moment of inertia about the horizontal midline is given by

Similarly, the radius of gyration about a vertical axis passing through the center of the cross-section is . 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.

#### Circular Cross-Section

For a circular cross-section of radius , Eq.(B.11) tells us that the squared radius of gyration about any line passing through the center of the cross-section is given by

Using the elementrary trig identity , we readily derive

The first two terms of this expression contribute zero to the integral from 0 to , while the last term contributes , yielding

Thus, the radius of gyration about any midline of a circular cross-section of radius is

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

 (B.12)

Next Section:
Two Masses Connected by a Rod
Previous Section:
Area Moment of Inertia