### 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

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):

#### 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

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

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

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Two Masses Connected by a Rod

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Area Moment of Inertia