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Rectangular Cross-Section

For a rectangular cross-section of height $ h$ and width $ w$ , area $ S=hw$ , the area moment of inertia about the horizontal midline is given by

$\displaystyle I_w
= w\int_{-h/2}^{h/2} y^2 dy
= w\left.\frac{1}{3}y^3\right\vert _{-h/2}^{h/2}
= \frac{Sh^2}{12}.

The radius of gyration about this axis is then

$\displaystyle R_g = \sqrt{\frac{I_w}{S}} = \sqrt{\frac{h^2}{12}} = \frac{h}{2\sqrt{3}}.

Similarly, the radius of gyration about a vertical axis passing through the center of the cross-section is $ R_g=w/(2\sqrt{3})$ .

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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University