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• To: <seaint(--nospam--at)seaint.org>
• Subject: Re: Radius of Gyration
• From: Christopher Wright <chrisw(--nospam--at)skypoint.com>
• Date: Sat, 20 Jun 2009 14:54:31 -0500

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On Jun 20, 2009, at 1:58 PM, Richard Calvert wrote:

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Does anyone know the origin of this term? Why it refers to a radius and/or what is gyrating? It just seems to specific to be simply obscure… yet I’ve never found anyone who knows. So, anyone on here know?
My god, man--Never found *anyone* who knows? Have you never studied strength of materials or physics?
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There are two definitions, the first having to do with rotational motion, the second applies to beam theory by extension.
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Applying Newton's 2nd law to rotational motion about a fixed axis, it turns out that the torque required
```to produce an angular acceleration is
Torque = moment of inertia x angular acceleration or T = I x alpha.

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The moment of inertia, I, is the sum of the squares of the masses of the particles making up the body x the square of the distance, R, from the fixed axis to each particle. It's expressed as an integral:
```I = integral(R^2dm).

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As an example (look this up in your physics book) the moment of inertia of a thin rod with mass, m, and length, L, about an axis at one end is mL^2/3.
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By definition the radius of gyration, Rg^2 x mass = I or Rg = sqrt(I/ mass). So T = mass x Rg^2 x alpha. Lots of times it's convenient to characterize rotational inertia with the radius of gyration. You can think of a rotating physical object as a particle with the same mass as the body placed at a distance Rg from the axis of rotation.
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Extending the notion to the moment of inertia of a cross-section of a beam or column you define the radius of gyration as I = sqrt(moment of inertia/area). For example the radius of gyration of a rectangular area with base, b, and depth, d, as r = sqrt[(bd^3/12)/bd] = d/sqrt(12). It turns out that Euler buckling and the natural frequencies of beams are expressed conveniently in terms of the ratio of the span to the radius of gyration of the section. You should verify this in your strength of materials book if you ever hope to convert that EIT into a PE any time soon.
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The rotation, BTW, refers to rotation of a particle about a fixed axis in regard to angular motion. I supose you can think of the rotation in respect to beam and column design as the rotation of a member cross section when a beam bends.
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This is really basic stuff you should have learned as a sophomore.

Christopher Wright P.E. |"They couldn't hit an elephant at
chrisw(--nospam--at)skypoint.com   | this distance" (last words of Gen.
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.......................................| John Sedgwick, Spotsylvania 1864)
```http://www.skypoint.com/members/chrisw/

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