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Radius of gyration is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis.

Applications in structural engineering

In structural engineering, the two-dimensional radius of gyration is used to describe the distribution of cross sectional area in a beam around its centroidal axis. The radius of gyration is given by the following formula

$R_{\mathrm{g}}^{2} = \frac{I}{A},$

or

$R_{\mathrm{g}} = \sqrt{ \frac {I} {A} },$

where I is the second moment of area and A is the total cross-sectional area. The gyration radius is useful in estimating the stiffness of a beam. However, if the principal moments of the two-dimensional gyration tensor are not equal, the beam will tend to buckle around the axis with the smaller principal moment. For example, a beam with an elliptical cross-section will tend to buckle around the axis with the smaller semiaxis.

It also can be referred to as the radial distance from a given axis at which the mass of a body could be concentrated without altering the rotational inertia of the body about that axis.

In engineering, where people deal with continuous bodies of matter, the radius of gyration is more usually calculated as an integral.

[edit] Applications in mechanics

The radius of gyration about a given axis can be computed in terms of the moment of inertia I around that axis, and the total mass M;

$R_{\mathrm{g}}^{2} = \frac{I}{M},$

or

$R_{\mathrm{g}} = \sqrt{ \frac {I} {M} }.$

It should be noted that I is a scalar, and is not the moment of inertia tensor. ^[1]

[edit] Molecular applications

In polymer physics, the radius of gyration is used to describe the dimensions of a polymer chain. The radius of gyration of a particular molecule at a given time is defined as:

$ R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2}, $

where $\mathbf{r}_{\mathrm{mean}}$ is the mean position of the monomers. As detailed below, the radius of gyration is also proportional to the root mean square distance between the monomers:

$ R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2N^{2}} \sum_{i,j} \left( \mathbf{r}_{i} - \mathbf{r}_{j} \right)^{2}. $

As a third method, the radius of gyration can also be computed by summing the principal moments of the gyration tensor.

Since the chain conformations of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time. That is, the radius of gyration which is measured is an average over time or ensemble:

$ R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \langle \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} \rangle, $

where the angular brackets $\langle \ldots \rangle$ denote the ensemble average.

An entropically governed polymer chain (i.e. in so called theta conditions) follows a random walk in three dimensions. The radius of gyration for this case is given by

$R_{\mathrm{g}} = \frac{1}{ \sqrt 6\ } \ \sqrt N\ a.$

Note that, although $a N$ represents the contour length of the polymer, a is strongly dependent of polymer stiffness, and can vary over orders of magnitude. N is reduced accordingly.

One reason that the radius of gyration is an interesting property is that it can be determined experimentally with static light scattering as well as with small angle neutron- and x-ray scattering. This allows theoretical polymer physicists to check their models against reality. The hydrodynamic radius is numerically similar, and can be measured with Size exclusion chromatography.

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A picture taken by J.S. Henrardi on June 15, 1990 during the Belgian UFO wave.

^[1]

^{The most generic definition of an Unidentified Flying Object, or UFO, is any flying object or phenomenon that cannot be identified by the observer. Various studies show that after investigation, the majority of UFOs are usually identified, and are relabeled IFOs or}^{Identified Flying Objects}^{. Therefore, some stricter definitions reserve the label "UFO" for only those instances where the objects remain unexplained after a proper investigation.}^{^[2] The percentages of IFOs vs. UFOs varies with the researchers, study, and case sample, ranging from only 5% to 10% being UFOs, according to The J. Allen Hynek Center for UFO Studies, [6] to 20% to 30% being UFOs according to earlier U.S. Air Force statistics or the later Condon Committee.^[3]

UFOs have been spotted in many different places around the world. Reports of unusual aerial phenomena date back to ancient times,^[4]^[5]^[6]^[7] but modern reports and first official investigations began during World War II with sightings of so-called foo fighters by Alliedairplane crews and in 1946 with widespread sightings of European "ghost rockets." UFO reports became even more common after the first widely publicized United States UFO sighting, by private pilot Kenneth Arnold in the summer of 1947. Many tens of thousands of UFO reports have since been made worldwide. ^[8]}

Major civilian UFO groups in the U.S that have conducted extensive investigations were/are NICAP, APRO, MUFON, and CUFOS.

French space agency CNES releases UFO files

Wikinews

has related news:

French Space Agency CNES releases UFO files

Pilots spot 'UFOs' near the Channel Islands

In March 2007, the French Centre National d'Études Spatiales (CNES) published an archive of UFO sightings and other phenomena

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