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The radius of gyration is a defined measure for the dimension of an object, a surface, or an ensemble of points. It is calculated as the mean square distance of the objects parts' from its center of gravity.

## Contents

Depending on the field, a series of different formulae can be applied to calculate the radius of gyration.

In polymer physics the radius of gyration can be calculated as the sum over the N individual monomers of the polymer: Polymer physics is the field of physics associated to the study of polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively. ... In chemistry, a monomer (from Greek mono one and meros part) is a small molecule that may become chemically bonded to other monomers to form a polymer. ... A polymer is a substance composed of molecules with large molecular mass composed of repeating structural units, or monomers, connected by covalent chemical bonds. ...

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

where $mathbf{r}_{mean}$ is the mean position of the particles. As detailed below, the radius of gyration is also proportional to the root mean square distance between the particles In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ...

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

As a third method, the radius of gyration can also be computed by summing the principal moments of the gyration tensor. The gyration tensor is a tensor that describes the second moments of position of a collection of particles where is the Cartesian component of the position of the particle and which has been defined such that In the continuum limit, where represents the number density of particles at position . ...

In engineering, where people deal with continous bodies of matter, the radius of gyration is more easily calculated as an integral. Engineering is the design, analysis, and/or construction of works for practical purposes. ...

## Molecular applications

In polymer physics, the radius of gyration is used to describe the dimensions of a polymer chain. Since the chain conformations of a polymer sample are quasi infinite and subject to constant change over time, the "radius of gyration" measured given in the context of polymer physics must be understood as a mean over all polymer molecules of the sample and over time. This allows calculation of same using an average location of monomers over time or ensemble: Polymer physics is the field of physics associated to the study of polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively. ... A polymer is a substance composed of molecules with large molecular mass composed of repeating structural units, or monomers, connected by covalent chemical bonds. ... An ideal chain (or freely-jointed chain) is the simplest model to describe a polymer. ... Conformation generally means structural arrangement. ...

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

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

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 xray scattering. This allows theoretical polymer physists to check their models against reality. The hydrodynamic radius is numerically similar, and can be measured with Size exclusion chromatography. Static light scattering is a technique in physical chemistry that uses the intensity traces at a number of angles to derive information about the radius of gyration and molecular mass of the polymer or polymer complexes, for example, micellar formation (1-5). ... This article or section is in need of attention from an expert on the subject. ... Schematic representation of SAS experiment Small-angle scattering (SAS) of X-rays (SAXS) and neutrons (SANS) is a fundamental method for structure analysis of materials. ... The hydrodynamic radius of a collection of particles (typically belonging to a polymer) is defined as where is the distance between particles and , and where the angular brackets represent an ensemble average. ... Equipment for running size exclusion chromatography. ...

## 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 Taipei 101, the worlds tallest building as of 2004. ... A 3-D view of a beverage-can stove with a cross section in yellow. ...

$R_{g}^{2} = frac{I}{A}$

or

$R_g = sqrt{ frac {I} {A} }$

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. The second moment of area, also known as the area moment of inertia and less precisely as the moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ... The gyration tensor is a tensor that describes the second moments of position of a collection of particles where is the Cartesian component of the position of the particle and which has been defined such that In the continuum limit, where represents the number density of particles at position . ... In engineering, buckling is a failure mode characterised by a sudden failure of a structural member that is subjected to high compressive stresses where the actual compressive stresses at failure are smaller than the ultimate compressive stresses that the material is capable of withstanding. ... For other uses, see Ellipse (disambiguation). ...

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.

## Applications in mechanics

The radius of gyration can be computed in terms of the second moment of inertia I and the total mass M: The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending. ...

$R_{g}^{2} = frac{I}{M}$

or

$R_g = sqrt{ frac {I} {M} }.$

It should be noted that I is a scalar, and is not the moment of inertia tensor. In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

## Derivation of identity

To show that the two definitions of $R_{g}^{2}$ are identical, we first multiply out the summand in the first definition

$R_{g}^{2} stackrel{mathrm{def}}{=} frac{1}{N} sum_{k=1}^{N} left( mathbf{r}_{k} - mathbf{r}_{mean} right)^{2} = frac{1}{N} sum_{k=1}^{N} left[ mathbf{r}_{k} cdot mathbf{r}_{k} + mathbf{r}_{mean} cdot mathbf{r}_{mean} - 2 mathbf{r}_{k} cdot mathbf{r}_{mean} right].$

Carrying out the summation over the last two terms and using the definition of $mathbf{r}_{mean}$ gives the formula

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

Similarly, we may multiply out the summand of the second definition

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

which can be written

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

Substituting the definition of $mathbf{r}_{mean}$ and carrying out one of the summations in the final term (and renaming the remaining summation index to k) yields

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

proving the identity of the two definitions.

## References

• Grosberg AY and Khokhlov AR. (1994) Statistical Physics of Macromolecules (translated by Atanov YA), AIP Press. ISBN 1563960710
• Flory PJ. (1953) Principles of Polymer Chemistry, Cornell University, pp. 428-429 (Appendix C o Chapter X).

Results from FactBites:

 Radius of gyration - Wikipedia, the free encyclopedia (427 words) The radius of gyration (possibly multiplied by a constant factor) is a useful estimate of the size of a molecule. 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 gyration radius is useful in estimating the stiffness of a beam.
 Kin 241b (1797 words) The radius of gyration is used to calculate the rotational inertia of a body whereas the radius of rotation is used to calculate the linear velocity of a rotating point. The radius of gyration is defined as a measure of the average distribution of the mass of an implement. It is part of the implement's rotational inertia, thus, the greater the radius of gyration, the greater the inertia and the smaller the angular velocity generated by a given torque.
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