Radius of gyration
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''Radius of gyration'' or gyradius of a body about the
axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
is defined as the radial distance to a point which would have a
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
the same as the body's actual distribution of mass, if the total mass of the body were concentrated there. Mathematically the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of
gyration In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold cor ...
is the
root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
distance of the object's parts from either its
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
or a given axis, depending on the relevant application. It is actually the perpendicular distance from point mass to the axis of rotation. One can represent a trajectory of a moving point as a body. Then radius of gyration can be used to characterize the typical distance travelled by this point. Suppose a body consists of n particles each of mass m. Let r_1, r_2, r_3, \dots , r_n be their perpendicular distances from the axis of rotation. Then, the moment of inertia I of the body about the axis of rotation is :I = m_1 r_1^2 + m_2 r_2^2 + \cdots + m_n r_n^2 : If all the masses are the same (m), then the moment of inertia is I=m(r_1^2+r_2^2+\cdots+r_n^2). Since m = M/n (M being the total mass of the body), :I=M(r_1^2+r_2^2+\cdots+r_n^2)/n From the above equations, we have :MR_g^2=M(r_1^2+r_2^2+\cdots+r_n^2)/n : Radius of gyration is the root mean square distance of particles from axis formula :R_g^2=(r_1^2+r_2^2+\cdots+r_n^2)/n : Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation. It is also known as a measure of the way in which the mass of a rotating rigid body is distributed about its axis of rotation.


Applications in structural engineering

In
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
, the two-dimensional radius of gyration is used to describe the distribution of cross sectional area in a column around its
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
al axis with the mass of the body. The radius of gyration is given by the following formula: :R_^ = \frac or :R_ = \sqrt Where I is the
second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
and A is the total cross-sectional area. The gyration radius is useful in estimating the stiffness of a column. If the principal moments of the two-dimensional
gyration tensor In physics, the gyration tensor is a tensor that describes the second moments of position of a collection of particles : S_ \ \stackrel\ \frac\sum_^ r_^ r_^ where r_^ is the \mathrm Cartesian coordinate of the position vector \mathbf^ of the \ ...
are not equal, the column will tend to
buckle The buckle or clasp is a device used for fastening two loose ends, with one end attached to it and the other held by a catch in a secure but adjustable manner. Often taken for granted, the invention of the buckle was indispensable in securing tw ...
around the axis with the smaller principal moment. For example, a column with an
elliptical Elliptical may mean: * having the shape of an ellipse, or more broadly, any oval shape ** in botany, having an elliptic leaf shape ** of aircraft wings, having an elliptical planform * characterised by ellipsis (the omission of words), or by conc ...
cross-section will tend to buckle in the direction of the smaller semiaxis. In
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, where continuous bodies of matter are generally the objects of study, the radius of gyration is usually calculated as an integral.


Applications in mechanics

The radius of gyration about a given axis (r_) can be calculated in terms of the
mass moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
I_\text around that axis, and the total mass ''m''; :r_^ = \frac or :r_ = \sqrt I_\text is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, and is not the moment of inertia
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
.


Molecular applications

In polymer physics, the radius of gyration is used to describe the dimensions of a
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A c ...
. The radius of gyration of a particular molecule at a given time is defined as: : R_\mathrm^2 \ \stackrel\ \frac \sum_^ \left, \mathbf_k - \mathbf_\mathrm \^2 where \mathbf_\mathrm is the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
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^2 \ \stackrel\ \frac \sum_ \left, \mathbf_i - \mathbf_j \^2 As a third method, the radius of gyration can also be computed by summing the principal moments of the
gyration tensor In physics, the gyration tensor is a tensor that describes the second moments of position of a collection of particles : S_ \ \stackrel\ \frac\sum_^ r_^ r_^ where r_^ is the \mathrm Cartesian coordinate of the position vector \mathbf^ of the \ ...
. 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 as an ''average'' over time or
ensemble Ensemble may refer to: Art * Architectural ensemble * ''Ensemble'' (album), Kendji Girac 2015 album * Ensemble (band), a project of Olivier Alary * Ensemble cast (drama, comedy) * Ensemble (musical theatre), also known as the chorus * ''En ...
: : R_^2 \ \stackrel\ \frac \left\langle \sum_^ \left, \mathbf_k - \mathbf_\mathrm \^2 \right\rangle where the angular brackets \langle \ldots \rangle denote the
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
. 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 = \frac \ \sqrt\ a Note that although aN represents the
contour length Contour length is a term used in molecular physics. The contour length of a polymer chain (a big molecule consisting of many similar smaller molecules) is its length at maximum physically possible extension Extension, extend or extended may refer ...
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 Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight ''Mw'' of a macromolecule like a polymer or a protein in solution. Measurement of the scattering ...
as well as with small angle neutron- and
x-ray scattering X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystal structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observ ...
. This allows theoretical polymer physicists to check their models against reality. The
hydrodynamic radius The hydrodynamic radius of a macromolecule or colloid particle is R_. The macromolecule or colloid particle is a collection of N subparticles. This is done most commonly for polymers; the subparticles would then be the units of the polymer. R_ ...
is numerically similar, and can be measured with
Dynamic Light Scattering Dynamic light scattering (DLS) is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers in solution. In the scope of DLS, temporal fluctuations are usually analyzed using ...
(DLS).


Derivation of identity

To show that the two definitions of R_^ are identical, we first multiply out the summand in the first definition: : R_^ \ \stackrel\ \frac \sum_^ \left( \mathbf_ - \mathbf_ \right)^ = \frac \sum_^ \left \mathbf_ \cdot \mathbf_ + \mathbf_ \cdot \mathbf_ - 2 \mathbf_ \cdot \mathbf_ \right Carrying out the summation over the last two terms and using the definition of \mathbf_ gives the formula : R_^ \ \stackrel\ -\mathbf_ \cdot \mathbf_ + \frac \sum_^ \left( \mathbf_ \cdot \mathbf_ \right)


Applications in geographical data analysis

In data analysis, the radius of gyration is used to calculate many different statistics including the spread of geographical locations. These locations have recently been collected from social media users to investigate the typical mentions of a user. This can be useful for understanding how a certain group of users on social media use the platform. : R_ = \sqrt


Notes


References

* Grosberg AY and Khokhlov AR. (1994) ''Statistical Physics of Macromolecules'' (translated by Atanov YA), AIP Press. {{ISBN, 1-56396-071-0 * Flory PJ. (1953) ''Principles of Polymer Chemistry'', Cornell University, pp. 428–429 (Appendix C of Chapter X). Solid mechanics Polymer physics