Editing Radius of gyration (section)

==Formulation==
Mathematically the [[radius]] of [[gyration]] is the [[root mean square]] distance of the object's parts from either its [[center of mass]] 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 <math>n</math> particles each of mass <math>m</math>. Let <math>r_1, r_2, r_3, \dots , r_n</math> be their perpendicular distances from the axis of rotation. Then, the moment of inertia <math>I</math> of the body about the axis of rotation is
  
:<math>I = m_1 r_1^2 + m_2 r_2^2 + \cdots + m_n r_n^2</math>
:
 
If all the masses are the same (<math>m</math>), then the moment of inertia is <math>I=m(r_1^2+r_2^2+\cdots+r_n^2)</math>.
 
Since <math>m = M/n</math> (<math>M</math> being the total mass of the body), 
 
:<math>I=M(r_1^2+r_2^2+\cdots+r_n^2)/n</math>
 
From the above equations, we have 
 
:<math>MR_g^2=M(r_1^2+r_2^2+\cdots+r_n^2)/n</math>
: 
Radius of gyration is the root mean square distance of particles from axis formula 
 
:<math>R_g^2=(r_1^2+r_2^2+\cdots+r_n^2)/n</math>
: 
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.{{Quote box
| title = [[International Union of Pure and Applied Physics|IUPAP]] definition
| quote = '''Radius of gyration''' (in polymer science)(<math>s</math>, unit: nm or SI unit: m): For a macromolecule composed of <math>n</math> mass elements, of masses <math>m_i</math>, <math>i</math>=1,2,…,<math>n</math>, located at fixed distances <math>s_i</math> from the centre of mass, the radius of gyration is the square-root of the mass average of <math>s_i^2</math> over all mass elements, i.e.,

:<math>
s=\left(\sum_{i=1}^{n} m_i s_i^2 / \sum_{i=1}^{n} m_i \right)^{1/2}
</math>

Note: The mass elements are usually taken as the masses of the skeletal groups constituting the macromolecule, e.g., –CH<sub>2</sub>– in poly(methylene).<ref>{{cite journal |author1=Stepto, R. |author2=Chang, T. |author3=Kratochvíl, P. |author4=Hess, M. |author5=Horie, K. |author6=Sato, T. |author7=Vohlídal, J. |title=Definitions of terms relating to individual macromolecules, macromolecular assemblies, polymer solutions, and amorphous bulk polymers (IUPAC Recommendations 2014). |journal=Pure Appl Chem |date=2015 |volume=87 |issue=1 |page=71 |doi=10.1515/pac-2013-0201 |url=https://www.degruyter.com/downloadpdf/j/pac.2015.87.issue-1/pac-2013-0201/pac-2013-0201.pdf}}</ref>
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