Radius of gyration of a body about an axis of rotation is defined as radial distance of a point ,from the axis of rotation at which, if whole mass of any body is assumed to be concentrated, its moment of inertia about the given axis would be same as with its actual distribution of mass. It is denoted by{displaystyle R_{g}}.
Radius of gyration is generally defined as the distance from the axis of rotation to a point where total mass of any body is supposed to be concentrated, so that the moment of inertia about the axis may remain same. Simply, gyration is the distribution of the components of an object.
Mathematically the radius of gyration is root mean square distance of the object’s part from either its centre of mass or a given axis, depending on the relevant application. It is actually perpendicular distance from point mass to the axis of rotation.
Suppose a body consists of n{displaystyle n} particles each of mass {displaystyle m}m. Let {displaystyle r_{1},r_{2},r_{3},dots ,r_{n}} be their perpendicular distance from the axis of rotation. Then, the moment of inertia {displaystyle I}І of the body about the axis of rotation can be-
{displaystyle I=m_{1}r_{1}^{2}+m_{2}r_{2}^{2}+cdots +m_{n}r_{n}^{2}}
If all the masses are same as ({displaystyle m}m), then the moment of inertia is {displaystyle I=m(r_{1}^{2}+r_{2}^{2}+cdots +r_{n}^{2})}.
Since {displaystyle m=M/n}m=M/n ({displaystyle M}M being the total mass of the body),
{displaystyle I=M(r_{1}^{2}+r_{2}^{2}+cdots +r_{n}^{2})/n}
From the above equations, we have-
{displaystyle 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 the particles from axis formula, we have-
{displaystyle 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.
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