Radius of Gyration - What is Radius of Gyration, Derivation, Formula

The radius of gyration is a very important concept in physics and engineering. It is used to describe how well the mass of an object is distributed about an axis of rotation. It is, so to speak, the distance from the axis where the whole body mass can be assumed to be concentrated to impose the same moment of inertia. Like many other parameters, it also finds wide application in mechanics, structural analysis, and material science in analyzing the stability and dynamics of rotating bodies around an axis. The radius of gyration thus helps determine the design of beams, calculate stress in structures, study molecular dynamics, and all such applications where the effects of rotating masses must be considered.

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Radius of Gyration Definition

The meaning of radius of gyration or gyration is the distance from an axis at which the mass of a body could also be assumed to be concentrated and at which the instant of inertia is going to be adequate to the instant of inertia of the particular mass about the axis, adequate to the root of the quotient of the instant of inertia and therefore the mass.

The moment of inertia of a body about an axis is usually represented using the radius of gyration. Now, the Radius of gyration is defined as the distance axis of rotation to some extent where the entire body is meant to concentrate. The radius of gyration is a constant quantity.

Symbol of radius of gyration or it is denoted by k.

SI unit: meter

CGS unit: cm

Dimensional analysis: L

Radius of Gyration Formula

The formula of moment inertia in terms of the radius of gyration is given as follows:

$$
I=m k^2(1)
$$

where $l$ is the moment of inertia and $m$ is the mass of the body
Accordingly, the radius of gyration is given as follows

$$
k=\sqrt{\frac{I}{m}}
$$
The unit of the radius of gyration is mm. By knowing the radius of gyration, one can find the moment of inertia of any complex body equation (1) without any hassle.

Consider a body having an $n$ number of particles each having a mass of $m$. Let the perpendicular distance from the axis of rotation be given by $r_1, r_2, r_3, \ldots, r_n$. We know that the moment of inertia in terms of radius of gyration is given by the equation (1). Substituting the values in the equation, we get the moment of inertia of the body as follows

$$
I=m_1 r_1^2+m_2 r_2^2+m_3 r_3^2+\ldots+m_n r_n^2
$$
If all the particles have the same mass, then equation (3) becomes

$$
\begin{aligned}
& I=m\left(r_1^2+r_2^2+r_3^2+\ldots r_n^2\right) \\
& =\frac{m n\left(r_1^2+r_2^2+r_3^2+\ldots+r_n^2\right)}{n}
\end{aligned}
$$
We can write $m n$ as $M$, which signifies the total mass of the body. Now the equation becomes

$$
I=M \frac{\left(r_1^2+r_2^2+r_3^2+\ldots+r_n^2\right)}{n}
$$

We can write $m n$ as $M$, which signifies the total mass of the body. Now the equation becomes

$$
I=M \frac{\left(r_1^2+r_3^2+r_3^2+\ldots+r_n^2\right)}{n}
$$
From equation (4), we get

$$
M K^2=M\left(\frac{r_1^2+r_2^2+r_3^2+\ldots+r_n^2}{n}\right)
$$

or, $K=\sqrt{\frac{r_1^2+r_2^2+r_3^2+\ldots+r_n^2}{n}}$
From the above equation, we can infer that the radius of gyration can also be defined as the root-mean-square distance of various particles of the body from the axis of rotation.

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Radius of Gyration Applications

The applications of the radius of gyration are mentioned as follows:

• It is used to determine the pressure exerted by a body.
• It is also used to estimate the strength of a body.
• It is also used to estimate the cross-section area of a body.
• It is used in structural engineering.
• It is also used in molecular physics to determine the dimensions of a polymer chain.