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

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

Vishal kumarUpdated on 02 Jul 2025, 05:02 PM IST

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.

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

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

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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.

Frequently Asked Questions (FAQs)

Q: What's the significance of the radius of gyration in earthquake engineering?
A:
In earthquake engineering, the radius of gyration is used to analyze the torsional behavior of buildings during seismic events. It helps engineers understand how the distribution of mass in a structure affects its
Q: How does the radius of gyration relate to the concept of reduced mass in two-body problems?
A:
While the radius of gyration and reduced mass are distinct concepts, they both deal with simplifying complex mass distributions. In two-body problems, the reduced mass allows us to treat the system as a single body with an effective mass. Similarly, the radius of gyration allows us to treat a distributed mass as if it were concentrated at a single point for rotational calculations.
Q: Can the radius of gyration be used to optimize the design of wind turbines?
A:
Yes, the radius of gyration is an important consideration in wind turbine design. A larger radius of gyration for the turbine blades can increase the moment of inertia, which affects the turbine's ability to maintain consistent rotation in variable wind conditions. However, this must be balanced against other factors like material strength and manufacturing constraints.
Q: How does the concept of radius of gyration apply to fluid dynamics?
A:
In fluid dynamics, the radius of gyration is sometimes used to describe the distribution of mass in a fluid system. It's particularly useful in analyzing the rotational motion of fluids, such as in vortices or rotating tanks. The concept helps in understanding how the distribution of fluid mass affects the system's rotational inertia and stability.
Q: What's the relationship between radius of gyration and angular acceleration?
A:
The radius of gyration affects angular acceleration through its relationship with moment of inertia. For a given torque, a larger radius of gyration results in a smaller angular acceleration. This is because α = τ / I, where α is angular acceleration, τ is torque, and I is moment of inertia. Since I = Mk², where k is the radius of gyration, a larger k leads to a smaller α for a given τ and M.
Q: How is the radius of gyration related to the parallel axis theorem?
A:
The parallel axis theorem relates the moment of inertia about any axis to the moment of inertia about a parallel axis through the center of mass. It states that I = I_cm + Md², where I_cm is the moment of inertia about the center of mass, M is the total mass, and d is the distance between the axes. This theorem allows us to calculate how the radius of gyration changes when we consider different axes of rotation.
Q: Can the radius of gyration be used to compare the rotational behavior of different objects?
A:
Yes, the radius of gyration is an excellent tool for comparing the rotational behavior of different objects. Objects with similar masses but different radii of gyration will behave differently under rotation. Those with larger radii of gyration will have greater rotational inertia and be more resistant to changes in their rotational motion.
Q: How does the concept of radius of gyration apply to planetary motion?
A:
In planetary motion, the radius of gyration helps describe the distribution of mass within a planet or star. It's used in calculations involving rotational dynamics, such as determining a planet's moment of inertia. This is important for understanding phenomena like precession, nutation, and tidal interactions between celestial bodies.
Q: How does the shape of a molecule affect its radius of gyration?
A:
The shape of a molecule significantly affects its radius of gyration. Linear molecules tend to have larger radii of gyration compared to compact, spherical molecules of the same mass. In polymer science, the radius of gyration is used to characterize the size of polymer chains, with extended conformations having larger radii than coiled ones.
Q: What's the significance of the radius of gyration in sports equipment design?
A:
In sports equipment design, the radius of gyration is crucial for optimizing performance. For example, in tennis rackets or baseball bats, a larger radius of gyration can increase the "sweet spot" and reduce vibration. In golf clubs, it affects the club's moment of inertia, influencing the club's resistance to twisting during off-center hits.