Radius of Gyration

Radius of Gyration, or the radius of a body, is always centered on its rotational axis. It is a geometric characteristic of a rigid body and is described as the distance between the axis of the body to the point where the body’s moment of inertia is the same as the body’s total moment of inertia. The S.I. unit of the gyration radius is a meter denoted by ‘m’.

In this article, we will discuss about Radius of Gyration, its derivation, formulas, and radius of gyration of a thin rod, circle, and disc along with some applications and significance of radius of gyration.

What is the Radius of Gyration?

Radius of Gyration is the imaginary distance from the centroid of the given axis to a point where the whole mass of a body is supposed to be concentrated. Its moment of inertia about the given axis is equal to the actual distribution of mass. It is represented by ‘k’.

Mathematically, it can be described as the root mean square distance from the given axis,

k = √r²1 + r²2 + r²3 + …….. + r²n/n

Radius of Gyration Definition

Radius of Gyration is defined as the radial distance between the axis of rotation to the point where whole mass of a body is assumed to be concentrated. In simple words, it is the distribution of the components of the object.

Radius of Gyration Formula

The radius of gyration can be expressed numerically as the root mean square distance of particles from the axis of rotation. The Formula to calculate the radius of gyration can be derived as follows:

The moment of inertia of a body with mass (m) can be given by:

 I = mk² ….(1)

Here, k denotes the radius of gyration.

So, the formula for radius of gyration can be given by:

K = √I/m ….(2)

Note: Equation (1) can be used to determine the moment of inertia of any rigid body by knowing its radius of gyration.

Now consider a body that consists of n particles, each with mass m, and the perpendicular distance of particles from the axis of rotation be r₁, r₂, r₃,…, r_n.

Substituting these values in equation 1 we get:

I = m₁r₁² + m₂r₂² + m₃r₃² + ……. + m_nr_n² ……(3)

But we have to consider the mass of each particle as ‘m’ so our equation becomes:

I = m(r₁²+ r₂² + r₃² + …….. + r_n²)

Multiply the right-hand side by n/n we get,

 I = mn(r₁²+ r₂² + r₃² + …….. + r_n²)/n

Now, we can write mn as the M, which represents the body’s overall mass.

So, the equation will become:

I = M(r₁²+ r₂² + r₃² + …….. + r_n²)/n  ……..(4)

Substituting I = mk² in equation (4) we get,

Mk² = M(r²1 + r²2 + r²3 + …….. + r²n)/n

⇒ k² = (r²1 + r²2 + r²3 + …….. + r²n)/n

⇒ k = √r²1 + r²2 + r²3 + …….. + r²n/n

From the above equation, we can say that the root-mean-square distance of the various body particles is referred as the radius of gyration of that body.

Unit of Radius of Gyration

The radius of gyration represents a distance, so the SI unit of the radius of gyration is meter denoted by m. The dimensional formula of the radius of gyration is M⁰ L¹ T⁰.

Factors Affecting Radius of Gyration

The radius of gyration depends on three key factors:

Shape and size of the object: Different shapes have different mass distributions, causing changes in the mass elements’ average distance from the rotational axis. E.g., Even though their overall sizes may be comparable, a sphere’s radius of gyration is smaller than that of a long, thin rod with the same mass.

Configuration and Position of the axis of rotation: The axis that is chosen significantly affects the calculated radius of gyration. E.g., A cylinder’s value will vary depending on whether it is rotated around its short or long axis.

Mass distribution within the object: Uneven distribution of mass will influence the radius of gyration. The radius of gyration of an object will be greater for denser or heavier parts that are positioned farther from the axis of rotation than for objects with a more uniform mass distribution.

Radius of Gyration of Various Objects

The radius of gyration varies with distinct objects. The explanation of few of them is below:

• Radius of Gyration of Compound Pendulum.
• Radius of Gyration of Solid Sphere
• Radius of Gyration of Circular Ring
• Radius of Gyration of Disc
• Radius of Gyration of Thin Rod

Radius of Gyration of Compound Pendulum

The radius of gyration of a compound pendulum depends upon its mass distribution. It is described as the distance from the axis of rotation to the point where the mass of the whole body is concentrated to produce the same moment of inertia as the actual mass distribution.

The formula to calculate the radius of gyration of a compound pendulum is given by:

K = √(I/M)

Where,

• I is Moment of Inertia of the pendulum and
• M is Mass of the pendulum

Radius of Gyration of Solid Sphere

Moment of Inertia of a solid sphere about an axis passing through its center and perpendicular to its plane with mass M and radius R is given by:

I = 2 MR² / 5

Radius of Gyration of a Sphere

But we know I = MK², so equation 1 becomes:

MK² = 2 MR² / 5

On canceling M from both sides and solving, we get

K² = 2R²/5

By taking the square root, we get:

K = √2/5 R

Hence, the radius of gyration of sphere about its center is given by K = √2/5 R.

Radius of Gyration of Circular Ring

The Moment of Inertia of a Circular ring about an axis passing through the tangent of the ring in its plane with radius R can be calculated by the parallel axis theorem. According to this, we have:

I = I_com + Md² , Here d = R,

⇒ I = MR²/2 + MR²

I = 3/2 MR²

Radius of Gyration of a Circular Ring

We know I = MK², so equation 1 becomes:

MK² = 3/2 MR²

On canceling M from both sides and solving, we get

K² = 3/2 . R²

By taking the square root on both sides, we get:

K = √3/2 . R

Hence, the radius of gyration of a Circular ring about its tangent is given by K = √3/2 .R

Radius of Gyration of Disc

The Moment of Inertia of a disc about an axis passing through its center and perpendicular to its plane with radius R and mass M is:

I = MR²/2

Radius of Gyration of a Disc

We know I = MK², so equation 1 becomes:

MK² = MR² / 2

On canceling M from both sides and solving, we get

K² = R² / 2

By taking the square root, we get:

K = R / √2

Hence, the radius of gyration of a disc is given by K = R/√2.

Radius of Gyration of Thin Rod

The moment of inertia of a uniform rod about an axis passing through its center and perpendicular to its plane with mass M and length L is given by:

I = ML²/12

Radius of Gyration of a Thin Rod

If K is the radius of the thin rod about an axis, then the equation will be

I = Mk²

By equating the value of the moment of Inertia, we get

Mk² = ML²/12

On canceling M from both sides and solving, we get

k² = L²/12

By taking the square root, we get:

K = L/√12

Hence, the radius of gyration of a thin rod is given by K = L/√12.

Radius of Gyration Solved Questions

Example 1: Radius of gyration of a compound pendulum about the point of suspension is 100 mm. The distance between the point suspension and the centre of mass is 250 mm. Considering the acceleration due to gravity is 9.81 m/s². What will be the natural frequency (in radian/s) of the compound pendulum.

Solution:

Given, Radius of Gyration(k) = 100mm = 0.1 m

We know that Moment of Inertia(I) = mk²

Put k = 0.1 in the above equation, we get

I = m(0.1)² kg m²

Also, we know

Here sinθ ≈ θ

we have ,

I = m(0.1)²

a = 250 mm = 2.5 m

g = 9.81 m/s²

ω_n=√245.25=15.660 rad/s

Hence, the natural frequency of the compound pendulum is 15.660 rad/s.

Example 2: Find the radius of gyration of a disc of mass M and radius 2m rotating about an axis passing through the center of mass and perpendicular to the plane of the disc.

Solution:

We know that, the radius of gyration of a disc is given by K = R/√2.

Given R = 2 m, putting value in the above equation we get

K = 2/ √2

K = √2 m

Hence, the radius of gyration of the given disc is √2 m.

Significance of Radius of Gyration

The significance of the radius of gyration is listed below:

• Rotational Dynamics: Simplifies calculations in physics and mechanics for rotational motion and angular momentum.

• Biomechanical Analysis: Analyzes mass distribution in body segments, aiding in understanding human movement.

• Aerospace Engineering: Used to assess stability and control of aircraft, guiding design for predictable responses.

• Sports Equipment Design: Optimizes mass distribution of equipment for enhanced athlete performance and control.

• Material Behavior Prediction: Characterizes mass distribution in materials, predicting their behavior under different conditions.

• Molecular Structure Understanding: Provides insights into the distribution of atoms within molecules, contributing to understanding molecular structures and stability.

Applications of Radius of Gyration

The radius of gyration is a crucial parameter in various fields and applications due to its ability to characterize the distribution of mass in a given object or system.

• Structural Engineering: Assess the efficiency and stability of structures, especially in resisting buckling.

• Mechanical Engineering: Influences performance of rotating machinery by determining moment of inertia.

• Biomechanics: Analyzes mass distribution in body segments for understanding human movement, applicable in sports science and rehabilitation.

• Aerospace Engineering: Used to assess stability and control of aircraft by considering mass distribution in different components.

• Sports Equipment Design: Optimizes mass distribution of equipment like golf clubs and tennis rackets for enhanced athlete performance.

• Material Science: Characterizes mass distribution in materials such as polymers and fibers, predicting material behavior under various conditions.

Radius of Gyration: FAQs

What is meant by Radius of Gyration?

The radius of Gyration (k) is the distance of the point where we can consider all mass of the body to be concentrated to the axis of rotation.

On what Factors Does Radius of Gyration Depend?

The radius of gyration depends upon the shape and size of the body, the position, and configuration of the axis of rotation, and also on the distribution of the mass of the body concerning the axis of rotation.

What is the SI Unit of Radius of Gyration?

The SI unit of the radius of gyration is the meter.

What are the Uses of Radius of Gyration?

The radius of gyration is used to compare how various structural shapes will behave under compression along an axis. It is used to predict buckling in a compression beam.

What is the Difference Between Radius of Gyration and Moment of Inertia?

The main difference between the moment of inertia and radius of gyration is about an object’s rotation:

Moment of inertia (I): This is a measure of how much effort it takes to change the rotational speed of an object. It is the required resistance to change when spinning something.

The radius of gyration (k): This is a radial distance from the axis to a point at which all the object’s mass is concentrated to get the same moment of inertia. It is the “spread out” mass of a body.

What is the Symbol for Radius of Gyration?

The symbol commonly used to represent the radius of gyration is “k.” It is a standard notation in physics and engineering when referring to this parameter.

What is Radius of Gyration Range?

The term “radius of gyration range” refers to the distribution of mass in an object around an axis.

How does Radius of Gyration Increase?

The radius of gyration increases when:

• Mass is farther from the axis of rotation
• Mass is less concentrated
• The object is larger