Radius of Gyration: Definition, Formula, Mathematical Derivation and Real-Life Applications

The Radius of gyration is one of the important concepts in rotational motion. It helps us know how the mass of an object is spread around its axis of rotation. This idea is useful while studying wheels, fans, flywheels, gears, and many other rotating objects.

Have you ever noticed that two objects with the same mass do not always rotate in the same way? The reason lies in how their mass is distributed. This article provides complete insights into the radius of gyration, its formula, derivation, what is the unit of radius of gyration, factors, and reasons behind why it plays an important role in physics.

Table of Contents

• What is Radius of Gyration
• Mathematical Derivation of Radius of Gyration Formula
• Factors Affecting Radius of Gyration
• Radius of Gyration of Different Objects
• Applications of Radius of Gyration
• Real-Life Examples of Radius of Gyration
• Frequently Asked Questions about Radius of Gyration

What is Radius of Gyration

Imagine two school bags that have the same weight. In one bag, all the books are packed close to the center. In the other bag, the books are spread towards the sides. Although both bags have the same mass, they feel different when you try to rotate them.

The same idea applies to rotating objects. The radius of gyration tells us how far the mass of an object is distributed from its axis of rotation.

The radius of gyration definition states:

“The radius of gyration is the distance from the axis of rotation at which the entire mass of a body can be assumed to be concentrated so that its moment of inertia remains the same.”

It gives an idea of how the mass of a body is spread around the axis of rotation.

If most of the mass is farther from the axis, the radius of gyration is larger. If the mass is closer to the axis, its value becomes smaller.

Interestingly, the radius of gyration formula is directly related to the moment of inertia. The relation between moment of inertia and radius of gyration is, 

$I = Mk^2$

Where I is the moment of inertia, M is the total mass of the object, and k is the radius of gyration. 

After rearranging the equation gives the radius of gyration formula, 

$k=\sqrt{\frac{I}{M}}$

This formula shows that the radius of gyration depends on both the object's mass and its moment of inertia.

Now, there is an interesting question that comes into the picture. How is the radius of gyration formula derived? Let's discuss this in detail. 

Mathematical Derivation of Radius of Gyration Formula

Consider an object made up of many small particles.

Suppose, 

• The masses of the particles are m₁, m₂, m₃, ...
• Their perpendicular distances from the axis are r₁, r₂, r₃, ...

The moment of inertia of the object is,

$I=m_1r_1^2+m_2r_2^2+m_3r_3^2+\cdots$

If all particles have the same mass m, then

$I=m(r_1^2+r_2^2+r_3^2+\cdots+r_n^2)$

Let consider the total number of particles be n. So the total mass becomes

M=mn

After substituting this into the equation, we get

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

But we already know that

$I=Mk^2$

After comparing both equations, we can write

$k^2=\frac{r_1^2+r_2^2+r_3^2+\cdots+r_n^2}{n}$

After taking the square root,

${k=\sqrt{\frac{r_1^2+r_2^2+r_3^2+\cdots+r_n^2}{n}}}$

So, in simple words, the radius of gyration is the root mean square (RMS) distance of all the particles from the axis of rotation.

But what is the unit of radius of gyration? 

Since the radius of gyration represents a distance, its SI unit is the same as the unit of length. The SI unit of radius of gyration is metre (m).

The radius of gyration is a length, its dimensional formula is  $[M^0L^1T^0].$

Factors Affecting Radius of Gyration

The value of the radius of gyration depends on how the mass of an object is arranged and the axis about which it rotates. Even two objects with the same mass can have different values if their shape or mass distribution is different. 

• Shape of the Object: Different shapes spread their mass differently. For example, a solid sphere has a smaller radius of gyration than a long rod of the same mass.
• Position of the Axis of Rotation: Moving ahead, the value changes if the axis of rotation changes. For example, a cylinder has a different radius of gyration values when it rotates about its central axis and about an axis through its edge.
• Distribution of Mass: If more mass is located farther from the axis, the radius of gyration becomes larger. If most of the mass is close to the axis, the radius of gyration becomes smaller.
• Size of the Body: Larger objects generally have a greater average distance between their mass and the axis of rotation. As a result, they may have a larger radius of gyration than smaller objects with a similar shape.

Radius of Gyration of Different Objects

The radius of gyration is not the same for every object. It changes depending on the shape of the object and the axis about which it rotates.

Radius of Gyration of a Compound Pendulum

A compound pendulum is a rigid body that swings freely about a fixed horizontal axis. Unlike a simple pendulum, its mass is spread throughout the body.

The radius of gyration of a compound pendulum is the distance from the axis of rotation where the entire mass can be assumed to be concentrated without changing its moment of inertia.

The formula is, 

${k=\sqrt{\frac{I}{M}}}$

Where k is the radius of gyration, I is the moment of inertia of the pendulum, and M is the mass of the pendulum.

Do you know that the value of k depends on how the mass is distributed with respect to the axis of rotation?

Radius of Gyration of a Solid Sphere

Imagine a cricket ball spinning about its centre. Since the mass of a solid sphere is spread evenly in all directions, its radius of gyration can be found using its moment of inertia.

The moment of inertia of a solid sphere is

$I=\frac{2}{5}MR^2$

Using the relation

$I=Mk^2$

we get, 

$Mk^2=\frac{2}{5}MR^2$

Cancelling M from both sides, we get 

$k^2=\frac{2}{5}R^2$

Taking the square root,

${k=\sqrt{\frac{2}{5}}R}$

So, the radius of gyration of a solid sphere about its centre is:

${k=\sqrt{\frac{2}{5}}R}$

Radius of Gyration of a Circular Ring

For a thin circular ring rotating about a tangent in its plane, the moment of inertia is obtained using the parallel axis theorem.

The moment of inertia is

$I=\frac{3}{2}MR^2$

By using

$I=Mk^2$

we get

$Mk^2=\frac{3}{2}MR^2M$

Cancelling M,

$k^2=\frac{3}{2}R^2$

Taking the square root,

${k=\sqrt{\frac{3}{2}}R}$

Hence, the radius of gyration of a circular ring about its tangent is

${k=\sqrt{\frac{3}{2}}R}$

If the ring rotates about its centre, then k=R.

Radius of Gyration of a Disc

The moment of inertia of a disc about an axis passing through its centre and perpendicular to its surface is

$I=\frac{1}{2}MR^2$

Using,

$I=Mk^2$

we get

$Mk^2=\frac{1}{2}MR^2$

After cancelling M,

$k^2=\frac{R^2}{2}$

Taking the square root,

${k=\frac{R}{\sqrt2}}$

Thus, the radius of gyration of a disc is

${k=\frac{R}{\sqrt2}}$

Radius of Gyration of a Thin Rod

Imagine a ruler rotating about its centre.

 For a uniform thin rod, the moment of inertia about an axis passing through its centre is

$I=\frac{1}{12}ML^2$

Using

$I=Mk^2$

we get,

$Mk^2=\frac{1}{12}ML^2$

Cancelling M,

$k^2=\frac{L^2}{12}$

Taking the square root,

${k=\frac{L}{\sqrt{12}}}$

Hence, the radius of gyration of a thin rod is

${k=\frac{L}{\sqrt{12}}}$

Applications of Radius of Gyration

The radius of gyration helps engineers and scientists study how mass is distributed in rotating objects. It is used in many fields to design safer structures and improve the performance of machines.

• Structural Engineering: The radius of gyration helps engineers design strong and stable structures. It is used to check the strength and stability of columns in buildings and bridges. It helps in designing beams that can safely support different loads.

• Mechanical Engineering: Many machines contain rotating parts, and the radius of gyration helps improve their performance. It is used in the design of engines, motors, turbines, and gears. Engineers also use it while designing flywheels, which store rotational energy and ensure smooth machine operation.

• Aerospace Engineering: Aircraft and spacecraft must remain stable during motion. The radius of gyration helps engineers distribute mass properly so that aircraft can fly safely and remain balanced during take-off, flight, and landing.

• Sports Science: The radius of gyration helps athletes improve their performance in activities such as Gymnastics, Diving, Figure skating, and the high jump. By changing their body position, athletes can rotate faster or slower.
• Robotics: Robotic arms are designed to move quickly and accurately. Engineers use the radius of gyration to distribute the mass of robotic parts, making their movements smoother and more precise.
• Material and Product Design: The radius of gyration is also considered while designing products such as bicycle wheels, ceiling fans, wind turbines, and industrial rotating machines. Proper mass distribution improves efficiency and reduces vibration.

Real-Life Examples of Radius of Gyration

The radius of gyration is not just a concept in physics. It can be seen in many everyday activities and objects around us. These examples show how the distribution of mass affects the speed and ease of rotation. 

• Ice Skater Spinning: When an ice skater pulls their arms close to the body, the radius of gyration decreases. As a result, the skater spins faster.
• Rotating Chair: When you stretch your arms outward, spinning becomes slower. When you pull your arms inward, you spin faster because the radius of gyration becomes smaller.
• Dumbbell: Imagine holding a dumbbell. If the weights are moved closer to the centre, it becomes easier to rotate. If the weights are farther away, rotating the dumbbell requires more effort because the radius of gyration increases.
• Flywheel of an Engine: A flywheel stores rotational energy in vehicles and machines. Its mass is distributed carefully so that it rotates smoothly and maintains a steady motion.
• Figure Skater or Gymnast: When a gymnast tucks their body while performing a flip, they reduce their radius of gyration. This helps them complete the rotation more quickly before landing.
• Ceiling Fan: The blades of a ceiling fan are designed with balanced mass distribution. This allows the fan to rotate smoothly with less vibration.
• Bicycle Wheel: A bicycle wheel rotates efficiently because its mass is distributed evenly around the axle, making the ride smoother and more stable.
• Wind Turbine: Wind turbine blades are designed by considering the radius of gyration. Proper mass distribution helps the blades rotate efficiently even at lower wind speeds.

Did You Know? Athletes such as divers, gymnasts, and figure skaters change their body position to control their radius of gyration. This allows them to spin faster or slower without changing the force acting on them.

In this article, we have learned that the radius of gyration shows how the mass of an object is distributed about its axis of rotation. This concept plays an important role in rotational motion and helps explain the behaviour of many everyday rotating objects.