Physics
Study Material

# Radius of Gyration- Derivation and Formula

A small way to remember radius of gyration is that, take its moment of inertia, divide it by its mass and take square root. Read full article to know concept in a very interesting way.

3 minutes long
Posted by Mahak Jain, 13/9/2021

Hesap Oluştur

Got stuck on homework? Get your step-by-step solutions from real tutors in minutes! 24/7. Unlimited.

Radius of gyration has always been a difficult and confusing topic for students. Here in this blog I have tried to explain it in a very simple way. Below here you will see the bookish definition followed radius of gyration formula followed by beautiful explanation. Thus, read this small blog till end.

According to books

Radius of gyration is the distance from the axis of a mass point whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis.

Moment of inertia in terms of radius of gyration is written as :-

I = mk2

⇒ k = √(I/m)

k is radius of gyration. This I over here is moment of inertia of the object about the axis from centroid perpendicular to its plane. This is that moment of inertia that we usually use in questions, just to explain clearly. For example, for ring I is MR2, for disc I is MR2/2, etc.

## What is Radius of Gyration

Now let’s come to the interesting part of understanding concept.

See the first equation under heading formula above. The equation is I = mk2. Now, let’s think where this equation came from.

This equation came from definition. But the definition is so confusing. How do I understand it in simple words? Let’s understand.

Take a body of mass m of which you want to know radius of gyration. Let its moment of inertia about the axis passing through centroid and perpendicular to the plane be I.

Now, take a point mass of same mass m and an imaginary axis. Keep the particle at distance x from the imaginary axis. So, its moment of inertia is mx2.

The value of mr2 should be such that it is equal to I. When this condition satisfies we call r, radius of gyration.

Thus, x = k

Therefore, I = mk2

⇒ k = √(I/m)

Let’s take an example,

Moment of inertia of a ring of mass m and radius r about the axis passing through centroid and perpendicular to plane is mr2.

So, by formula k = √(I/m)

⇒ k = √(mr2/m) = r

This is to be noted that, it is jus a coincidence that here k = r. So, do not get confused that k is always equal to radius of every object. It is only in the case of ring, not in general.