**Radius of
Gyration**

**What is Radius of Gyration?**

Radius of Gyration is defined as the distance from the axis of rotation to a point where the total mass of the body is supposed to be concentrated so that the moment of inertia about the axis may remain the same. It is denoted by K.

**Radius of Gyration Formula:**

In terms of Radius of Gyration, the moment of inertia of the body of mass M is given by:

I = MK² … (1)

Where,

I = Moment of Inertia,

M = Mass of the body.

Accordingly, the radius of gyration is given as follows:

Radius of Gyration (K) = √(I/ M).

The unit of radius of gyration is mm. By knowing the radius of gyration, we can find the moment of inertia of any complex body.

Suppose a body consists of n particles each mass m. Let, r₁, r₂, r₃, r₄, … r_{n} be their perpendicular distances from the axis of rotation. Then, the moment of inertia (I) of the body about the axis of rotation is:

I = m₁r₁² + m₂r²₂ +
m₃r²₃ + m₄r²₄ + … + m_{n}r_{n}²

If all the particles are of same mass m, then:

I = m (r₁² + r²₂ + r²₃
+ r²₄ + … + r_{n}²)

I = mn (r₁² + r²₂ + r²₃
+ r²₄ + … + r_{n}²)/ n

Since mn = M, total mass of the body:

∴ \(I=M\left( \frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}+……+r_{n}^{2}}{n} \right)\) … (2)

From equations (1) and (2), we get:

\(M{{K}^{2}}=M\left( \frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}+……+r_{n}^{2}}{n} \right)\).

\(K=\sqrt{\left( \frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}+……+r_{n}^{2}}{n} \right)}\).

Therefore, from the above relation we can say that, the radius gyration can also be defined as the root mean square distance of the various particles of the body from the axis of rotation.