## Radius of Gyration of Thin Rod

The radius of gyration of a body is referred to as the radial distance from the rotational axis at which, the entire body mass is supposed to be concentrated. The moment of inertia (MOI) about any given axis will be the same as the actual body mass distribution.

When seen in terms of mathematics, the “radius of gyration” is denoted as the square root of the mean square distance of parts of the object from the middle region of body mass or a specified axis that depends on the appropriate application. In other words, the radius of gyration is calculated as the perpendicular distance noted from the rotational axis to the point mass.

The actual radial distance between the rotational axis and the point where the body mass is joined to it keeps the inertia of a rotating object fixed. As the rotational body mass is focused on the point mass, it implies that the radius of gyration is measured as the distance by taking the mid-point of the rotational axis and measuring its distance with the mass of the body.

In terms of physics, the radius of gyration is referred to as the method of the distribution of different components of the object present around an axis. When seen with respect to the moment of inertia, the radius of gyration is calculated as the perpendicular distance taken from the rotational axis to a specific point mass. This provides inertia equivalent to the original object.

If the moment of Inertia is represented by I, then its value is MK2. I = MK2

If a body has n particles, each of them has mass m.

Let r1, r2, r3, ... , rn be the perpendicular distance of the object from the rotation axis. Then, MOI or the moment of inertia of the body on its rotational axis is calculated as

I = m1r12 + m2r22 + m3r32 + ….. + mnrn2
If the mass of all the particles is the same as m, then the equation can be written as:
Moment of Inertia (I) = mr12 + mr22 + mr32 + ….. + mrn2
It can also be written as I = m (r12 + r22 + r32 + ….. + rn2)
If we multiply and divide the equation by n, then the equation will look as:
I = mn ((r12 + r22 + r32 + ….. + rn2))/n

Here mn is M or total body mass.

Replacing mn by M makes the equation like

I = M ((r12 + r22 + r32 + ….. + rn2))/n
By substituting the value of I by MK2 from equation 1
Then the equation can also be written as:

MK2 = M ((r12 + r22 + r32 + ….. + rn2))/n
If we take square root both the sides, then the equation becomes:

K

This clearly shows that K or the radius of gyration of a body about an axis is the root of the mean square distance of several different body particles from the rotational axis.

How is the Radius of Gyration is calculated for a Thin Rod

The moment of inertia (MOI) of any uniform rod of length l and mass M about an axis through the center and forming a 90-degree angle to the length is shown as:

I or moment of Inertia = MI2 / 12

If K = the radius of the thin rod about an axis, then the equation will become

I = MK2
By equating the value of I or moment of Intertia from the above equations, we have
MK2 = MI2 / 12,

On cancelling M from both the sides, we now have,

K2 = I2 / 12

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

K = I / sq. root of 12

How is the radius of gyration calculated for a solid sphere?

The moment of inertia or MOI for any solid sphere with a mass M and radius Ris given by:

I = 2/5 MR2 ………….. (1)

If K implies the radius of a solid sphere, then

I = MK2…………………. (2)

On combining both the equations 1 and 2, the equation can also be written as

MK2 = 2/5 MR2

Canceling M and taking square root on both the sides, our equation now becomes:
K = square root of 2/5 R

The radius of gyration and slenderness ratio

Radius of Gyration is defined as the sq. root of the ratio of Inertia to the area of the material. The value obtained from it denotes the imaginary distance calculated from the point at which the cross-sectional area is supposed to be concentrated at a point. This will help you obtain the same inertia. It is calculated by measuring the slenderness of an area of the cross-section of a column.

The radius of any sphere that touches a point in the curve and has got the same curvature and tangent at that point is regarded as the radius of curvature. In terms of mathematics, the “radius of gyration” is regarded as the square root of the mean square radius of the different parts of the object from the central point of its mass or any given axis. This depends on any relevant application.

Applications in the field of structural engineering

In the field of structural engineering, the two-dimensional gyradius helps in describing the distribution of any cross-sectional area around the centroidal axis in the body mass.

Application in the field of polymer physics

The term radius of gyration describes the spatial dimensions of a given polymer chain.

The SI unit for gyradius is the length (that is shown in inches or millimeters or feet). It is shown as the sq. root of inertia divided by the object’s area

On what factors radius of gyration depends?

Some of the factors that influence the value of the radius of gyration are the size and shape of the body, configuration of the rotational axis and position. It also depends on body mass distribution with respect to the rotational axis.

How is the gyradius termed as a constant quantity?

The value of “Radius of gyration" or radius is not fixed. Its value depends on the rotational axis and the distribution of body mass about the axis.

How is the radius of gyration defined for a regular solid sphere?

The “radius of gyration” is the square root of the average squared distance of a sphere object from its midpoint of mass.

What is the application of radius of gyration?

Calculation of the radius of gyration is beneficial in various ways. It is useful in comparing how different structural shapes behave under the body's compression along a rotational axis. It is also used in forecasting buckling in a beam or compression member. One thing that you need to take into account is that the SI unit of gyradius measurement is mm.

The significance of “radius of gyration”

• 1. The radius of gyration is significant in the calculation of the clasping load of a beam or compression.

• 2. It is also useful in the distribution of power between the cross-sections of a given column.

• 3. The radius of gyration is helpful in comparison to the performance of different kinds of structural shapes at the time of the compression.

• 4. Lesser value of “radius of gyration” is effective in performing structural analysis.

• 5. Lesser value of “radius of gyration” displays that the rotational axis at which the column clasps.