Important Topic of Physics: Radius Of Gyration

A moment of inertia is the resistance of a body to angular acceleration, which is the sum of the mass of each particle and the square of its distance from an axis of rotation. Or, to put it simply, it’s a quantity that determines the amount of torque needed for a specific angular acceleration in a rotational axis.

Angular mass and rotational inertia are both terms for moment of inertia. Defining the unit of moment of inertia as SI is kg m². Typically, moment of inertia is measured in relation to a chosen axis of rotation.

Inertia about an axis of a body is sometimes expressed using its radius of gyration. A radius of gyration is the distance from the centroid at which we imagine the area of cross-section to be focused at a point in order to obtain the same moment of inertia. It is denoted by k.

Radius of Gyration Formula:

The formula of moment inertia in terms of the radius of gyration is given below:

I = mk² → equation (1)

where I is the moment of inertia and m is the mass of the body. Therefore, the radius of gyration is given below:

k=√(I/m) → equation (2)

The radius of gyration is measured in millimetres. Regardless of the body equation (1), one can find the moment of inertia by knowing the radius of gyration without any hassle.

Imagine a body with n particles, each with mass m. The perpendicular distances from the axis of rotation are r₁ , r₂ , r₃ , …, r_n. Equation (1) gives the moment of inertia in terms of the radius of gyration. Substituting the values in the equation, we get the moment of inertia of the body as follows:

I = m₁ r₁² + m₂ r₂² + m₃r₃² + m_nr_n² → equation (3)

If all the particles have the same mass then equation (3) becomes :

I = m(r₁² + r₂² + r₃²+ r₄² + r_n²)

=m n(r₁² + r₂² + r₃²+ r₄² + r_n²)/n

We m n is equal to M, which is the body’s total mass. Now the equation becomes

I = M (r₁² + r₂² + r₃²+ r₄² + r_n²)/n → equation (4)

From equation (4), we get

MK² = M (r₁² + r₂² + r₃²+ r₄² + r_n²)/n

Or,

K = √ (r₁² + r₂² + r₃²+ r₄² + r_n²)/n

What is the use of the radius of gyration?

The radius of gyration is used to compare how different structural shapes will behave under compression along an axis. In compression beams or members, it is used to predict buckling.

We’ll calculate the radius of gyration of a thin rod and a solid sphere.

The (MOI) of a uniform rod of length L and the mass M about its axis through the center, forming a 90-degree angle to the length is:

I = ML²/12

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

I=Mk²

By equating the value of the moment of Inertia,

we get Mk² = ML²/12

On canceling M from both sides we get,

K² = L²/12

By taking the square root, we get:

K= L/√12

Solid Sphere:

Solid spheres with mass M and radius R, having a moment of inertia or MOI, have the following MOI:

I _axis = M k ²

I for solid sphere about tangent = 2 / 5 MR² + MR² = 7 / 5 MR²

By equating both the moment of inertias

k= (7 /5) ^1/2 R

Radius of Gyration polymer:

In polymer physics, the radius of gyration describes the dimensions of polymer chains. Molecular gyration radius is defined as:

R_g² = 1/ N ^N ∑ _k=1 (r _k – r _mean )²

where r is the mean position of the monomers. The radius of gyration is proportional to the root mean square distance between the monomers:

R_g² = 1/ 2N² ∑ _{i j} (r _i – r _j )²

By summing the principal moments of the gyration tensor, the radius of gyration can also be calculated.

Considering that the chain conformations of a polymer sample are quasi-infinite in number and constantly change over time, “radius of gyration” discussed in polymer physics is usually understood as a mean overall polymer molecule of the sample. It is, therefore, a measurement of the average of the radius of gyration overtime or an ensemble.

R_g² = 1/ N 〈 ^N ∑ _k=1 (r _k – r _mean )² 〉

Here the angular brackets symbolize ensemble average.

An en-tropically governed polymer chain follows a random walk in three dimensions. The radius of gyration for this case is given by the following equation:

R _g = (N /6) ^1/2 ×a

Note that, although an N represents the contour length of the polymer, a is strongly dependent on polymer stiffness, and can vary over orders of magnitude. N is reduced accordingly.

What factors determine the radius of gyration?

Size, shape, rotational axis configuration, and position are some factors influencing the radius of gyration. Also, it depends on how the body mass is distributed in relation to the rotational axis.

1. Variation in a radius of gyration is directly related to the size of the object.
2. It is related to the shape of a body that determines the radius of gyration of that body.
3. Rotation axis configuration: The axis along which a body rotates will be an important factor in determining the radius of gyration for a particular body in a dimensional plane.
4. An object’s position in a plane determines the radius of its rotation.

The radius of gyration can be used to determine the pressure exerted at a particular point.It is very useful in estimating the intensity and strength between two cross-sections of a given area. Angular acceleration is caused by the resistance a body has to angular acceleration, which is equal to the mass of each particle divided by its square distance from an axis of rotation. A torque value is a measure of how much force is required for a specific angular acceleration in a rotational axis. The closer the mass is to the axis of rotation, the smaller the radius of gyration for that point. Gyration radius varies from point to point. It varies with the changing axis of rotation.

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Frequently Asked Questions:

Mention the factors on which the radius of gyration depends?

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

Why do we calculate the radius of gyration?

The gyration radius is useful in determining the stiffness of a column. If the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment.

Does the angular velocity of the body affect its radius of gyration?

The radius of gyration of a body does not depend on the angular velocity of the body.

Is the radius of gyration of a body a constant quantity?

No, the radius of gyration of a body depends on the axis of rotation and also on the distribution of mass of the body on this axis.