A Short Note on The Use of Radius of Gyration

When an object or item moves along a straight line, it experiences inertia that restricts a change in its state of motion. It is the inertia of linear motion.

Similarly, suppose the object rotates about an axis. In that case, the inertia does not allow a change in its rotational motion state, which is called the object’s rotational inertia or moment of inertia.

The moment of inertia of a body about an axis is sometimes represented using the radius of gyration. It is a parameter that tells the way the distribution of the mass of an object occurs along its axis of rotation.

Let’s take a deeper look into the radius of gyration.

What is the Radius of Gyration?

Before going into the definition of the radius of gyration, let’s understand it by taking the example of a ceiling fan. The fan blades rotate about an axis and have a moment of inertia.

Suppose you are to combine the mass of all the blades at a single point in such a way that the moment of inertia remains the same. Therefore, the distance between the point and its axis at which the moment of inertia remains the same after combining the mass of all the blades is called the radius of gyration.

Hence, the radius of gyration is the imaginary distance from the axis of rotation of an object to the point where the entire mass of the object is concentrated. At this point, the moment of inertia remains the same after combining the whole object’s mass.

You must note that the distance is calculated based on the moment of inertia such that it remains the same before and after combining the entire mass at one point.

Factors on Which Moment of Inertia Depends

The following factors determine the moments of inertia of a rigid object:

• Total mass of the object
• Size of the object
• Shape of the object
• Distribution of mass about the axis of rotation
• Position of the axis of rotation
• Orientation of the axis of rotation

Mathematical Representation of Radius of Gyration

The radius of gyration of a rotating object is a measure of its moment of inertia and total mass. The moment of inertia can be written as

I = Mk2

Where M = mass of the body

k = radius of gyration

So, the radius of gyration is the square root of the quotient or ratio of inertia and mass.

Note that the radius of gyration is a geometric property, and it varies with the shape and axis of the object.

1. The radius of gyration for a thin circular ring with radius, R and the axis of rotation at the centre perpendicular to the plane is given by

k = R

1. The radius of gyration for a thin circular ring with radius, R and diameter as the axis of rotation is given by

k = R/√2

1. The radius of gyration for a thin rod of length, l and axis of rotation at the midpoint perpendicular to the rod is given by

k = L/√12

1. The radius of gyration for a circular disc of radius, R with the axis of rotation perpendicular to the disc at the centre is given by

k = R/√2

1. The radius of gyration of a circular disc of radius, R with the axis of rotation along its diameter is given by

k = R/2

1. The radius of gyration of a solid cylinder of radius, R with the axis of rotation along the axis of the cylinder is given by

k = R/√2

1. The radius of gyration of a solid sphere of radius, R, with the axis of rotation along the diameter of the sphere, is given by

k = R√2 /√5

The Use of Radius of Gyration

The radius of gyration has a vital role to play in mechanical engineering. For example, several situations like the rotation of the earth on its axis, rotation of flywheels in engines, and the rigid spokes in the axis of bicycle wheels increase the moment of inertia exponentially to reduce its effect.

• Determining the radius of gyration helps determine when a machine’s structure or component will buckle or bend as you compress it.
• So, the radius of gyration indicates the stiffness and strength of a material.
• It is a comparing quantity that helps you understand the behaviour of objects with different shapes under compression along the object’s axis.

Conclusion

The moment of inertia is a fundamental property of every rotating object. The moment of inertia of a body about an axis is sometimes represented using the radius of gyration. Therefore, determining ways to reduce the effect of inertia is an essential part of mechanical engineering. The measure we can use to evaluate the way the mass is distributed in a rotating object along its axis of rotation is called the radius of gyration.

It is a geometric property of a rotating object that depends on the distribution of mass and axis of rotation. It is a useful parameter in determining the stiffness and buckling potential of materials in machines.