What is the radius of gyration?

Short Answer:

The radius of gyration is the distance from the axis of rotation at which the whole mass (or area) of a body can be assumed to be concentrated without changing its moment of inertia. It helps to express the distribution of mass or area about an axis in a simplified form. Mathematically, it is given as:

where is the moment of inertia, is the mass, and is the area. The radius of gyration provides a simple way to describe how far the body’s mass or area is spread from its axis of rotation.

Detailed Explanation :

Radius of Gyration

In Engineering Mechanics, the radius of gyration is an important concept used to simplify the analysis of a body’s rotational properties. It is a measure that represents how the area or mass of a body is distributed with respect to an axis. Instead of dealing with the actual irregular distribution of particles, the radius of gyration assumes that the entire mass (or area) of the body is concentrated at a single point located at a certain distance — this distance is called the radius of gyration (k).

The radius of gyration is directly related to the moment of inertia (I), which represents the body’s resistance to rotation. Thus, the radius of gyration provides a simpler and more intuitive understanding of how a body’s geometry affects its rotational behavior.

Definition

The radius of gyration can be defined as:
“The distance from the axis of rotation at which the entire mass or area of a body can be assumed to be concentrated so that its moment of inertia remains the same as that of the actual distribution.”

Mathematically,

 

where,

  • = radius of gyration,
  • = moment of inertia,
  • = total mass of the body,
  • = total area of the plane figure.

Physical Meaning

The physical meaning of the radius of gyration is that it indicates how far the mass or area of the body is spread from the axis of rotation.

  • If the mass (or area) is concentrated closer to the axis, the radius of gyration will be small.
  • If the mass (or area) is distributed farther from the axis, the radius of gyration will be large.

For example:

  • A solid cylinder has a smaller radius of gyration compared to a hollow cylinder of the same mass and radius because more mass in the hollow cylinder lies away from the axis.

Thus, the radius of gyration provides a numerical way to compare how mass distribution affects rotational motion and bending resistance.

Derivation

Consider a body made up of several small particles of masses situated at perpendicular distances from a given axis.

The moment of inertia (I) of the body about that axis is given by:

If all the mass is assumed to be concentrated at a distance (the radius of gyration) from the axis, then:

where is the total mass of the body.

Now,

Hence, the radius of gyration depends on the distribution of mass and the geometry of the body.

Formulas of Radius of Gyration for Common Shapes

  1. Thin Rod (length )
    • About an axis through its center:
    • About an axis through one end:
  1. Rectangular Area (width , height )
    • About centroidal x-axis:
    • About centroidal y-axis:
  1. Circular Section (radius )
    • About its centroidal axis:
  1. Solid Cylinder (radius )
    • About its own axis:
  1. Hollow Cylinder (inner radius , outer radius )
    • About its axis:

These standard results are used in engineering calculations for beams, shafts, and rotating machinery.

Importance of Radius of Gyration

The radius of gyration is important in both static and dynamic analysis because it simplifies the study of moments of inertia, bending, and rotation.

  1. In Structural Engineering:
    • It helps to determine the buckling strength of columns.
    • Columns with larger radii of gyration are more stable and less likely to buckle under load.
  2. In Mechanical Engineering:
    • It assists in analyzing rotational motion of flywheels, shafts, and gears.
    • The radius of gyration is used to express mass distribution in rotating bodies for dynamic balance.
  3. In Design of Machines:
    • Used in determining energy storage capacity of flywheels.
    • Helps in reducing vibrations in rotating systems by maintaining uniform mass distribution.
  4. In Moment of Inertia Calculations:
    • Simplifies calculation of polar moment of inertia for torsional stress analysis.
  5. In Stability Analysis:
    • Provides an easy measure to compare how mass or area affects resistance to bending and twisting.

Example

Example:
A thin rod of mass 3 kg and length 1.2 m rotates about an axis through its center. Find the radius of gyration.

Solution:
Moment of inertia about the center:

Now,

Hence, the radius of gyration is 0.346 m.

Factors Affecting Radius of Gyration

  1. Shape and Size of the Body: Different shapes (like rods, plates, or cylinders) have different distributions of mass, hence different radii of gyration.
  2. Axis of Rotation: The distance of the axis from the mass distribution affects .
  3. Mass Distribution: Concentrating mass farther from the axis increases the radius of gyration.

Relation Between Moment of Inertia and Radius of Gyration

The relation between moment of inertia and radius of gyration is:

This shows that the radius of gyration depends on both the total mass (or area) and how that mass (or area) is distributed about the axis. For bodies of the same mass, the one with the larger will have a higher moment of inertia and will be harder to rotate.

Conclusion

In conclusion, the radius of gyration is a measure that simplifies the concept of moment of inertia by assuming that the entire mass or area of a body is concentrated at a single point at a specific distance from the axis. It provides insight into how mass or area is distributed relative to an axis of rotation. The radius of gyration is crucial in analyzing stability, designing rotating systems, and understanding the bending and torsional behavior of structures and machines.