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Home / Physics / Radius of Gyration

Radius of Gyration: Learn its Unit, Formula and Applications

Last Updated on Feb 20, 2025
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Moment of Inertia can be defined as the quantity which expresses the tendency of a body to avoid angular acceleration. It gets created by the sum of a product of the mass of every particle present in a body with the square of the distance of the body from the axis of its rotation in rotational motion. The moment of inertia is also represented by the radius of gyration.

So read on to learn more about its unit, formula and applications along with a few solved examples like uses in particle accelerator.

Radius of Gyration

Radius of Gyration can be defined as the distance from the rotating axis from where the whole mass of the body is presumed to be concentrated and due to which the moment of the inertia always remains the same. The radius of gyration of a body is also known as the radius of a body. The diagram given below depicts the radius of gyration.

To understand this we can take an example of the centre of mass of a body. It is always on the axis of rotation. It is also characterized as the spiral distance of a body to a point where it would be having a moment of inertia using laws of inertia. It is also considered to be the geometric property of a rigid body. It is also equivalent to the real dissemination of the body of having a mass only if all the outward mass present in the body gets concentrated.

The S.I. unit of the radius of gyration is a meter which is denoted by m. The dimensions of radius of gyration are as follows:

Radius of Gyration Formula

The formula for the radius of gyration is given as follows:

Where I is the moment of inertia.

m is the mass of the body.

Steps to Calculate Radius of Gyration

The steps to calculate radius of gyration are as follows:

If we consider a body which is having an ‘n’ number of particles and each of them have a mass equal to that of m.

If the perpendicular distance of the body from the axis of its rotation is given by,

.

So from this, we know that the moment of inertia of the body in terms of the radius of gyration will be given by the following equation.

Now if we substitute the values of the body in the above equation, then we will get its moment of inertia as follows:

If all the particles are having same mass then the above equation will change into the following,

Now if we write mn as M then the above equation will be changed into the following,

So from simplifying the above equation we will get the following equation.

Now from the above equation, we can conclude that the radius of gyration can also be defined as the root-mean-square distance of several particles of a body from its axis of rotation.

Radius of Gyration Applications

The applications of the radius of gyration are mentioned as follows:

  • It is used to determine the pressure exerted by a body.
  • It is also used to estimate the strength of a body.
  • It is also used to estimate the cross-section area of a body.
  • It is used in structural engineering.
  • It is also used in molecular physics to determine the dimensions of a polymer chain.
  • It is also used to determine the behaviour of several structural shapes under compression.
  • It is also used to determine the figure of a heap of beams.
  • It is also used to determine the segment of a body.

Solved Examples on Radius of Gyration

Example 1- Find the radius of gyration of a disc having mass M and radius r, and is rotating about an axis passing through its centre of mass and perpendicular to its plane.

Solution. Given

Since we know,

Therefore,

As,

So now,

.

.

We can see that the radius for gyration of a rigid body is always a geometrical property like length, breadth and radius, or their combination with a positive numerical multiplier.

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Radius of Gyration FAQs

The radius of gyration is a measure of how spread out an object's mass is from its axis of rotation.

It is the distance from the axis of rotation to a point where the object's entire mass could be concentrated and have the same moment of inertia.

It is used to determine the behavior of rotating objects. It can be used to calculate the moment of inertia, which is a key factor in determining how an object will respond to external forces.

The radius of gyration depends on the shape and size of an object. For example, a thin rod has a smaller radius of gyration than a thick rod of the same length because its mass is concentrated closer to the axis of rotation.

The radius of gyration is an important concept in sports, particularly in gymnastics and figure skating. Athletes who perform aerial maneuvers need to control their body's radius of gyration to maintain balance and stability in the air.

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