An Introduction
The radius of gyration or gyroradius of anybody about the axis of rotation is defined as that radial distance from the axis of rotation, at which the entire mass of that body is concentrated. Thus, the point will have a moment of inertia at this particular point also.
If we describe the radius of gyration in mathematics, then we can say that the radius of gyration is the root -mean square distance of the part of the object from either its centre of mass or given axis. This depends upon the relevant application. This is actually called the perpendicular distance from the point mass to the axis of rotation. The radius of gyration can be used to characterize the particular distance travelled by a point.
This is also referred to as the measure of the way in which the mass of a rotating rigid body can be distributed about its axis of rotation.
The radius of Gyration of Thin Rod
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 midpoint 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+ ….. + mnn2
If the mass of all the particles is the same as m, then the equation can be written as:
Moment of Inertia (I) = mr21+ mr22 + mr32 + ….. + mr2n
It can also be written as I = m (r21 + r22 + r23 + ….. + r2n)
If we multiply and divide the equation by n, then the equation will look as:
I = mn \[\frac {(r^2_1+r^2_2+r^2_3+....+r^2_n)} {n}\]
Here mn is M or total body mass.
Replacing mn by M makes the equation like
I = M \[\frac {(r^2_1+r^2_2+r^2_3+....+r^2_n)} {n}\]
By substituting the value of I by MK2 from equation 1
Then the equation can also be written as:
MK2= M \[\frac {(r^2_1+r^2_2+r^2_3+....+r^2_n)} {n}\]
If we take square root both the sides, then the equation becomes: K
This clearly shows that K is 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 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 centre and forming a 90-degree angle to the length is shown as:
I or moment of Inertia = \[\frac {MI^2} {12}\]
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 Inertia from the above equations, we have
MK2 = \[\frac {MI^2} {12}\],
On cancelling M from both sides, we now have,
K2 = \[\frac {I^2} {12}\]
By taking square root on both sides, we have:
K = \[\frac {I} {\sqrt 12}\]
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 R is given by:
I = \[\frac {2} {5}\] MR2………….. (1)
If K implies the radius of a solid sphere, then I = MK2…………………. (2)
On combining both equations 1 and 2, the equation can also be written as
MK2 = \[\frac {2} {5}\] MR2
Cancelling M and taking square root on both the sides, our equation now becomes:
K = \[\sqrt \frac {2} {5}\] R
The radius of gyration and slenderness ratio
The 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.
How does radius differ from the radius of gyration?
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 gyroradius helps in describing the distribution of any cross-sectional area around the centroidal axis in the body mass. In the field of structural engineering, it is used to estimate the stiffness of a column. If the principle moment of two-dimensional gyration is not equal, the column will tend to buckle around the axis having the smaller principle moment. If we talk about engineering where continuous bodies of matter are generally the objects that we study, the radius of gyration will be calculated as an integral.
Application in the field of polymer physics
The term radius of gyration describes the spatial dimensions of a given polymer chain. At times you can be provided with a polymer chain in a dimension that is spatial, meaning a dimension for the measure of spatial extent, especially with height or length. In such cases, you can use the radius of gyration for the problems.
Application in the field of mechanics
The radius of gyration about a given axis can be calculated in the terms of the moment of inertia around the axis, and the total mass m.
Application in geographical data analysis
In data analysis, this radius of gyration is used to calculate various different statistics including the spread of geographic locations. Recently these locations have been collected from the social media users to investigate the typical mention of a user. This can be useful for you to understand how certain groups of users on social media use the given platform.
What is the international unit of gyroradius or radius of gyration?
The SI unit for gyroradius is the length (that is shown in inches or millimetres or feet). It is shown as the sq. root of inertia divided by the object’s area.
On what factors does the radius of gyration depend?
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.
Size: The radius of gyration varies with the change in the size of the object.
Shape: The shape of the body affects the radius of gyration of that particular body.
Configuration of rotation axis: The axis of rotation, that is, the axis along which a body is rotating will be an important factor that will decide the radius of gyration of a particular body in the dimensional plane.
Position: The radius of gyration for a body depends upon the position of the particular object in the plane.
How is the gyroradius 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. Since the rotational axis for a particular body in the plane will differ or change then the radius or radius of gyration will also change simultaneously.
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.
The significance of “radius of gyration”
The radius of gyration is significant in the calculation of the clasping load of a beam or compression.
It is also useful in the distribution of power between the cross-sections of a given column
The radius of gyration is helpful in comparison to the performance of different kinds of structural shapes at the time of the compression.
Lesser value of “radius of gyration” is effective in performing structural analysis.
Lesser value of “radius of gyration” displays the rotational axis at which the column clasps.
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. Besides this, you use the radius of gyration in the field of geographical analysis or other analysis purposes. You can also use this radius of gyration for application in polymer physics, structural engineering, and mechanics. One thing that you need to take into account is that the SI unit of gyroradius measurement is mm.
FAQs on Radius of Gyration
1. What does the radius of gyration mean?
A radius of gyration is a length that represents the distance in a rotating system between the point about which it is rotating and the point to or from which the transfer of energy is going to have a maximum effect. For the radius of gyration, the symbol we use is particularly k or r. The radius of gyration is an important topic that finds its use in a number of fields like polymer physics, mechanics, systematics, etc. In all such fields, we can find the use of the radius of gyration.
2. What is gyration in terms of physics?
In physics, the radius of gyration for a body around a particular axis is termed the radial distance of a point from its axis of rotation. At this particular point, you can assume that the whole mass of a body is concentrated, that is at its moment of inertia. The concentration of mass in the body at this given axis of the moment of inertia is the same as that of the distribution of mass in the body.
3. From where can I get notes for the radius of gyration?
The radius of gyration is a tricky topic for which you will require a brief understanding and along with this, you need to work with your basic concepts. You cannot memorize the whole information that is provided in your books and for that, revision notes or simple study notes will help you. You can get such study notes easily from the internet. You can download it simply from the Vedantu app or website of Vedantu, that is vedantu.com. Vedantu provides you with notes that will help you to clear your concept well.
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5. Is the radius of gyration an important concept?
Yes, if you are one who is preparing for their board examinations and JEE examinations or even for NEET examinations, this topic is of great importance. This chapter carries direct questions with a good weightage of marks. To improve your score in the examination and your performance, you need to learn this concept by heart and you should have good knowledge of this topic.
