Radius of Gyration - What is Radius of Gyration, Derivation, Formula

Edited By Vishal kumar | Updated on Sep 24, 2024 09:42 PM IST

Radius of gyration geometry depends on moment of inertia of the body, mass of the body, figure and size of the body, and position of the axis of rotation. In this article we will discuss about, what is Radius? Write SI units for Radius. What is radius of gyration? Define radius of gyration. What is physical significance of radius of gyration? Give its SI unit or unit and dimension. What are the functions of radius of gyration? What is dimensional formula for radius of gyration? What is the function of gyrase? So, let’s see,

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What is radius?
What is radius of gyration?
Formula of Radius of gyration:
Radius of gyration is the root mean sq. distance of particles from axis formula
Derivation
Physical significance:

Radius of Gyration - What is Radius of Gyration, Derivation, Formula

What is radius?

A line segment expanding from the middle of a circle or sphere to the circumference or bounding surface. SI unit of radius, radius is one dimensional length, so you can measure radius in meters (m).

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What is radius of gyration?

The meaning of radius of gyration or gyration is the distance from an axis at which the mass of a body could also be assumed to be concentrated and at which the instant of inertia are going to be adequate to the instant of inertia of the particular mass about the axis, adequate to the root of the quotient of the instant of inertia and therefore the mass. The meaning of radius of gyration in hindi is “परिभ्रमण त्रिज्या” .

Definition: The moment of inertia of a body about an axis is usually represented using the radius of gyration. Now, Radius of gyration is defined as the distance axis of rotation to some extent where the entire body is meant to concentrate. Radius of gyration is a constant quantity.

In physics, Class 11, the radius of gyration defines as the imaginary distance from the centroid at which the world of cross-section is alleged to be focused at some extent so as to get an equivalent moment of inertia. Now we see, what is moment of inertia? moment of inertia, in physics, quantitative measure of the rotational inertia of a body—i.e., the opposition that the body exhibits to having its speed of rotation about an axis altered by the appliance of a torque (turning force). The axis could also be internal or external and should or might not be fixed.

Symbol of radius of gyration or it is denoted by k.

SI unit: meter

CGS unit: cm

Dimensional analysis: L

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Formula of Radius of gyration:

The formula of moment inertia in terms of the radius of gyration is represented as:

I=mK2

Where, I = Moment of Inertia

m = mass of body

K = Radius of gyration

By knowing the radius of gyration, we can find easily the moment of inertia of any complex body

Therefore, the formula of Radius of gyration is represented as:

K= Im

Mathematically the radius of gyration of mechanics is that the foundation mean sq. distance of the object's segments from either its center of mass or a particular axis, looking forward to the relevant application. It is evenly the perpendicular distance from point mass to the axis of rotation. One can act a trajectory of a moving point as a body. Then radius of gyration also is used to specify the typical distance travelled by this point.

Let it consider, a body consists of n particles each of mass m. Let r1, r2, r3, r4…….rn be their perpendicular axis be there. Then, the moment of inertia I of the body concerning the axis of rotation is

I= m1r12+ m2r22+…+mnrn2

If all the masses are the same m, then the moment of inertia is I= m(r12+ r22+…+rn2)

Since m=M/n (M existence to the total mass of the body),

I= M(r12+ r22+…+rn2)/n

We get,

MRg2= M(r12+ r22+…+rn2)/n

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Radius of gyration is the root mean sq. distance of particles from axis formula

Rg2= (r12+ r22+…+rn2)/n

Therefore, the radius of gyration of a body a couple of given axis might also be outlined because the root mean sq. distance of the assorted particles of the body from the axis of rotation. It’s additionally called a live of the manner during which the mass of a rotating rigid body is distributed concerning its axis of rotation.

Example: The ratio of the radii of gyration of a circular disc about a tangential axis within the plane of the disc and of a circular ring of the equivalent to radius about a tangential axis within the plane of the ring is_____.

Solution: Idisk=54MR2= MKdisk2

Iring=32MR2= MKring2KdiskKring = 56

Derivation

To identify the 2 definitions of Rg2 are identical, first multiply out the sum within the first definition:

1641288946283

Sew up the summation over the last two terms and using the definition of r mean gives:

1641288946368

Physical significance:

1. If the particles of the body are distributed on the brink of the axis of rotation, the radius of gyration is a smaller amount.

2. If the particles are distributed far away from the axis of rotation, the radius of gyration is more.

The knowledge of mass and radius of gyration of the body a few given axis of rotation gives the worth of its moment of inertia about an equivalent axis, albeit we don't know the particular shape of the body. The radius of gyration is employed to match how various structural shapes will behave under compression along an axis. it's wont to predict buckling during a compression beam or member.

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NCERT Physics Notes:

Frequently Asked Questions (FAQs)

1. How does radius differ from the radius of gyration?

The radius of any sphere that touches some extent within the curve and possesses an equivalent curvature and tangent at that time is considered the radius of curvature. In terms of mathematics, the “radius of gyration” is considered the root of the mean square radius of the various parts of the thing from the central point of its mass or any given axis. This relies on any relevant application.

2. How is that the gyradius termed as a constant quantity?

The value of “Radius of gyration" or radius isn't fixed. Its value depends on the rotational axis and therefore the distribution of body mass about the axis.

3. Applications within the field of structural engineering

In the field of structural engineering, the two-dimensional gyradius helps in describing the distribution of any cross-sectional area round the centroidal axis within the body mass.

4. How is that 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 = 2/5 MR2 ………….. (1)

If K implies the radius of a solid sphere, then

I = MK2…………………. (2)

On combining both the equations 1 and a couple of, the equation also can be written as

MK2 = 2/5 MR2

Canceling M and taking root on both the edges, our equation now becomes:

K = root of 2/5 R

5. What is the appliance of radius of gyration and Application within the field of polymer physics?

Calculation of the radius of gyration is useful in various ways. it's useful in comparing how different structural shapes behave under the body's compression along a rotational axis. It’s also utilized in forecasting buckling during a beam or compression member. One thing that you simply got to take under consideration is that the SI unit of gyradius measurement is mm. The terminology of radius of gyration explain the spatial dimensions of a given polymer chain.