System of Particles and Rotational Dynamics

Radius of Gyration

The snapshot of the moment of inertia of a body about an axis is in some cases spoke to utilize the radius of gyration. Now, what actually is the radius of gyration? We can characterize the radius of gyration as the fanciful good ways from the centroid at which the territory of cross-area is envisioned to be engaged at a point to get a similar moment of inertia. It has a symbol k.

radius of gyration

What is the Radius of Gyration?

The radius of gyration or gyradius of a body is always about an axis of rotation. It is characterized as the spiral distance to a point which would have a moment of inertia. The radius of gyration is a geometric property of a rigid body. For example, the centre of mass.  It is equivalent to the body’s real dissemination of mass. If the all-out mass of the body is concentrated.

Applications of the Radius of Gyration in Structural Engineering

Regarding material science in physics, the radius of gyration refers to the technique for the dispersion of various segments of the item present around an object. When seen regarding the moment of inactivity, the radius of gyration is the perpendicular distance taken from the rotational axis to an explicit point of mass.

In primary designing, two-dimensional range of gyration is in use to portray the dispersion of cross-sectional zone. It is in a segment around its centroidal pivot with the mass of the body. The radius of gyration is given by the accompanying formula, i.e.

\(R^{2}\) = \(\frac{I}{A}\)

Or R=\(\sqrt{\frac{I}{A}}\)

Where І is the second moment of area and A is the total cross-sectional area of the object.

The gyration radius is helpful in the assessment of the solidness of a segment. On the off chance that the vital snapshots of the two-dimensional gyration tensor are not equivalent. The segment will, in general, be clasp around the pivot with the more modest head second. For instance, a segment with a curved cross-section will, in general, be lock toward the more modest semi pivot.

In designing, where constant groups of the issue are commonly the objects of the examination. The radius of gyration is generally determined and calculated as a fundamental integral.

The Radius of Gyration Formula

Numerically the radius of gyration is the root mean square distance of the object’s parts from either its focal point of mass or a given axis. It is contingent upon the important application. It is the perpendicular distance from the mass toward the pivot of rotation. The radius of gyration can represent a trajectory of the direction of a moving point as a body. At that point range of gyration can be in use to portray the regular distance travelled by this point.

The formula of moment inertia i.e. in terms of the radius of gyration is:

I = \(mk^{2}\) …..(1)

here I is the moment of inertia and m is the mass of the body

Thus, the radius of gyration is as follows

K = \(\sqrt{\frac{I}{m}}\) ………(2)

The unit of the radius of gyration is mm. By knowing the radius of gyration, the moment of inertia of any rigid body by equation (1).

Consider a body having n number of particles each with a mass of m. Let the perpendicular distance from the pivot of the rotation. It is given by \(r_{1}\), \(r_{2}\), \(r_{3}\) … \(r_{n}\). We realize that the moment of inertia as far as radius of gyration is given by the condition (1). Subbing the qualities in the condition, we get the moment of inertia of the body as follows

I = \(m_{1}r_{1}^{2} + m_{2}r_{2}^{2} + m_{3}r_{3}^{2}\) + ……. + \(m_{n}r_{n}^{2}\)……… (3)

If all the particles have the same mass then equation (3) can be rewritten as:

I = \(m(r_{1}^{2} + r_{2}^{2} + r_{3}^{2}\) + …….. + \(r_{n}^{2}\)

= \(\frac{mn(r_{1}^{2} + r_{2}^{2} + r_{3}^{2} +…….. + r_{n}^{2} )}{n}\)

Thus, we can write mn as the M which shows the total mass of the body.

So, the equation will be,

= \(\frac{M(r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + ……….+ r_{n}^{2} )}{n}\) ………… (4)

From equation (4) given above, we get

\(Mk^{2}\) = \(\frac{M(r_{1}^{2} + r_{2}^{2} + r_{3}^{2} +………+ r_{n}^{2} )}{n}\)

Or, k^{2} = \(\sqrt{\frac{r_{1}^{2} + r_{2}^{2} + r_{3}^{2} +  ……..+ r_{n}^{2}}{n}}\)

From the equation above, we can say that the radius of gyration actually the root-mean-square distance of different particles of the body. It is from the axis of rotation.

Use of Radius of Gyration

The radius of gyration is in use compare how the various structural shapes will behave under compression along an axis. It is used to predict buckling in a compression beam or member.

The Radius of Gyration for a Thin Rod

The moment of inertia (MOI) of a uniform rod of length I and the mass M. It is about an axis through the centre. Forming a 90-degree angle to the length it is:

I or moment of Inertia = \(\frac{Ml^{2}}{12}\)

If K is the radius of the thin rod about an axis, then the equation will be

I = \(Mk^{2}\)

By equating the value of moment of Inertia, we get \(Mk^{2}\) = \(\frac{Ml^{2}}{12}\)

On cancelling M from both the sides and solving, we get

\(k^{2}\) = \(\frac{l^{2}}{12}\)

By taking the square root, we get:

K = \(\frac{l}{\sqrt{12}}\)

L = \(2\sqrt{3}K\)

The radius of gyration of a solid sphere

The moment of inertia or MOI of any solid sphere that has a mass M and radius R is:

\(I_{axis}\) = \(Mk^{2}\) (k is the radius of gyration) ………..(1)

I for solid sphere about tangent = \(\frac{2}{5}MR^{2}+MR^{2}\) = \(\frac{7}{5}MR^{2}\)

By equating both the moment of inertias

Therefore, \(\frac{7}{5}MR^{2}\) = \(Mk^{2}\)

k = \(\sqrt{\frac{7}{5}}R\)

FAQs about the Radius of Gyration

Q.1. The significance of “radius of gyration”

Answer: Following are the significant points:

  1. The radius of gyration is huge in the figuring of the catching heap of a beam or pressure.
  2. It is additionally valuable in the appropriation of intensity between the cross-areas of a given section.
  3. The range of gyration is useful in contrast with the exhibition of various types of primary shapes at the hour of the pressure.
  4. Lesser estimation of the radius of gyration is successful in performing the primary investigation.
  5. Lesser estimation of the radius of gyration shows that the rotational axis at which the segment catches.

Q.2. What is the international unit of gyradius or radius of gyration?

Answer: The SI unit for the radius of gyration is the length or in inches or millimetres or in feet. It is the sq. root of inertia divided by the area of the object.

Q.3. On what factors radius of gyration depends?

Answer: A portion of the elements that impact the estimation of the radius of gyration is the size and state of the body. It is the arrangement of the rotational axis and position. It likewise relies upon the mass appropriation regarding the rotational axis of the body.

Q.3. How is the radius of gyration termed as a constant quantity?

The value of the radius of gyration or the radius is not fixed or constant. Its value depends upon the rotational axis and the distribution of the mass of the body about the axis.

Q.4. How is the “radius of gyration” defined for a regular solid sphere?

Answer: The radius of gyration is the square root of the average squared distance of a sphere object from the midpoint of the mass of the body.

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