Radius of Gyration: Formula & Moment of Inertia

The radius of gyration can be defined as an imaginary distance from the centroid wherein the area of the cross-section is assumed to be focused at a point to yield the same moment of inertia. It is represented by the letter k. The radius of gyration is sometimes used to express a body’s moment of inertia about an axis. The radius of gyration can be represented by:

\(k = \sqrt{\frac{I}{A}}\)

The radius of gyration unit is mm. In case the radius of gyration is known, one can determine the moment of inertia of any complex body. The radius of gyration is characterized as the spiral distance to a point which could possess a moment of inertia. It is typically a geometric property of a rigid body.

Key Terms: Radius of Gyration, Axis of Rotation, Mass, Spiral Distance, Moment of Inertia, Centroid

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Radius of Gyration 

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A radius of gyration, or gyradius of a body, is always centred on an axis of rotation.

• Radius of Gyration can be defined as the spiral distance to a point with a moment of inertia.
• A rigid body’s radius of gyration is a geometric attribute.
• Consider the centre of mass. It is analogous to the body’s actual mass dissemintion, if the entire body’s mass is concentrated. 

Radius of Gyration

Moment of Inertia

Moment of inertia can be expressed as “the quantity defined by the body resisting angular acceleration, that is the mass’ sum of the product of every particle with its square of a distance from the axis of rotation.”

• It can simply be defined as a quantity which decides the amount of torque required for a specific angular acceleration in a rotational axis.
• Moment of Inertia is also often termed angular mass or rotational inertia. 
• The SI unit of moment of inertia is kg m².

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Significance of Radius of Gyration

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The following are the key features of Radius of Gyration:

• The radius of gyration is important when calculating the catching heap of a beam or pressure.
• It is also useful in appropriating intensity between cross-sections of a given section.
• The range of gyration is useful in contrast to the display of various forms of fundamental shapes at the pressure hour.
• The primary investigation is successful with a lower estimate of the radius of gyration.
• A lesser estimation of the radius of gyration also reveals the rotational axis at which the segment catches.

Circular Rod and Rectangular Beam

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Radius of Gyration Formula

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The radius of gyration is measured as the root mean square distance of the object’s components from its focal centre of mass or a specific axis. It is the perpendicular distance from the mass to the rotational pivot. A trajectory of the direction of a moving point as a body can be represented by the radius of gyration. The range of gyration at that location can be used to depict the regular distance travelled by this point.

The moment inertia formula, expressed in terms of the radius of gyration, is as follows:

I = mk² … (1)

Here,

• I = moment of inertia
• m = mass of the body

Thus, the radius of gyration is as follows:

The radius of gyration is measured in millimetres (mm). If the radius of gyration is known, the moment of inertia of any rigid body can be calculated (1).

Consider a body with n particles, each with a mass of m. Allow for the rotation’s perpendicular distance from the pivot. It is represented by r₁, r₂, r₃,… r_n. We see that the condition gives the moment of inertia as far as the radius of gyration is concerned (1). By substituting the qualities in the condition, we get the body’s moment of inertia as follows.

\(\begin{array}{l}I = m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + …. + m_nr_n^2 \,\,\, (3)\end{array}\)

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,

= \(I = M \frac{(r_1^2 + r_2^2 + r_3^2 + … \, + r_n^2)}{n}\,\,\, (4)\)

From equation (4) given above, we get

\(MK^2 = M \left ( \frac{r_1^2 + r_2^2 + r_3^2 + … \, + r_n^2}{n} \right )\)

Or, \( K = \sqrt{ \frac{r_1^2 + r_2^2 + r_3^2 + … \, + r_n^2}{n}}\)

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

Applications of Radius of Gyration in Structural Engineering

The radius of gyration refers to the strategy for dispersing multiple segments of the item present around an object in material science.

• The radius of gyration is the perpendicular distance between the rotational axis and an explicit point of mass when viewed in terms of the moment of inactivity.
• The dispersion of a cross-sectional zone is depicted in primary design using a two-dimensional range of gyration.
• It is in a segment with the mass of the body around its centroidal pivot.
• The radius of gyration is calculated using the formula below:

Where,

• I is the object’s second moment of area
•  A is the object’s total cross-sectional area

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Slenderness Ratio

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The square root of the ratio of inertia to the area of the material is the radius of gyration.

• The imaginary distance determined from the place where the cross-sectional area is meant to be concentrated at a point is denoted by the value obtained from it.
• This assists in achieving the same level of inertia.
• It is calculated by determining the slenderness of a column’s cross-sectional area.

The radius of gyration usually shows up in two places:

1. Strength of Materials: Herein, a two-dimensional radius of gyration is utilized and is further considered as area property. Radius of gyration can be given by the relation: \(k=\sqrt{\frac{I}{A}}\)
2. Mechanics: The radius of gyration, here, is about an axis of rotation that can be estimated using mass moment of inertia. The relation can be shown by: \(k=\sqrt{\frac{I}{M}}\)

Radius of Gyration and Slenderness Ratio

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Things to Remember:

• The radius of gyration is defined as the spiral distance to a point with a moment of inertia.
• Radius of Gyration can be determined by moment of inertia, mass, shape, size and axis rotation. 
• Radius of Gyration can be represented by K. 
• It is used to examine how different structural shapes will behave when compressed along one axis. For example: Catching heaps of pressure, etc. 
• The formula of Radius of Gyration is \(k=\sqrt{\frac{I}{m}}\,\,\, \).

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Previous Year Questions

1. Let the moment of inertia of a hollow cylinder of length … [JEE Mains 2019]
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4. In a physical balance working on the principle of moments … [JEE Mains 2017]
5. An L-shaped object, made of thin rods of uniform … [JEE Mains 2019]
6. Consider a thin uniform square sheet made of a rigid material … [JEE Mains 2015]
7. One quarter section is cut from a uniform circular disc … [JEE Advanced 2000]
8. A thin wire of length L and uniform linear mass density p … [JEE Advanced 2000]
9. The torque τ on a body about a given point is found to be equal … [JEE Advanced 1998]
10. Two balls, having linear momenta … [JEE Advanced 2008]