Radius of Gyration

Rotational Dynamics · Maharashtra HSC

Theory

Definition

The radius of gyration of a body about a given axis is defined as the distance between the axis of rotation and a point at which the whole mass of the body is supposed to be concentrated, so as to possess the same moment of inertia as that of the body about the same axis.

$I = MK^2, \qquad K = \sqrt{\dfrac{I}{M}}$

Its SI unit is the metre and its dimensions are $[M^0L^1T^0]$ .

Physical Significance

Theoretical calculation of moment of inertia is possible only for mathematically integrable geometrical shapes.
However, experimentally one can determine the moment of inertia of any object.
It depends upon the mass of that object and upon how that mass is distributed from or around the given axis of rotation.
If one is interested in knowing only the mass distribution around the axis of rotation, the moment of inertia of any object can be expressed as $I = MK^2$ , where $M$ is the mass of that object.
It means that the mass of that object is effectively at a distance $K$ from the given axis of rotation.
In this case, $K$ is defined as the radius of gyration of the object about the given axis of rotation.
In other words, if $K$ is the radius of gyration for an object, $I = MK^2$ is the moment of inertia of that object.

So the value of $K$ is decided purely by the distribution of the mass: the larger the value of $K$ , the farther the mass is from the axis.

Comparison of a Ring and a Disc

Consider a uniform ring and a uniform disc of the same mass $M$ and the same radius $R$ , with moments of inertia $I_r$ and $I_d$ and radii of gyration $K_r$ and $K_d$ about their own axes.

$I_r = MR^2 = MK_r^2 \quad\Rightarrow\quad K_r = R$

$I_d = \dfrac{1}{2}MR^2 = MK_d^2 \quad\Rightarrow\quad K_d = \dfrac{R}{\sqrt2}$

$\therefore\ K_d < K_r$

This shows mathematically that $K$ is decided by the distribution of mass. In a ring the entire mass is distributed at the distance $R$ , while in a disc the mass is distributed between $0$ and $R$ . Among any objects of the same mass and radius, the ring has the largest radius of gyration and hence the maximum moment of inertia.

What K Depends On, and What It Does Not

It depends on the shape of the body, on the distribution of its mass about the chosen axis, and on the position and orientation of that axis.
It does not depend on the mass of the body. Two bodies of identical shape and size but different masses have the same $K$ about corresponding axes.
Being a length, it can be compared with the centre of mass: both are single equivalent stand-ins for a whole distribution of matter. The centre of mass is the point where the whole mass may be taken to act for translational purposes; the radius of gyration is the distance at which the whole mass may be taken to be concentrated for rotational purposes about a given axis. The difference is that the radius of gyration is always defined with respect to an axis, and it changes when the axis changes, whereas the centre of mass is a fixed point of the body.
If a uniform ring and a uniform disc have the same radius of gyration, then $R_{ring} = \dfrac{R_{disc}}{\sqrt2}$ , i.e. the disc must have $\sqrt2$ times the radius of the ring.

Useful Values

$K_{ring} = R, \qquad K_{disc} = \dfrac{R}{\sqrt2}, \qquad K_{solid\ sphere} = R\sqrt{\dfrac{2}{5}}, \qquad K_{hollow\ sphere} = R\sqrt{\dfrac{2}{3}}$

$K_{rod\ about\ centre} = \dfrac{L}{\sqrt{12}}, \qquad K_{rod\ about\ one\ end} = \dfrac{L}{\sqrt3}$

Standard Results

$I = MK^2$ , $\quad K=\sqrt{\dfrac{I}{M}}$ , SI unit metre
Parallel axes theorem in terms of $K$ : $K_O^2 = K_C^2 + h^2$
If $I = n(MR^2)$ , then $\dfrac{K^2}{R^2} = n$
$K$ is independent of the mass of the body; it depends only on shape, mass distribution and the axis

Theory Questions Asked on This Subtopic

Real board questions. Answers are at the end of the page.

1
Explain the physical significance of radius of gyration.
Maharashtra HSC board question — asked in July 2017Answer 1 ↓
2
Define radius of gyration and give its physical significance.
Maharashtra HSC board question — asked in July 2018; March 2008; October 2013Answer 2 ↓
3
Define radius of gyration. Write its physical significance.
Maharashtra HSC board question — asked in March 2019Answer 3 ↓

Answers & Solutions

Work the questions above first, then check yourself here. Each answer is set out the way it should be written in the paper.

1.↑ back to the question
i. Calculation of moment of inertia is possible only for mathematically integrable geometrical shapes. ii. However, experimentally one can determine the moment of inertia of any object. iii. It depends upon mass of that object and how that mass is distributed from or around the given axis of rotation. iv. If one is interested in knowing only the mass distribution around the axis of rotation, we can express moment of inertia of any object as I = MK $^2$ , where M is mass of that object. v. It means that the mass of that object is effectively at a distance K from the given axis of rotation. vi. In this case, K is defined as the radius of gyration of the object about the given axis of rotation. vii. In other words, if K is radius of gyration for an object, $I = MK^2$ is the moment of inertia of that object.
2.↑ back to the question
Definition: Radius of gyration of a body about its given axis is defined as the distance between the axis of rotation and a point at which the whole mass of the body is supposed to be concentrated, so as to possess the same moment of inertia as that of body about the same axis. Physical significance: [From 1.6, Q1]
i. Calculation of moment of inertia is possible only for mathematically integrable geometrical shapes. ii. However, experimentally one can determine the moment of inertia of any object. iii. It depends upon mass of that object and how that mass is distributed from or around the given axis of rotation. iv. If one is interested in knowing only the mass distribution around the axis of rotation, we can express moment of inertia of any object as I = MK $^2$ , where M is mass of that object. v. It means that the mass of that object is effectively at a distance K from the given axis of rotation. vi. In this case, K is defined as the radius of gyration of the object about the given axis of rotation. vii. In other words, if K is radius of gyration for an object, $I = MK^2$ is the moment of inertia of that object.
3.↑ back to the question
[From 1.6, Q1 (Definition)]
i. Calculation of moment of inertia is possible only for mathematically integrable geometrical shapes. ii. However, experimentally one can determine the moment of inertia of any object. iii. It depends upon mass of that object and how that mass is distributed from or around the given axis of rotation. iv. If one is interested in knowing only the mass distribution around the axis of rotation, we can express moment of inertia of any object as I = MK $^2$ , where M is mass of that object. v. It means that the mass of that object is effectively at a distance K from the given axis of rotation. vi. In this case, K is defined as the radius of gyration of the object about the given axis of rotation. vii. In other words, if K is radius of gyration for an object, $I = MK^2$ is the moment of inertia of that object. [From 1.6, Q2 (Physical significance)] Definition: Radius of gyration of a body about its given axis is defined as the distance between the axis of rotation and a point at which the whole mass of the body is supposed to be concentrated, so as to possess the same moment of inertia as that of body about the same axis. Physical significance: [From 1.6, Q1]
i. Calculation of moment of inertia is possible only for mathematically integrable geometrical shapes. ii. However, experimentally one can determine the moment of inertia of any object. iii. It depends upon mass of that object and how that mass is distributed from or around the given axis of rotation. iv. If one is interested in knowing only the mass distribution around the axis of rotation, we can express moment of inertia of any object as I = MK $^2$ , where M is mass of that object. v. It means that the mass of that object is effectively at a distance K from the given axis of rotation. vi. In this case, K is defined as the radius of gyration of the object about the given axis of rotation. vii. In other words, if K is radius of gyration for an object, $I = MK^2$ is the moment of inertia of that object.