Radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there.
Mathematically the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application. It is actually the perpendicular distance from point mass to the axis of rotation. One can represent a trajectory of a moving point as a body. Then radius of gyration can be used to characterize the typical distance travelled by this point.
Suppose a body consists of
n
{\displaystyle n}
particles each of mass
m
{\displaystyle m}
. Let
r
1
,
r
2
,
r
3
,
…
,
r
n
{\displaystyle r_{1},r_{2},r_{3},\dots ,r_{n}}
be their perpendicular distances from the axis of rotation. Then, the moment of inertia
I
{\displaystyle I}
of the body about the axis of rotation is
I
=
m
1
r
1
2
+
m
2
r
2
2
+
⋯
+
m
n
r
n
2
{\displaystyle I=m_{1}r_{1}^{2}+m_{2}r_{2}^{2}+\cdots +m_{n}r_{n}^{2}}
If all the masses are the same (
m
{\displaystyle m}
), then the moment of inertia is
I
=
m
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
{\displaystyle I=m(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})}
.
Since
m
=
M
/
n
{\displaystyle m=M/n}
(
M
{\displaystyle M}
being the total mass of the body),
I
=
M
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
/
n
{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
From the above equations, we have
M
R
g
2
=
M
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
/
n
{\displaystyle MR_{g}^{2}=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
Radius of gyration is the root mean square distance of particles from axis formula
R
g
2
=
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
/
n
{\displaystyle R_{g}^{2}=(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation. It is also known as a measure of the way in which the mass of a rotating rigid body is distributed about its axis of rotation.