Chapter 31 : Kinetics of Motion of Rotation „„„„„ 627

=

523

4

2

• 253

r

r

mxrx

rx

+

π⎡⎤

⎢⎥+

⎣⎦

=

52

(^82) 0.4 2
15 5
mr Mr
Mr
π

4 3
...
3
mr
M
⎛⎞π
⎜⎟⎜⎟=
⎝⎠
Note: Since the sphere is symmetrical, therefore the mass moment of inertia of a sphere
about any axis is the same.
31.12.UNITS OF MASS MOMENT OF INERTIA
We have already discussed that the mass moment of inertia of a body is numerically equal to
its mass and the square of distance between the centre of gravity of the mass and the point about
which the mass moment of inertia is required to be found out. Therefore units of mass moment of
inertia depend upon the mass of the body and distance. If mass is in kg and the distance in metres,
then the units of mass moment of inertia will be kg-m^2. Similarly, it may be kg-mm^2 etc.
31.13.RADIUS OF GYRATION
If the entire mass of a given body be assumed to be concentrated at a certain point, at a
distance k from the given axis, such that
Mk^2 =I ...(where I is the mass moment of inertia of the body)
∴ k=
I
m
The distance k is called radius of gyration. Thus the radius of gyration of a body may be
defined as the distance from the axis of reference where the whole mass (or area) of a body is
assumed to be concentrated.
The suffixes such as X-X or Y-Y are, usually attached to k, which indicate the axis about which
the radius of gyration is evaluated. Thus kXX will indicate the radius of gyration about X-X axis. In
the following lines, we shall discuss the radius of gyration of some important sections :

1. Radius of gyration of a thin circular ring

We know that mass moment of inertia of a thin circular ring about the central axis,

IZZ=Mr^2 or Mk^2 = Mr^2

∴ k=r

2. Radius of gyration of a circular lamina

We know that mass moment of inertia of a circular lamina about the central axis,

IZZ=

2

2

Mr

or

2

2

2

Mr

Mk =

∴ k=

2

r

or

2

(^22) 0.5
2
r
kr==

3. Radius of gyration of a solid sphere

We know that the mass moment of inertia of a solid sphere about any axis,

I= 0.4 Mr^2 or Mk^2 = 0.4 Mr^2

∴ k=r× 0.4 or k^2 = 0.4 r^2