Kinetic energy of a particle or body represents the form of energy that arises from motion. We are aware that kinetic energy of a particle in translation is given by the expression :

K
=
1
2
m
v
2
K
=
1
2
m
v
2

(7)In pure rotation, however, the rigid body has no "over all" translation of the body. However, the body in rotation must have kinetic energy as it involves certain motion. A closer look on the rotation of rigid body reveals that though we may not be able to assign translation to the rigid body as a whole, but we can recognize translation of individual particles as each of them rotate about the axis in circular motion with different linear speeds. The speed of a particle is given by :

v
i
=
ω
r
i
v
i
=
ω
r
i

Thus, kinetic energy of an individual particle is :

K
i
=
1
2
m
i
v
i
2
K
i
=
1
2
m
i
v
i
2

where "
K
i
K
i
" is the kinetic energy of "i" th particle having a speed "
v
i
v
i
". In terms of angular speed, the kinetic energy of an individual particle is :

K
i
=
1
2
m
i
(
ω
r
i
)
2
K
i
=
1
2
m
i
(
ω
r
i
)
2

Now, the kinetic energy of the rigid body is sum of the kinetic energies of the particles constituting the rigid body :

K
=
∑
K
i
=
∑
1
2
m
i
ω
2
r
i
2
K
=
∑
K
i
=
∑
1
2
m
i
ω
2
r
i
2

We note here that angular speeds of all particles constituting the body are same. Hence, the constant "1/2" and "
ω
2
ω
2
" can be taken out of the summation sign :

K
=
1
2
ω
2
∑
m
i
r
i
2
K
=
1
2
ω
2
∑
m
i
r
i
2

However, we know that :

I
=
∑
m
i
r
i
2
I
=
∑
m
i
r
i
2

Combining two equations, we have :

K
=
1
2
I
ω
2
K
=
1
2
I
ω
2

(8)
This is the desired expression of kinetic energy of a rigid body rotating about a fixed axis i.e. in pure rotational motion. The form of expression of the kinetic energy here emphasizes the correspondence between linear and angular quantities. Comparing with the expression of kinetic energy for translational motion, we find that "moment of inertia (I)" corresponds to "mass (m)" and "linear speed (v)" corresponds to "angular speed (ω)".

We can also interpret the result obtained above from a different perspective. We could have directly inferred that expression of kinetic energy in rotation should have an equivalent form as :

K (Kinetic energy)
=
1
2
x
(inertia)
x
(speed)
2
K (Kinetic energy)
=
1
2
x
(inertia)
x
(speed)
2

In rotation, inertia to the rotation is "moment of inertia (I)" and speed of the rigid body is "angular speed (ω)". Substituting for the quantities, we have the expression for kinetic energy of rigid body in rotation as :

K
=
1
2
I
ω
2
K
=
1
2
I
ω
2

Comparing this equation with the expression of the sum of kinetic energy of individual particles as derived earlier :

K
=
1
2
ω
2
∑
m
i
r
i
2
K
=
1
2
ω
2
∑
m
i
r
i
2

Clearly,

I
=
∑
m
i
r
i
2
I
=
∑
m
i
r
i
2

This conclusion, thus, clearly establishes that the expression as given by
∑
m
i
r
i
2
∑
m
i
r
i
2
indeed represents the inertia of the rigid body in rotation.

Though it is clear, but we should emphasize that expressions of kinetic energy of rotating body either in terms of angular speed or linear speed are equivalent expressions i.e. two expressions measure the same quantity. Two expressions do not mean that the rotating body has two types of kinetic energy.