A flywheel rotating about a fixed axis has a kinetic energy of \[360{\text{ }}J\]. When its angular speed is \[30{\text{ }}rad/sec\] . The moment of inertia of the wheel about the axis of rotation is.
A. \[0.6kg{m^2}\]
B. \[0.15kg{m^2}\]
C. \[0.8kg{m^2}\]
D. \[0.75kg{m^2}\]

Hint: The moment of Inertia is given by the product of mass and the square of the radius. \[I = M{R^2}\]
2)Use the angular speed and rotational kinetic energy is given from this given data to calculate I.
\[E = \dfrac{1}{2}I{\omega ^2}\]

Complete step by step answer:
Here, we are given the Angular speed (\[\omega \]) of the flywheel, and the kinetic energy \[\left( E \right)\] of the rotating flywheel.

\[\omega \]\[ = {\text{ }}30{\text{ }}rad{\text{ }}/s\]
Rotational kinetic energy,
\[E = {\text{ }}360{\text{ }}J\]

Assuming that the moment of inertia of the flywheel as we know from the relation between Kinetic energy and moment of Inertia as follows:
Rotational kinetic energy: = \[E = \dfrac{1}{2}I{\omega ^2}\]
Putting the given values from above in this equation, we have,
\[
  360 = \dfrac{1}{2}I{(30)^2} \\
  I = \dfrac{{360 \times 2}}{{{{30}^2}}} \\
  I = 0.8kg{m^2} \\
\]

We found out the value of moment of Inertia of the flywheel is \[0.8kg{m^2}\] .

So, the correct answer is “Option C”.

Note:
The moment of Inertia is different for different shaped bodies, Hence the kinetic energy which depends on the moment of inertia will be different.
As an alternative approach, we can calculate the moment of inertia if the mass of the body was given as well as the shape of it, or the angular momentum of the body. But, we are given with rotational kinetic energy and angular speed, which is justified.