Is there an upper limit on the radius of a rotating wheel? Is there an upper limit on the radius of a real wheel which is rotating at an Angular frequency of $\,\omega \,$ along its axis, such that we just require a finite amount of energy to rotate it? Why/why not?
e.g. if a wheel which is rotating at an angular speed of $\omega  = 3 \times {10^5}\,{{rad} \over {\sec }}$ could have a radius of $r = 2 \times {10^3}\,meters$?? 
 A: Yes, there is. The numbers you gave would result in the outer rim of the wheel moving at twice the speed of light. That's just not possible.
A: I'm gonna go along with the "may have been caused by a misunderstanding" and tell you about something else that will help clarify it.
Consider an arbitrarily long spoon. You can hold and flick it at such angular velocity that its endpoint, it may seem to you, will certainly exceed the speed of light.
But in reality, there is no spoon. There are atoms, perhaps silver, making up that spoon and they interact at finite speeds, therefore it takes time for the flick to traverse the entire length of the spoon.
A rotating wheel may be a different object, but the argument is basically the same.
A: One of the results of special relativity as a result of (1) light cones and (2) the speed limit on the speed of light is that there is no such thing as a "rigid body" in special relativity. The reason is that a "rigid body" is one where the displacement vector between any two points is a constant. But because you have a spinning wheel, they have different velocities and the proper distance changes with the velocity.
see here: http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html for more discussion on this.
A: Rotating an object faster than the speed of light just isn't possible. First of all, it would have to accelerate to that speed, and the atoms in an object cannot interact with each other faster than the speed of sound(The speed the vibrations and force can travel through the object), so it would take an awfully long time to get moving. Once the wheel even started to approach the speed of sound, nuclear fusion would occur, destroying everything. Even if none of this happened, the speed of the wheel would be capped, just under the speed of light. It couldn't even get near the speed you had in mind, Unfortunately... :(
