$v=\omega * r$, so with $\omega = 1 rad/s$ and $r=3*10^8 m$ the v would be c. I know that the information of the rotation travels with the speed of sound in the rod.
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If you can't reach the speed of light, then what you have proven is that a rod of that length cannot reach 1 rad/s.
The issue in achieving it would be that, as you spin it faster and faster, the mass of the tips of the rod increase due to their kinetic energy. As you approach the speed of light, that mass asymptotically approaches infinity, meaning accelerating that rod would require an amount of energy that also asymptotically approaches infinity.
Cort Ammon
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