What prevents an orbiting object from getting a speed which is greater than $c$? Consider an object orbiting around a point with radius $r$ and angular velocity $\omega$. Here its linear velocity is $v=\omega r$. If we choose a large enough $r$ and reasonable $\omega$, $v$ might be greater than $c$.
If this fact is impossible, what prevents it from happening?
 A: (This was intended to be a comment but ended up being too long)
A related "paradox": Consider the diurnal motion of the stars as they appear to rise and set each night (the non-circumpolar stars at least) traversing circular paths on the celestial sphere. An obvious back-of-the-envelope value for their angular velocity $\omega=\frac{360^\circ}{1\,\text{day}}$. Combine this with the distance to a particular star, say Barnard's Star which happens to be the fifth closest star to Earth, $r\approx 6\,\text{light years}.$
Naively applying the velocity relation $v=\omega r$ to these values gives you a velocity on the order of 14,000 times the speed of light! What gives??? 
A: If you tell me that an object has velocity $\omega r$, it will have velocity $\omega r$. There's nothing stopping me from imagining a nonphysical/theoretical point moving this fast. If you tell me that a physical particle of something, or something that moves at this speed, has this velocity, I will tell you that you're wrong when $\omega>c/r$, and I'll give answers like I am answering the question "why can't an object travel faster than $c$". It's a consequence of special relativity that an object travelling at less than the speed of light, with only finite forces applied to it, will always travel at less than the speed of light.
For an object with mass to start orbiting, you have to accelerate it. To accelerate it, you have to pump energy into it. As you exhaust the Earth's power grid by pumping more and more energy into it, you find that its velocity increases: $.99c$, $.999c$, $.9999c$, $.99999c$, and so on, until Earth's energy resources are exhausted and the particle is still moving slower than light.
To answer your question, you can apply the above formula to see that $\omega$ can only approach $c/r$ from below. Special relativity is what prevents it from happening!
