# Angular velocity and c: Is this a paradox? Can you explain it?

I thought of this today while studying relativity.

Imagine a very large disc spinning with an arbitrary angular velocity. Perhaps w=57 rad/s (this is the speed of a CD, got this from a random physics text, whatever, the speed is really not important)..

The speed of light is c=3*10^8 m/s.

Since tangential velocity v=rw, if one were to sit anywhere on the disc such that r > 5,260,000m, then you will be traveling faster than the speed of light.

But this is impossible.

How can you explain this? Is there a constraint on how large the disc can be?

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Dale
Mentor
Is there a constraint on how large the disc can be?
Yes.

Yes.
ok, then what is it? ~5million meters is not very far in the grand scheme of things.

Dale
Mentor
r<c/w

r<c/w
that's not exactly a size constraint, more like an angular velocity constraint. i don't see any reason why it's theoretically impossible to build a disc of arbitrarily large size.

so you claim that no matter what size the disc is, it's angular velocity will be constrained such that v at the very edge will be < c ?

Dale
Mentor
so you claim that no matter what size the disc is, it's angular velocity will be constrained such that v at the very edge will be < c ?
Yes.

The reason is that solid bodies aren't held together by magic; they're held together by internal intermolecular forces. The internal forces required to maintain that rigidity become infinite as you spin the disk that fast. So it shatters.

Of course, any real disk will shatter long before its edge reaches c, but even a disk of pure unobtanium will shatter at that speed/radius.

Drakkith
Staff Emeritus
The reason is that solid bodies aren't held together by magic; they're held together by internal intermolecular forces. The internal forces required to maintain that rigidity become infinite as you spin the disk that fast. So it shatters.

Of course, any real disk will shatter long before its edge reaches c, but even a disk of pure unobtanium will shatter at that speed/radius.
Also, remember that as the velocity increases it takes more and more energy to continue to accelerate the disc. You could not physically get it to c because that would take infinite energy.

The reason is that solid bodies aren't held together by magic; they're held together by internal intermolecular forces. The internal forces required to maintain that rigidity become infinite as you spin the disk that fast. So it shatters.

Of course, any real disk will shatter long before its edge reaches c, but even a disk of pure unobtanium will shatter at that speed/radius.
well i guess there certainly is a size constraint, in that the part of the disc that is too large will just fly apart.

sorry for doubting you dale, and thanks for explaining it everybody else.