# 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 $r = 2 \times {10^3}\,meters$??

• This question seems to assume an underlying inertial system by which geometric relations such as "radius $r$", "angular frequency $\omega$", "wheel circumfence" etc. are defined and evaluated in the first place. The speed of any element of the wheel rim travelling between members of this inertíal system is of course limited (already purely kinematically) by the speed of any signal between them: $|\omega\cdot r| \lt c_{\, 0}$. (Importantly, the question requires the wheel elements to be real and not, say, just some laser pointer dot; because there's no corresponding limit on phase speed.) – user12262 Sep 20 '13 at 5:49
• @user12262: Yes, the underlying frame to define those relations is any frame which is not accelerated with respect to the point at which center of the wheel is located. Of course required energy could not be infinite but I may have to modify my question, I want to see if the upper limits (upper bound for the speed of light and therefore the limitation on radius of the wheel) of these frames is different?? – 2physics Sep 20 '13 at 9:36
• Frames of reference are completely irrelevant here. If there is a point on the wheel that is moving at $c$ in some frame of reference, then it's moving at $c$ in all frames of reference. Since SR doesn't allow material objects to move at $c$, this is impossible. – user4552 Sep 20 '13 at 14:53
• @2physics: Your question, especially after the recent edit, seems to be focussed on how to define/measure "speed" (i.e. values $|\beta|$) in the first place, and to clarify the (possible) role which inertial systems may have in this definition. (This may have been conclusively addressed already somewhere; but it surely seems worth to ask the question explicitly.) – user12262 Sep 23 '13 at 21:04

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.

• you think so? but it's not an inertial frame... Thank you for your answer. – 2physics Sep 19 '13 at 22:06
• Inertial frames got nothing to do with it. There cannot be any object moving faster than light in any reference frame, inertial or otherwise. This is one of the key points of special relativity. I can't give you a better explanation why, it's basically a postulate. But I'm 100% positive that it's not possible. Just as Schlomo Steinbergerstein suggested, the wheel would probably break apart while accelerating. Otherwise you would need to put in tremendous amounts of energy as the rim approaches the speed of light. – Jonas Greitemann Sep 19 '13 at 22:24
• But Special relativity just talks about inertial frames of reference.. and this frame has a rotational motion , so shouldn't we expect things in it not to work according known physical laws of our inertial frame? – 2physics Sep 19 '13 at 22:37
• @2physics: It's not true that SR can only deal with inertial frames of reference. That's the way Einstein posed the distinction between the two theories a century ago, but it turns out that it's simply not true. SR deals with accelerated frames just fine. And in any case, you can consider a noninertially moving object in an inertial frame; we do it all the time in Newtonian mechanics. – user4552 Sep 20 '13 at 3:33
• @2physics: The WP quote is referring to a coordinate velocity. Coordinate velocities are of no fundamental interest in relativity. They're not observable. What is observable is the velocity of a particle in a local inertial frame (LIF). In any LIF, $c$ is the same. – user4552 Sep 20 '13 at 15:58

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.

• Thanks;You mean it's impossible to rotate this wheel at such a speed?? – 2physics Sep 19 '13 at 22:04
• You're welcome! :) My point was that many paradoxes of such kind are resolved simply by considering real objects, made up of matter, not mathematically idealized rigid structures. I don't have a precise answer, but by thinking about atoms, I'm sure that either you would create disturbances in the matter travelling at finite speeds (waves) or it would simply break apart at some point for any physical material. – user20250 Sep 19 '13 at 22:18
• Yup you are right , physical instances could be really helpful. Your spoon example was a good idea and I got it. But my question is about the probable difference between an inertial and non-inertial frame. If it doesn't make difference to be in an inertial or non-inertial frame, so why does Einstein's second principle of special relativity emphasize on constancy of light speed in all "inertial frames" not all frames?? – 2physics Sep 19 '13 at 22:51
• :I've edited my question , please have a look at it. – 2physics Sep 20 '13 at 11:36
• I'm sorry, I'm not exactly sure what you mean, I just wanted to point out these things that I pointed out! – user20250 Sep 20 '13 at 11:46

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.

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... :(

• Once the wheel even started to approach the speed of sound, nuclear fusion would occur, destroying everything. This is wrong. – user4552 Sep 25 '13 at 23:08
• @BenCrowell Oops, meant speed of light – PQx Sep 26 '13 at 23:24
• It's still wrong if you replace speed of sound with speed of light. – user4552 Sep 27 '13 at 4:30