Spinning disk goes from 0 to relativistic speed. What happens to RPM and angular momentum?

Factual Questions
1

Inspired by the pizza thread, I had a question.

Say a disk spinning about its internal axis (say at 100 RPM) is slowly accelerated to the speed of light. When I mean accelerates, I mean the center of mass of the disk accelerates, the disk Is not spun faster -( much like a frisbee.)

Since there is time dilation, will an observer on the disk feel the disk slow down ? That is will the 100RPM reduce ? What about an outside observer ?

Also as the relativistic mass of the disk will increase With speed, will the angular momentum increase too?

2

An observer on the disk probably won’t notice much. As far as he is concerned he’s just hanging out on a spinning frisbee as the rest of the universe accelarates past him. I’m not sure how it will look from an outside observer, but given that spatial distortion is a common side effect of things traveling near the speed of light, my intuition is that the whole thing will be warped out of its nice spinning disk arrangement into something not quite so simple.

3

“Relativistic mass” is an antiquated concept that never makes anything simpler to understand. The angular momentum of the disk would remain unchanged, to all observers.

Whether the disk is distorted will depend on its orientation.

4

How would that work in detail though? I’m trying to do some of the math in my head and I come up confused.

Say observer A has a spinning hoop with a certain RPM. He uses that as his clock.

To observer B A is moving, so time should be moving slower for A. So in B’s reference frame the RPM is reduced.

Do we just conclude that to calculate the angular momentum of a moving disk we have to take its velocity into consideration, or am I missing something else?

5

Its orientation with respect to what?

Not fighting the hypothetical, but just pointing out that this is not something that will happen in the real world. Unless the disk is made of unobtainium, it’s going to disintegrate and fly apart long before any relativistic effects become significant.

6

The disk is made of steel. It’s spinning about its axis at 100RPM . The spin is constan;t it is not spun faster. The center of the disk is accelerated to relativistic speed, say To 0.0001c or something like that. Why would it disintegrate ?

Maybe think of a yoyo that is spinning and traversing too .

7

OK, I misunderstood. I thought you were spinning the disk up to relativistic speeds. Instead you’re accelerating the disk as a whole to relativistic speeds. Never mind.

8

Given that it’s now the whole disk (rotating at 100RPM) being accelerated while spinning, I think it would matter whether or not it was accelerated in some direction along the plane of spin or “through the axle” (i.e., up or down).

9

I would not expect the rotation to make that much difference. The velocity suggested is about 30,000 m/sec and the speed of the outer rim of the disk will be very much less than that. For example, the rim of a disk 30 cm in diameter will be only going 15.7 m/sec relative to its center (if I didn’t slip a decimal point). That little bit of extra velocity should make little or no difference.

10

OK I thought about this some more, and I think I may know how it works. (Chronos will probably pipe in and tell me I’m wrong.)

The issue at hand is that when you are up near c, differences that would be significant when an object is not moving (say one side of the disk moving forward at 1m/sec and the other side moving backward at 1m/sec) are no longer significant because everything is moving at near c, not c + 1m/sec or c - 1m/sec.

For rotational ease let V the forward velocity of the disk (very close to c) and R be the rotational velocity of the edge of the disk V>>R.

Scenario 1, The axis of rotation is perpendicular to the direction of motion, like a drill

If we look at the atoms on the rim of the disk. They are all moving at the same speed with v in the direction of travel and r perpendicular to the direction of travel. If Newton’s laws applied then the total velocity vector would be sqrt(V^2+R^2) approximately equal to V+(r^2)/V. However since we are in a relativistic realm, the result of adding the additional r velocity will be much slower. So while we would observed the forward velocity v, the r velocity would appear very slow. Thus the disk would appear to be rotating much slower, almost as if time slowed down for the disk. Hmmn, didn’t I remember reading about time relativistic time distortion? That time appears to slow down as they move faster? It all works out to the same thing.

Scenario 2: The axis of rotation is perpendicular to the velocity, like a Frisbee.

For an atom of the front or back of the disk, the direction of rotation will be perpendicular to the direction of motion so you’ll get the same result, a general slowing that can be represented by a time distortion. However on the left and right edges, the nominal speed would be V+R much bigger jump than when it was perpendicular. Thus, to keep everything below c, the edges apparent velocity has to slow down even more, so that the side of the disk is just barely creeping forward relative to the center of the disk. Also for similar reasons the other side of the disk is also creeping backwards. Now in order to have the disk uniformly rotate s, while having the left and right sides move much slower than the front and back , without having a massive pile up somwehere in between, the atoms on the side must have a shorter distance to travel to rotate than the ones on the front and back, so rather than a circle, the disk must turn into an oval with short sides and a broad middle. Hey, didn’t I remember reading about relativistic distance distortion where things get crunched in the direction of travel? Again it all works out the same.

For angles in between the two the shape of the disk will be more smooshed the closer it is to a frisbee, and less smooshed the closer it is to a drill, while the rate of rotation will be uniformly slower regardless of its orientation.

11

The simplest way to see it is that a force acting on the center of mass exerts no torque, and therefore the angular momentum must remain unchanged.

12

So this post got me thinking, are there any instruments which spin fast enough for relativity to have to be taken into account?

The numbers don’t seem too ridiculous. e.g. a 15cm radius disc, spun at 3000 revolutions-per-second is 1% of speed of light by my calculation. The saw I buy at the hardware store isn’t going to reach that, but I can imagine some high-end scientific instrument reaching that, maybe?

13

I think you’re off by 3 orders of magnitude there. I get 2827 m/sec for that disk. The speed of light is approx 300,000,000 m/sec.

14

Ah yes, perils of back of the envelope calculations :slight_smile: 300,000 RPS seems a bit more ridiculous.

15

Define “relativity being taken into account”. Magnetism is a relativistic effect from the motion of charges, and can easily be detected even when those charges are only moving at millimeters per second, or even slower.

16

I mean in the actual motion of the disk, are there any cases were Newtonian laws of motion are not sufficient to describe the relationship between the forces applied and the acceleration.

17

Also Magnetism is relativistic, but so are all other forces. We just only require using the non-relativistic form (for magnetism that’s Maxwell’s equations, for gravity that’s Newton’s) in almost all cases we encounter every day. We could use relativity to predict the acceleration of ball falling off my desk, but it would not predict its behavior much better than Newton’s laws, unlike the behavior of a ball falling into a black hole.

Magnetism certainly affects a spinning metal disk (in fact its what causes it to spin, if you are using and electric motor), but does it reach the point where classical electromagnetic theory breaks down and you need to resort to relativistic ones?

18

Understood. But the mass of the disk goes up. So does the disk rpm slow down to keep the angular momentum the same ?

19

I think you missed Chronos’ point earlier that relativistic mass is not a real thing. It was something thought up by pop science writers to try to explain the behavior of things travelling at relativistic speeds.

20

Can someone explain this sentence to this unsharp mind? How is it possible to accelerate a disk without spinning it faster?