Punching a hole in a tape moving at relativistic speed

[FranzDiCoccio]

Homework Statement

a tape moves at half the speed of light wrt a cylinder of radius 1 cm and extremely small lenght, whose axis is perpendicular to the tape. The cylinder moves through the tape, punching a hole into it.
Then the tape is stopped, and the area of the hole is measured.
Find the area of the hole.

Relevant Equations

Length contraction

The proposed solution is that the hole is an ellipse that is wider than the cylinder in the direction of the original velocity of the tape (its major axis is along the tape, and its minor axis across it).
The reason would be that the piece of tape cut by the cylinder is circular, but the tape undergoes length contraction in the frame of reference of the cylinder. But since this piece of material should be "contracted" along the tape, the hole in the tape should be longer.
Therefore (when the tape stops) the hole looks "expanded" by the Lorentz factor.

I'm not sure the problem is well posed. I think that "extremely thin cylinder" means that the cutting of the hole is instantaneous... But this seems strange.

Also, I'm not really sure as to why one should take the point of view of the cylinder. The provided solution does not comment about this at all.
Is this because both the cylinder and the tape are at rest in the frame of reference of the former, in the end?

In the frame of reference of the moving tape, the cylinder is an ellipse whose shorter axis is along the direction of motion of the cylinder itself (minor axis along the tape, major axis across it).
Assuming "instantaneous punching", in this frame of reference the hole has the shape of the hole puncher, whose cross section is contracted along the tape length, and unchanged along the tape width.
The stopping of the tape does not change the size of the hole. Hence the hole is smaller than the hole puncher.

The only reason I see against this argument is that the hole puncher is moving wrt to the tape, and according to (nonrelativistic) intuition this would enlarge the hole in the direction of motion at least a little bit, if we assume that the punching is not instantaneous.

If the process is instantaneous, I do not see a reason to prefer "the bit cut by the cylinder is circular in the f.o.r. of the cylinder" over "the tool used to punch the hole has an ellyptic cross section in the f.o.r. of the tape".
Also, the question concerns the size of the hole, which is in the tape. The proposed solution measures it from assuming that the piece of material removed by the cylinder is circular. A measure of the hole itself seems more direct.

 

[FranzDiCoccio]

just another thought.
Perhaps we could imagine a second tool in the same f.o.r. of the cylinder. This tool measures the area of the hole punched by the cylinder. That is: the cylinder punches a hole, and at a position "downstream" from the cylinder, the second tool measures the size of the hole.
It seems fair to conclude that, according to this observer, the hole has the same length as the cylinder, along the direction of motion of the tape. Therefore the hole is circular.
This appears to support the original proposed solution.

 

[anuttarasammyak]

In the puncher’s FOR, the puncher makes a hole on contracted tape.
In the tape FOR, the contracted puncher makes a hole in not-simultaneous action, i.e. the end edge cut the tape first, the front edge cut the tape after.
As a result the hole proper shape becomes extended ellipse in moving direction for both FORs.

 

[phinds]

a cylinder of radius 1 cm and extremely small lenght, whose axis is perpendicular to the tape.

Uh ... I think you mean parallel. If the cylinder's axis is perpendicular to the tape then the cylinder is pointing away from the tape. That is, if the tape is flat on a table top, for example, then the cylinder's puncher is pointing at the vertical wall next to the table, not at the tape, so how can it punch a hole in the tape?

 

[FranzDiCoccio]

I'm not sure I'm understanding you correctly. Are you saying that the sides of the puncher (along the direction of motion of the tape) touch the tape simultaneously in the FoR of the puncher, but not simultaneously in the FoR of the tape?
Of course I do agree with the fact that two simultaneous events in the former FoR are not simultaneous in the latter FoR.
But what is the reason for assuming simultaneity in the former FoR rather than in the latter?
The problem does not contain any clue about this.
Or are you saying that the hole would be extended irrespective of the FoR in which simultaneity is assumed?

As to your perpendicular/parallel question: the problem says perpendicular, meaning perpendicular to the plane containing the tape. That is: if the tape is horizontal, the axis of the cylinder is vertical.

 

[phinds]

That is: if the tape is horizontal, the axis of the cylinder is vertical.

Which is exactly what I was pointing out. It won't punch holes in the tape. So ... the issue is in the statement of the problem.

 

[anuttarasammyak]

I'm not sure I'm understanding you correctly. Are you saying that the sides of the puncher (along the direction of motion of the tape) touch the tape simultaneously in the FoR of the puncher, but not simultaneously in the FoR of the tape?

Yes, it is what I say.

But what is the reason for assuming simultaneity in the former FoR rather than in the latter?

In our daily use of puncher, which part of edge touch the paper first does not matter. Even oblique shape of blade would work. Such a oblique blade would make ellipse hole even in non-relativistic slow motion of tape. If all parts of brade touch the tape simultaneously* we can avoid such non- relativistic case.

*: in non relativistic sense at first. In relativistic thought of common sense, puncher would be designed to work simultaneous in FOR of puncher.

 

[Hill]

I think that this how the thing works in the puncher FoR, on the left, and in the tape FoR, on the right:

 

[Orodruin]

Which is exactly what I was pointing out. It won't punch holes in the tape. So ... the issue is in the statement of the problem.

No. The problem is fine. Yes, if the tape is horizontal then the cylinder axis is vertical. It then moves down through the tape - making a hole.

So I do not agree with your assessment. Granted, the cylinder being very thin means it is more of a cutting ring than a cylinder.

I think that this how the thing works in the puncher FoR, on the left, and in the tape FoR, on the right:
View attachment 340186

Indeed. The cut is simultaneous in the frame where the cylinder (or rather - circle) is horizontal. In the tape’s rest frame, it won’t be.

 

[anuttarasammyak]

I think that this how the thing works in the puncher FoR, on the left, and in the tape FoR, on the right:
View attachment 340186

Say the puncher is going down, yes.

 

[FranzDiCoccio]

Which is exactly what I was pointing out. It won't punch holes in the tape. So ... the issue is in the statement of the problem.

I really do not understand. The tape is in a horizontal plane. The only way a cylinder can cut a round hole into it is when its axis is vertical, i.e. perpendicular to the plane.
The cylinder is above the tape, then it goes down through it and emerges below it. If the tape is not moving horizontally, the hole is there, and it is definitely round.
Now the question is: what is the shape of the hole if the tape is moving horizontally, and the cylinder moves through it vertically?
Of course, if the height of the cylinder is finite the answer is "a mess". I think this is the reason why the height of cylinder is assumed to be extremely small (like, the cylinder is really a circle, or a disk).

 

[Hill]

Say the puncher is going down, yes.

Sure. Here:

 

[Orodruin]

Sure. Here:
View attachment 340187

Not to scale, should be added. The cylinder should also be more of a line (negligible height) and the cylinder is moving down/left in the tape rest frame (it might be better with a single arrow).

 

[Hill]

Not to scale, should be added. The cylinder should also be more of a line (negligible height) and the cylinder is moving down/left in the tape rest frame (it might be better with a single arrow).

So?

 

[FranzDiCoccio]

ndeed. The cut is simultaneous in the frame where the cylinder (or rather - circle) is horizontal. In the tape’s rest frame, it won’t be.

Now I'm starting to see it, also thanks to the diagrams.
So the cutting ring is horizontal in the FoR where the tape is moving (horizontally), but it is tilted (and not moving vertically) in the FoR at rest with the tape. One of its ends touches the tape before the other end.
This must have to do with the fact that the ring is moving towards the tape along another direction (y).
If the ring did not approach the tape, it would appear parallel to it, wouldn't it?
So, in order to understand this better, should I consider both directions of motion (say x and y)?
Is it a matter of relativistic addition of velocities?
I sort of see what you mean, but I'm not really used to problems like this. Wouldn't be the tilting affected by the speed of approach of the cylinder (along y)?
The original problem does not give information on that speed, and the solution only depends on the Lorentz factor of the horizontal speed of the tape.

 

[Hill]

If the ring did not approach the tape, it would appear parallel to it,

Correct.
It helps to remember the phrase, "Leading clock lags." In the tape FoR, the left edge of the ring is leading (the ring moves to the left in this FoR.) So, the left edge lags behind the right edge as they move toward the tape.

 

[anuttarasammyak]

So, in order to understand this better, should I consider both directions of motion (say x and y)?

No, y-motion of puncher is by chance, not essential. Say, instead of puncher ring, series of small inkjets ring is nearby the tape. By simultaneous signal in ring FOR all inkjets eject pulse acid. Eliptic piece is cut out from the running tape. No mechanics or motion is involved here.

The same situation is often discussed as "relativity train" made of train center light source, front and back ends receivers, and groung observers. You may easily find it in Web.

 

[phinds]

So I do not agree with your assessment.

Clearly we are not envisioning it the same way.

 

[FranzDiCoccio]

it seems natural to assume that the amount of tilting (and hence the amount of delay) depends on the approach speed.
So, in the end, this effect should cancel out in some way, if only the horizontal velocity of the tape determines the area of the hole.
This is not obvious to me.

 

[Hill]

it seems natural to assume that the amount of tilting (and hence the amount of delay) depends on the approach speed.
So, in the end, this effect should cancel out in some way, if only the horizontal velocity of the tape determines the area of the hole.
This is not obvious to me.

The difference in time when two edges of the ring touch the tape, in the tape FoR, depends only on the horizontal speed of the ring relative to the tape. If this speed is zero, the difference is zero, regardless of the approach speed.

 

[Orodruin]

Clearly we are not envisioning it the same way.
View attachment 340190

The left is obviously what is intended. But what are even the things you have drawn onto the cylinders? There is nothing in this task about anything else than the cylinders.

 

[anuttarasammyak]

I think that this how the thing works in the puncher FoR, on the left, and in the tape FoR, on the right:
View attachment 340186

@Hill based on your interestiong illustration, I add touch and go sequence.

 

[FranzDiCoccio]

The difference in time when two edges of the ring touch the tape, in the tape FoR, depends only on the horizontal speed of the ring relative to the tape. If this speed is zero, the difference is zero, regardless of the approach speed.

Of course I understand that it must be so, in order for the solution to depend only on the translation speed of the tape. The contact times should not depend on the approach speed, in the FoR of the puncher.
However I cannot figure out an easy argument for this that does not involve knowing the correct solution.
The ring is untilted when its distance from the tape does not vary. It tilts when it approaches the tape.
I expect that the amount of tilting does depend on the approach speed. Is this wrong?

Sorry, where I live this discussion started around midnight, and now it is the early morning, I have not had the time to work out any calculation (I went to bed very late and I have to be at work in less than an hour).
I'm willing to do that calculation, but if that is very complex, this question is not as "trivial" as it initially seemed.

 

[anuttarasammyak]

I expect that the amount of tilting does depend on the approach speed. Is this wrong?

You are right. In FOR of tape as for tilted angle $\theta$
$$\tan \theta := \frac{h}{a} = \frac{\frac{V}{c} \frac{v}{c}}{ 1-\frac{V^2}{c^2} }$$
where a is contracted diameter of puncher ring, h is height of front edge when end edge touches the tape, V is horizontal speed and v is vertical speed of puncher, all in tape IFR.

When V=0 or v=0, $\theta$=0.
The larger v, the higher h and larger $\theta$. But the elipse hole shape and area do not depend on v.

 

[Hill]

an easy argument

Let the tape move to the right.
Let's consider two events. Event A, the right edge of the ring hits the tape. Event B, the left edge of the ring hits the tape.

In the ring FoR:
$t_A=t_B$
$x_A=x_B+2 \, cm$

In the tape FoR:
$t'_A=\gamma (t_A-.5x_A)$
$t'_B=\gamma (t_B-.5x_B)$
$x'_A=\gamma (x_A-.5t_A)$
$x'_B=\gamma (x_B-.5t_B)$

The approach speed does not enter calculations.

The hole length, in the tape FoR, is
$x'_A-x'_B=\gamma (x_A-x_B)$.

 

[Orodruin]

One important point here: The Lorentz transformation will lead you right. The length contraction formula will also be applicable, but … and it is a big but … you need to be aware of the conditions surrounding it to apply it correctly. Too many times you will see students apply the length contraction formula in ways in which it is not applicable. If in doubt, always go back to the Lorentz transformation.

 

[phinds]

The left is obviously what is intended. But what are even the things you have drawn onto the cylinders? There is nothing in this task about anything else than the cylinders.

Ah. NOW I get it. Everyone but me is talking about a cylinder which IS the punch. I was envisioning a cylinder with the punch thingies around the circumference of the cylinder.