Nail and cylinder problem in SR

[lalbatros]

Hello,

This may look like a homework, but it is rather a refresher for my hobby-time.
I just found a nice collection of exercices: http://uk.arxiv.org/PS_cache/physics/pdf/0605/0605057.pdf".

The exercise 2.2 is funny:

(see attached picture)
One end of a hollow cylinder is sealed by a paper. A nail with a large velocity V flies towards the open end. The hat of the nail is larger than the diameter of the cylinder, and their proper lengths are equal. In the cylinder's frame the nail is contracted y-times and consequently cannot perforate the paper. But in the nail's frame it's the cylinder that's length-contracted and consequently the paper is pierced. What occurs actually in each frame?

Sure this is no flaw of SR !
What's the explanation?
What are the interresting lessons from this exercice?

Michel

 

[Doc Al]

It wouldn't be much of a refresher if we didn't let you solve it for yourself, right? Hint: Do all parts of the nail stop instantly when the hat hits the cylinder?

 

[lalbatros]

Doc Al,

Do all parts of the nail stop instantly when the hat hits the cylinder?

For me, the easy side of the question is after the nail head and the cylinder get in contact.
It is clear that the compression wave cannot speed faster than light and causality therefore makes the answer easy.

What is less clear is the precise time when the nail and the cylinder just get into contact.

Lets assume the cylinder is equipped with a detector at both ends.
One detects the contact of the head of the nail with the cylinder.
The other detects the perforation at the other end.
These detectors send a signal to an "AND gate" or "coïncidence gate" at the middle of the cylinder.
Will the "coïncidence gate" trigger?


Michel

 

[Doc Al]

What is less clear is the precise time when the nail and the cylinder just get into contact.

Lets assume the cylinder is equipped with a detector at both ends.
One detects the contact of the head of the nail with the cylinder.
The other detects the perforation at the other end.
These detectors send a signal to an "AND gate" or "coïncidence gate" at the middle of the cylinder.
Will the "coïncidence gate" trigger?

According to the nail frame, the tip hits the paper before the head hits the cylinder.
According to the cylinder frame, the head hits first.

Either way, no coincidence gate triggers.

 

[lalbatros]

Doc Al,

Not sure what you mean.

I was not sure either. That's why I put the winking smile besides.
That's why I equipped the cylinder with two detectors.

Either way, no coincidence gate triggers.

Is this not surprising?

Michel

 

[MeJennifer]

It is clear that the compression wave cannot speed faster than light and causality therefore makes the answer easy.

What is a compression wave?

 

[Doc Al]

Is this not surprising?

Not to me. What would be surprising is if one frame said coincidence occurs while the other didn't!

 

[lalbatros]

MeJennifer

What is a compression wave?

Just like sound: The initial compression propagates through the medium.

Michel

 

[JesseM]

What is a compression wave?

There can't be any truly rigid objects in relativity--when the nail head hits the cylinder and rapidly accelerates to a state of rest relative to the cylinder, the rest of the nail keeps moving inertially as if nothing had happened until a compression wave traveling through the nail at the speed of sound reaches each part and accelerates it to a state of rest relative to the head. That's why the point of the nail will still be predicted to reach the end of the cylinder in the frame where the nail's length appears less than the cylinder, because the compression wave from the head won't reach the tip in time to stop it from hitting the back.