Relativistic Stick: Can the Hole Contract Enough?

[boardbox]

Homework Statement

A stick of proper length L moves at a speed v in the direction of its length. It passes over a thin sheet with a hole of diameter L cut into it. As the stick passes over the sheet is raised and the stick moves through the hole so that it is underneath the sheet.

Is this a reasonable scenario? In the lab frame the stick is contracted and should easily make it through. However the hole is contracted in the stick frame.

Homework Equations

The Attempt at a Solution

I know that events that are simultaneous in one frame are not in others. So I expect part of the sheet to rise before the rest in the stick frame. Because of that I also expect the stick to be able to pass through (despite being "longer" than the hole) but I'm having difficulty showing it quantitatively. In the stick frame the front of the hole moving and the back of the hole moving should be separated in time by Lv/c^2.

Now if I take that time, and multiply by v, get Lv^2 / c^2 as the distance that the hole would traverse in the stick frame which is less than L and would mean to me that the proposed scenario is impossible. What bothers me with this is it goes against my intuition. Can anyone see where I'm going wrong?

 

[tiny-tim]

… I know that events that are simultaneous in one frame are not in others. So I expect part of the sheet to rise before the rest in the stick frame. …

Hi boardbox!

If the sheet is at an angle, surely only the width of the stick matters?