Will There Be a Collision in the Shaw Paradox?

[briteliner]

Homework Statement

A very thin steel plate with a circular hole m

1 in diameter centered
on the y-axis lies parallel to the xz plane in frame S and moves in the +y
direction at a constant speed y
v (as illustrated in Figure 1-44 of the text).
A meterstick lying on the x-axis moves in the +x direction with c
v/
=
β .
The steel plate arrives at the 0
=
y plane at the same instant that the
center of the meterstick reaches the origin of S. Since the meterstick is
observed by observers in S to be contracted, it passes through the m

1
hole in the plate with no problem. A paradox appears to arise when one
considers that an observer in S′, the rest system of the meterstick,
measures the diameter of the hole in the plate to be contracted in the x
dimension and, hence, becomes too small to pass the meterstick,
resulting in a collision. Resolve the paradox. Will there be a collision?

 

[BigJDubs]

Homework Equations The Attempt at a Solution No, there will not be a collision. The paradox arises because the observer in S' sees the steel plate moving in the -y direction and the meterstick moving in the -x direction. In the rest system of the plate, the meterstick's diameter is not contracted, so it can pass through the hole. However, due to time dilation, the observer in S' will perceive the meterstick as passing through the hole before the steel plate reaches the origin, resulting in no collision between the two objects.