Special Relativity: Comparing Lengths in Parallel Motion

[RohanJ]

In Introduction to special relativity by Resnick,there is a thought experiment to compare lengths perpendicular to relative motion as given in the below image.
What if we try to perform such an experiment to compare lengths parallel to relative motion?
Suppose there are two horizontal rods moving relative to each other along x -x' axis. If both the rods have a marker at each end,then what will happen when they pass each other?

 

[PeroK]

A measurement of length requires simultaneous measurements of the position of both ends. When the rods are vertical, all four ends of the rods are at the same $x$ coordinate at the same time in both frames.

If the rods are horizontal, then the two ends are spatially separated along the direction of motion. And this makes all the difference. In this case, both rods will be contracted when measured in the other frame.

 

[Nugatory]

Suppose there are two horizontal rods moving relative to each other along x -x' axis. If both the rods have a marker at each end

If you google for "pole-barn paradox" you will find many good explanations of what's going on here.

So we have lab assistants, one stationed at each end of the two rods (that's four total) and holding loaded paintbrushes. They have been all been instructed to swipe out with their brushes to mark whatever is passing at the moment that the two origins coincide.

then what will happen when they pass each other?

Both rods will have one mark, at the origin, made by the lab assistant stationed at the origin end of the other rod. The two lab assistants at the non-origin ends of the two rods will find that at the moment that the origins coincide they are facing just empty space so they don't make any mark. This result is consistent with the moving rod being length-contracted no matter which rod we choose to consider to be at rest.

There is no contradiction because of the relativity of simultaneity. The events "origins coincide" and "lab assistant at end of rod A swipes their brush" happen at the same time in the frame in which rod A is at rest, but not the frame in which rod B is at rest, and vice versa. If you are not already comfortable with relativity of simultaneity, you'll want to learn about it (google for "Einstein train simultaneity") because it is essential to understanding how special relativity is consistent.