Why Does Length Contraction Affect Only Dimensions Parallel to Motion?

[georgeh]

I have a h.w. problem that states:
Explain WHY only the lengths parallel to motion shrink and why lengths perpendicular don't shrink.

I know that the vector component of the velocity for an object moving in straight line would be changing. This change only occurs on the x-axis and the not the y. But i am not sure how to extrapolate from that, Or i may be heading the wrong direction

 

[robphy]

From your question, I am assuming that you already understand why lengths parallel to motion shrink... and the issue is why just the parallel length and not the perpendicular lengths.

Assume horizontal motion on this page.
Consider two aligned vertically-oriented metersticks, call them A on the left and B on the right, with different horizontal velocities, approaching each other ...with nails at each end of their metersticks intended to scrape the other meterstick as they pass.

Study the problem from A's inertial frame.
Make your assumption about what should happen.
Then study from B's inertial frame.

 

[dimensionless]

I'm not sure what your teacher wants, but I would find the velocity perpendicular to the motion (v=0), and then use that value to calculate the perpendicular length contraction. You might get more credit if you use robphy's method though.

 

[Andrew Mason]

I have a h.w. problem that states:
Explain WHY only the lengths parallel to motion shrink and why lengths perpendicular don't shrink.

I know that the vector component of the velocity for an object moving in straight line would be changing. This change only occurs on the x-axis and the not the y. But i am not sure how to extrapolate from that, Or i may be heading the wrong direction

This is a good question. It brings out the fundmental reason for length contraction: Length measurement depends on simultanaeity of events (ie. where the ends are at a particular time).

Two events are required to measure length. These are necessarily simultaneous in the measuring frame. If the same events are not simultaneous in the frame of reference of the object being measured, the measurements of length will differ.

Events that occur simultaneously in the moving frame in positions separated by a distance perpendicular to the direction of travel will be perceived by an observer in the rest frame to be simultaneous also. This is because light signals from those points that are received simultaneously by an observer in the moving frame who is equidistant from those points will reach the rest observer who is equidistant from those points also at the same time.

However, events that occur simultaneouisly in the moving frame in positions separated by a distance parallel to the direction of travel will not be perceived by an observer in the rest frame to be simultaneous (the classic Einstein moving train gedanken experiment).

Because simultanaeity differs only for events separated by a distance in the direction of travel, measurement of distance will differ only for distances parallel to the direction of travel.

AM