## Exercise 3: Pole in a Barn

 View from Barn View from Pole

In this exercise, we explore how one goes about measuring the length of an object moving at relativistic speed.  When an object is stationary in our own frame of reference, we can put a ruler next to the object or mark off the ends against an available scale at leisure without considering any time element at all.  But what if the object is flying by us in another frame?  When marking the ends, what time-dependent condition must be satisfied to obtain the correct length?  Explain.

Consider a pole vaulter carrying her pole towards a barn as shown in the animation (position and time are both given in meters).  The length of the moving pole is 10 m as measured from the frame of the barn.  Also shown above the barn is the space-time diagram depicting the pole and the barn from two frames of reference: the pole and the barn.  When an object is stationary in a reference frame it is red and when it is moving it is green.  Restart

Select View from Barn and play.

1. How fast is the pole moving relative to the barn (measured in c)?
2. What is 1/slope of the red and green world-lines, respectively?  What does 1/slope represent in space-time diagrams?
3. What do events A, A', and B refer to?  Which if any of these events are simultaneous in this reference frame?

Select View from Pole and play.

1. How fast is the barn moving relative to the pole (measured in c)?
2. How long is the pole in this reference frame?  Why is it this long?
3. How long is the barn in this reference frame?  Why is it this long?
4. What do events A', B, and B' refer to?  Which if any of these events are simultaneous in this reference frame?

Finally, consider an experiment with automatic doors installed approximately 1 m outside the barn doorways that can be opened and closed simultaneously by a switch.  In each frame, does an instant in time exist when the pole is completely inside the doors and they can be closed without catching the pole?  Of course, you would need to reopen the doors very quickly, but at least momentarily, could you have the contracted pole entirely enclosed between the barn doors?  Explain your reasoning by referring back to the first question (and question c) in this exercise and your ensuing observations.  How would the door closing events be viewed in each frame?

Conflicting views from the perspectives of the barn and the pole can be reconciled by accounting for the lack of agreement on simultaneity.  In special relativity, events that appear simultaneous in one reference frame will generally not appear simultaneous in other reference frames that are moving relative to the first.

Authored by Mario Belloni, Wolfgang Christian, and Tim Gfroerer.
© 2004 by Mario Belloni and Wolfgang Christian.