Question

This chapter has assumed that lengths perpendicular to the direction of motion are not affected by the motion. That is, motion in the $x$-direction does not cause length contraction along the $y-$or $z-$axes. To find out if this is really true, consider two spray-paint nozzles attached to rods perpendicular to the $x-$axis. It has been confirmed that, when both rods are at rest, both nozzles are exactly 1 m above the base of the rod. One rod is placed in the $S$reference frame with its base on the $x-$axis; the other is placed in the $S\prime$reference frame with its base on the $x\prime -$axis. The rods then swoop past each other and, as FIGURE P36.60 shows, each paints a stripe across the other rod.

We will use proof by contradiction. Assume that objects perpendicular to the motion are contracted. An experimenter in frame $S$finds that the $S\prime$nozzle, as it goes past, is less than $1m$above the$x-$ axis. The principle of relativity says that an experiment carried out in two different inertial reference frames will have the same outcome in both.

a. Pursue this line of reasoning and show that you end up with a logical contradiction, two mutually incompatible situations.

b. What can you conclude from this contradiction?

Short answer

a.) The red bar will become shorter, and a red line will appear beneath the blue nozzle.

b.) Perpendicular lengths are unaffected by the motion.

Step-by-step solution

Part (a) Step 1: Given Information

We have to pursue the line of reasoning and show that we end up with a logical contradiction, two mutually incompatible situations.

Part (a) Step 2: Simplify

You may see the blue paint nozzle arriving at rapid speed if you're in the S frame. The blue nozzle will be less than 1 meter up from the x-axis if the perpendicular lengths contract, and the blue nozzle will paint a blue line under the red nozzle. You may see the red nozzle arriving at a high pace if you're in the S' frame. The red bar will then shorten, revealing a red line beneath the blue nozzle.

Part (b) Step 1: Given Information

We have to find the conclude from this contradiction.

Part (b) Step 2: Simplify

You can't have both of these things occurring at the same time. As a result, we can deduce that lengths parallel to the motion are unaffected.