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Chapter 36: Q. 22 (page 1059)
A cube has a density of while at rest in the laboratory. What is the cube鈥檚 density as measured by an experimenter in the laboratory as the cube moves through the laboratory at of the speed of light in a direction perpendicular to one of its faces?
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At what speed, as a fraction of , will a moving rod have a length that of an identical rod at rest?
A 30-m-long rocket train car is traveling from Los Angeles to New York at 0.50c when a light at the center of the car flashes. When the light reaches the front of the car, it immediately rings a bell. Light reaching the back of the car immediately sounds a siren.
a. Are the bell and siren simultaneous events for a passenger seated in the car? If not, which occurs first and by how much time?
b. Are the bell and siren simultaneous events for a bicyclist waiting to cross the tracks? If not, which occurs first and by how much time?
A firecracker explodes in reference frame S at . A second firecracker explodes at the same position at . In reference frame S鈥�, which moves in the x-direction at speed v, the first explosion is detected at and the second at .
a. What is the speed of frame S鈥� relative to frame S?
b. What is the position of the two explosions in frame S?
A -long rocket train car is traveling from Los Angeles to New York at when a light at the center of the car flashes. When the light reaches the front of the car, it immediately rings a bell. Light reaching the back of the car immediately sounds a siren.
a. Are the bell and siren simultaneous events for a passenger seated in the car? If not, which occurs first and by how much time?
b. Are the bell and siren simultaneous events for a bicyclist waiting to cross the tracks? If not, which occurs first and by how much time?
This chapter has assumed that lengths perpendicular to the direction of motion are not affected by the motion. That is, motion in the -direction does not cause length contraction along the or axes. To find out if this is really true, consider two spray-paint nozzles attached to rods perpendicular to the 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 reference frame with its base on the axis; the other is placed in the reference frame with its base on the 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 finds that the nozzle, as it goes past, is less than above the 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?
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