Throughout this problem the inertial rest frame of the rod will be referred to as the rod’s frame, while the inertial frame of the cylinder will be referred to as the cylinder’s frame.
A rod is traveling at a constant speed of v=\frac{4}{5} c to the right relative to a hollow cylinder. The rod passes through the cylinder, and then out the other side. The left end of the rod aligns with the left end of the cylinder at time t=0 and x=0 in the cylinder’s frame and time t^{\prime}=0 and x^{\prime}=0 in the rod’s frame.
The left end of the rod aligns with the left end of the cylinder at the same time as the right end of the rod aligns with the right end of the cylinder in the cylinder’s frame; in this reference frame the length of the cylinder is 15 \mathrm{~m}.
For the following, sketch accurate, scale diagrams of the motions of the ends of the rod and the cylinder on the graphs provided. The horizontal axis corresponds to x, the vertical axis corresponds to c t, where c is the speed of light. Both the vertical and horizontal gridlines have 5.0 meter spacing.
a. Sketch the world lines of the left end of the rod (RL), left end of the cylinder (CL), right end of the rod (RR), and right end of the cylinder (CR) in the cylinder’s frame.
b. Do the same in the rod’s frame.
c. On both diagrams clearly indicate the following four events by the letters A, B, C, and D.
A: The left end of the rod is at the same point as the left end of the cylinder
B: The right end of the rod is at the same point as the right end of the cylinder
C: The left end of the rod is at the same point as the right end of the cylinder
D: The right end of the rod is at the same point as the left end of the cylinder
d. At event \mathrm{B} a small particle \mathrm{P} is emitted that travels to the left at a constant speed v_{P}=\frac{4}{5} c in the cylinder’s frame.
i. Sketch the world line of \mathrm{P} in the cylinder’s frame.
ii. Sketch the world line of \mathrm{P} in the rod’s frame.
e. Now consider the following in the cylinder’s frame. The right end of the rod stops instantaneously at event B and emits a flash of light, and the left end of the rod stops instantaneously when the light reaches it. Determine the final length of the rod after it has stopped. You can assume the rod compresses uniformly with no other deformation.
Any computation that you do must be shown on a separate sheet of paper, and not on the graphs. Graphical work that does not have supporting computation might not receive full credit.
Following are the answer sheets for some of the graphical portions of the question.
The Cylinder’s Frame
