- #1
FranzDiCoccio
- 342
- 41
- Homework Statement
- a tape moves at half the speed of light wrt a cylinder of radius 1 cm and extremely small lenght, whose axis is perpendicular to the tape. The cylinder moves through the tape, punching a hole into it.
Then the tape is stopped, and the area of the hole is measured.
Find the area of the hole.
- Relevant Equations
- Length contraction
The proposed solution is that the hole is an ellipse that is wider than the cylinder in the direction of the original velocity of the tape (its major axis is along the tape, and its minor axis across it).
The reason would be that the piece of tape cut by the cylinder is circular, but the tape undergoes length contraction in the frame of reference of the cylinder. But since this piece of material should be "contracted" along the tape, the hole in the tape should be longer.
Therefore (when the tape stops) the hole looks "expanded" by the Lorentz factor.
I'm not sure the problem is well posed. I think that "extremely thin cylinder" means that the cutting of the hole is instantaneous... But this seems strange.
Also, I'm not really sure as to why one should take the point of view of the cylinder. The provided solution does not comment about this at all.
Is this because both the cylinder and the tape are at rest in the frame of reference of the former, in the end?
In the frame of reference of the moving tape, the cylinder is an ellipse whose shorter axis is along the direction of motion of the cylinder itself (minor axis along the tape, major axis across it).
Assuming "instantaneous punching", in this frame of reference the hole has the shape of the hole puncher, whose cross section is contracted along the tape length, and unchanged along the tape width.
The stopping of the tape does not change the size of the hole. Hence the hole is smaller than the hole puncher.
The only reason I see against this argument is that the hole puncher is moving wrt to the tape, and according to (nonrelativistic) intuition this would enlarge the hole in the direction of motion at least a little bit, if we assume that the punching is not instantaneous.
If the process is instantaneous, I do not see a reason to prefer "the bit cut by the cylinder is circular in the f.o.r. of the cylinder" over "the tool used to punch the hole has an ellyptic cross section in the f.o.r. of the tape".
Also, the question concerns the size of the hole, which is in the tape. The proposed solution measures it from assuming that the piece of material removed by the cylinder is circular. A measure of the hole itself seems more direct.
The reason would be that the piece of tape cut by the cylinder is circular, but the tape undergoes length contraction in the frame of reference of the cylinder. But since this piece of material should be "contracted" along the tape, the hole in the tape should be longer.
Therefore (when the tape stops) the hole looks "expanded" by the Lorentz factor.
I'm not sure the problem is well posed. I think that "extremely thin cylinder" means that the cutting of the hole is instantaneous... But this seems strange.
Also, I'm not really sure as to why one should take the point of view of the cylinder. The provided solution does not comment about this at all.
Is this because both the cylinder and the tape are at rest in the frame of reference of the former, in the end?
In the frame of reference of the moving tape, the cylinder is an ellipse whose shorter axis is along the direction of motion of the cylinder itself (minor axis along the tape, major axis across it).
Assuming "instantaneous punching", in this frame of reference the hole has the shape of the hole puncher, whose cross section is contracted along the tape length, and unchanged along the tape width.
The stopping of the tape does not change the size of the hole. Hence the hole is smaller than the hole puncher.
The only reason I see against this argument is that the hole puncher is moving wrt to the tape, and according to (nonrelativistic) intuition this would enlarge the hole in the direction of motion at least a little bit, if we assume that the punching is not instantaneous.
If the process is instantaneous, I do not see a reason to prefer "the bit cut by the cylinder is circular in the f.o.r. of the cylinder" over "the tool used to punch the hole has an ellyptic cross section in the f.o.r. of the tape".
Also, the question concerns the size of the hole, which is in the tape. The proposed solution measures it from assuming that the piece of material removed by the cylinder is circular. A measure of the hole itself seems more direct.