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About 60 years ago a thought experiment was proposed which is widely referenced, e.g. as "Man falling into grate variation" of "The ladder paradox" or, perhaps more poignantly, "The tank paradox":

Rindler, Wolfgang (1961). "Length Contraction Paradox". American Journal of Physics. 29 (6): 365–366; cmp. http://home.agh.edu.pl/~mariuszp/wfiis_stw/stw_rindler_lcp.pdf

A 10-in. long "rigid" rod moves longitudinally over a flat table [which has significant thickness]. In its path is a hole 10 in. wide [cut out of the table body].

[...] let us make the experiment more concrete. The hole shall be filled with a trap door which will be removed (downward, and with sufficient acceleration to allow the rod to fall freely) [...] at the instant [of table edge and adjacent trap door edge being passed by] the hind end of the rod [and all simultaneous instants of the trap door constituents].

This precaution eliminates the tendency of the rod to topple over the edge. [...] the rod will remain horizontal, in the frame of A [i.e. of the table]. [...] let it be understood that the rod is originally a rectangular parallelepiped

This descrption is accompanied by Fig. 1, with panels (a) and (b). Both panels seem to indicate that the far side wall inside the hole is about to be hit by the entire front side of the rod, where the separation in height between the wall constituents being hit by the lower front edge of the rod and the wall constituents being hit by by the upper front edge of the rod appears equal to the (original) height of the rod.

Apparently, therefore, lower and upper front edge of the rod were supposed to have started their free-falls simultaneously!

Isn't this in conflict with causality, namely with the idea that the upper front edge should respond only after some significant delay to the trap door being pulled from underneath the lower edge ?

For an acceptable answer, please

  • estimate the magnitude of gravity/downward acceleration according to the trajectory of the lower front edge of the object shown in Fig. 1, assuming the 10-in. long object was originally 5/3 in. high, crossing the hole and trap door with $\beta = \sqrt{1 - (1/4)^2}$, and consisting of fragile matter,

  • accordingly estimate the trajectory of the upper front edge of the object, and especially, its hight above the table surface upon having passed the hole,

  • discuss which trajectories may be expected instead for an object of significant trench-crossing capability (and related significant object-climbing ability), given the same magnitude of gravity/downward acceleration.


Notes:

  • An object consisting of fragile matter is for instance a pile of sand, or here more applicable: sandgrains arranged in the shape of a rectangular parallelepiped, which are "sustaining their shape against the force of gravity", i.e. given sufficient "support from underneath" for each of its constituents;
    where each constituent is expected to start "falling freely" as soon as it has lost "its support from underneath".

  • A tank is not thought as consisting of fragile matter, but instead as having significant practical trench-crossing capability and obstacle-climbing capability.

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  • $\begingroup$ So what you are saying is that the upper front edge has to continue its horizontal movement even after the lower front edge start falling down right? Indeed, it takes time for upper front edge to understand there is no trap door anymore. Probably the best thing we can do here is that we assume that trap door will be removed a little bit sooner than what Rindler proposed. $\endgroup$ – Paradoxy May 1 '19 at 15:03
  • $\begingroup$ @paradoxy: "So what you are saying is that the upper front edge has to continue its horizontal movement even after the lower front edge start falling down right?" -- At least "for a while", which should be very noticeable in Rindler's drawings. "Probably the best thing we can do here is that we assume that trap door will be removed a little bit sooner than what Rindler proposed." -- Which trajectories would you then expect for upper vs. lower front edge; for "fragile matter", or for "a tank" ?? $\endgroup$ – user12262 May 1 '19 at 16:04
  • $\begingroup$ Ofc, what i have said to assume won't remove this delay between upper and lower front edge, what it actually does is that although the upper front edge will continue its horizontal movement for a little while, it won't arrive at next wall at all, because before that the upper front edge has received signal from below, so it will fall as well. We would expect some kind of rotation here. Also if we imagine something like a paper (with almost no thickness) still Rindler's viewpoint might be available, kind of. $\endgroup$ – Paradoxy May 1 '19 at 23:15
  • $\begingroup$ @paradoxy: "Ofc [...]" -- I'd certainly appreciate a thorough quantitative treatment, foremost of the trap door experiment proposed by Rindler (and I've just updated/specified my question accordingly); but optionally also for cases with the trap door opening sooner (or entirely without trap door). "Also if we imagine something like a paper (with almost no thickness) still Rindler's viewpoint might be available" -- Indeed, but entirely expected/unsurprising/non-paradoxial. Men, and tanks, however, have finite height. $\endgroup$ – user12262 May 2 '19 at 1:17

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