alan123hk said:
I found the following article about the rod and hole paradox.
It's actually not the same.
The first part of the paper discusses a "car and hole paradox" which is quite different, since it assumes that the car is only supported by its wheels, not continuously over its length, and puts a spoiler at the front of the car that extends well beyond the front wheel, and is assumed not to start falling until not only the front wheel starts falling, but until the information that the front wheel has started falling propagates at finite speed to the front of the spoiler. This is obviously a different scenario from the standard rod and hole paradox and cannot be used to draw useful conclusions about it.
The second part of the paper discusses what purports to be the standard rod and hole paradox, but makes a crucial change: it
lowers the far end of the hole. It also ignores the crucial points already raised in this thread, that to realize this scenario with a rod of ordinary length, the "force of gravity" (or whatever force causes the rod to fall once its support is withdrawn) must be many, many orders of magnitude larger than 1 g, and also many, many orders of magnitude larger than the internal forces in the rod. So the claim in the paper that the rod's "proper stiffness" is unchanged does not justify the paper's conclusion that the rod will
not free-fall once its support is removed.
The paper does make one valid point: that if we assume the downward force is gravity, the rod will be compressed under its own weight while it is supported, so the initial motion of the rod once the support is removed will be for it to uncompress itself. This means that the top of the rod will initially
not be in free fall when the support is removed (I am describing things here in the hole's rest frame); it will still be experiencing an upward force from the layer of the rod below it, but this force will rapidly decrease to zero and the top of the rod
will start accelerating downward, just initially with less than the full "acceleration due to gravity".
However, the only way the paper can get this valid point to lead to a conclusion that the rod will
not fall into the hole is, as mentioned above, to
lower the far end of the hole. The drawings in the paper make it appear that the lowering is slight; but again, if you actually run numbers for a rod of ordinary size with the ##\gamma## values in question, you will find that the lowering has to be huge.
Also, given that the
bottom of the rod
does immediately start free-falling when the support is removed, if the speed of sound in the rod is slow, that just means the rod will end up
stretching vertically, not just "uncompressing" to its unstressed height. The paper does not take this possibility into account at all.
Finally, the paper assumes "exactly horizontal stress propagation", without ever realizing that
this can only be true in one frame. This issue has also been brought out in previous posts in this thread.
Unfortunately, this is a good illustration of why the "researchgate" site is not a trustworthy source (as we have often found in previous threads here on PF).