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You have discovered a pit of unknown depth with impervious smooth walls. You also have a 1/2" twisted nylon rope which is 5,000km long and marked every meter. It is anchored to the ground by an attachment point that can bear the full weight of the rope, which feeds the rope through it and down into the pit (an automatic winch). The attachment point also has a sensitive force gauge built in, so your plan to measure the depth of the pit is to feed rope into the pit until the weight stops increasing, which will indicate that you've hit the bottom. The rope can be fed through the winch at a maximum 5km/h, and does not stretch.

How deep of a pit are you able to measure with this apparatus? Is the only limit the length of your rope?

Hint:

There is another limit. Why?

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    $\begingroup$ What is your perceived difference between "lowering the rope" and "throw it down the hole"? $\endgroup$ – Ian MacDonald Aug 19 '20 at 17:47
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    $\begingroup$ @MacGyver88 Well of course the whole rope isn't right here, I left most of it infinitely far away. I'm not a madman. $\endgroup$ – Nuclear Hoagie Aug 19 '20 at 17:50
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    $\begingroup$ I don't get it. What is the puzzle here? I feel you have to be more specific about the world you are describing and what is possible and what is not. Is your rope infinitely heavy, are you allowed to go down the hole why would there be any limitations on how deep you can measure, if you can measure at all? I have no clue how you would measure a whole with finite rope either. $\endgroup$ – Helena Aug 19 '20 at 18:16
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    $\begingroup$ @NuclearWang "several others being objectively incorrect" which ones? And how are they "objectively" incorrect? MacGyver88's answer is the only one where I'm fairly sure that it's not correct because it doesn't focus on the limits. Another way to explain it is that the question seems to us to boil down to "here's a physical system. Guess which part of the physical system will fail first." However you don't give details about the physical system - the setup everyone else is imagining might completely avoid the issue you have in mind as the limit. $\endgroup$ – Rob Watts Aug 20 '20 at 16:53
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    $\begingroup$ @NuclearWang I think a good puzzle has an answer that is not easy to find but easy to validate. Answer should be clearly right or wrong, and not have very different answers that are "more-or-less correct" depending on the interpretation of the puzzle. I very much doubt that there is an answer to this questions where everyone would go "Ahh this makes sense, this is clearly a right answer" $\endgroup$ – Helena Aug 20 '20 at 17:33
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The measurement you make is limited by:

  1. Your lifespan.
  2. The lifespan of the universe.
  3. The lifespan of your patience.
  4. The speed at which the rope descends.
  5. The weight capacity of the pulley (if using a pulley).
  6. The friction endurance of the rope (if not using a pulley).
  7. The radius of the rope vs the radius of the hole.
  8. The number of species, and ultimately civilizations, that form on the rope as it descends.
  9. The number of extinction-level events that occur on the rope.
  10. The durability of your measuring aparatus.
  11. The number of bits within your computer (if using a computer).
  12. The sheet size of paper (if using paper).
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  • $\begingroup$ Some fun answers, one of which is not far from the intended answer. $\endgroup$ – Nuclear Hoagie Aug 19 '20 at 19:28
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    $\begingroup$ @NuclearWang So, is there any way for us to figure out which answer you specifically intend, or is it just "guess which physical constraint I'm thinking of"? I'm confused by what physical facts are important and what should be ignored. $\endgroup$ – Deusovi♦ Aug 20 '20 at 6:50
  • $\begingroup$ @Deusovi The intended answer relies on actual physical properties and should be achieved relatively quickly. It does not rely on personal qualities like "your patience", impossible-to-predict outcomes like species forming on the rope, or timescales that would limit any process like the end of the universe. $\endgroup$ – Nuclear Hoagie Aug 20 '20 at 12:34
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It seems the question discards all physical constraints that are not be part of the intended solution.

Anyway, here is an idea.

One magic property of the rope that is not given is that to be indefinitely strong.

Assuming the rope is not, I would say it can only be lowered as deep as it can
hold its own weight. You can measure the tension at the top. When it stops increasing, you reached the bottom.

But that is not the real limit. Because when you reach the point where the rope would break, you can make a big loop, tie one end to the rope to triple the rope so it can support more (you cannot double as it would imply to cut the rope). Further up you can end the loop and start a double loop to make it fivefold. etc. The real limitation will be when the knots and the strands of rope become to wide to fit in the hole.

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The rope will eventually snap under its own weight.

The breaking length of a rope is given by

length = (Tensile strength / density) / acceleration due to gravity.

For nylon rope, this works out to be (78 MPa/1.13 kg/m3)/9.8 m/s2 = 7 kilometres.

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My idea

You lower the rope, which has constant tension, down the hole.

When the tension stops, or the rope exhibits relaxed movement towards one side or the other, you have reached the bottom.

Then, measure the length you pull back up.

Note - I'm assuming after lowering the rope for a while, the weight of the rope in the hole will be so great that it will cease to sway back and forth and will be pulled directly down due to gravity's increasing effect on it. Thus any movement at the surface will mean the tension is relaxing.

I just noticed the second part of the question.

"Is there any limit?"

The limit is the radius of the planet you are working on. I'm not a scientist or anything, but my guess is that at the center of the planet, the pull of gravity will not be in play any longer.

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  • $\begingroup$ Is this a planetary physics question, because I'm starting to second guess my answer. I'm sensing this goes a little "deeper" than what I actually know at the moment. Does this question need the knowledge tag? $\endgroup$ – MacGyver88 Aug 19 '20 at 18:24
  • $\begingroup$ No planetary physics needed, you can consider yourself to be on an arbitrarily large planet with constant 1g gravity everywhere. $\endgroup$ – Nuclear Hoagie Aug 19 '20 at 18:27
  • $\begingroup$ >"reached the bottom" The question explicitly states the hole is bottomless, as well as infinite. $\endgroup$ – Dapianoman Aug 19 '20 at 19:03
  • $\begingroup$ @Dapianoman, presently, it says "seemingly bottomless." I take that as it could have a bottom, it just seems like it doesn't. $\endgroup$ – MacGyver88 Aug 19 '20 at 19:10
  • $\begingroup$ You were tantalizingly close with notions about the tension and weight of the rope, but your starting point was a little off track - the rope does not have constant tension as you lower it (or throughout it). $\endgroup$ – Nuclear Hoagie Aug 20 '20 at 17:42
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The length is limited by

the physical strength of the rope. As the rope is being lowered down, the weight of rope being supported by the new rope coming in is steadily increasing. Eventually, there will be so much rope inside the hole that its weight will snap the rope at the top of the hole. That is the limit of how deep you can measure.

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Here a few limitations:

If the rope is not weightless, then it may eventually break, or you may fall in. Also, the rope will stretch as you lower it, throwing off any measurements.

No other equipment appears to be available. Just you and the rope. So without a pulley and braking system, you won't have the strength to hold onto the rope beyond a certain depth. Without a strain gauge, you might not know if the rope has reached bottom yet. Without a measuring device, you won't have an accurate way to measure the rope.

If other equipment is available, then trade the infinite rope for a big silver space blanket, a "laser", an electronic timing device, and 100 billion dollars. Throw the space blanket down the hole, then zap the "laser" down the hole and measure how long it takes the reflected light to return. Wait one minute, and repeat the measurement. Keep doing this until the distance stops changing, indicating that the space blanket has reached the bottom.

If you don't know what to do with the 100 billion dollars, then give it to me.

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My expected lifespan is 80 years.
The rope can only be lowered with at least an infinitesimal amount of tension on it when the topmost portion is lowered at <9.81 m/s.
Assuming I lower the rope at 9.81 m/s for my entire life, I can measure to a depth of:

9.81 x 60 x 60 x 24 x 365.25 x 80 / 1000 km

~= 24,767,000 km.

This could further be constrained by needing to take breaks to eat, sleep, etc. But I think this is a decent approximation of the practical limit of me attempting to measure this hole.

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