3087: Pascal's Law

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Pascal's Law
Reductio ad absurdum fails when reality is absurd.
Title text: Reductio ad absurdum fails when reality is absurd.

    Explanation

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    Pascal's law states that when a change in pressure occurs in a static incompressible fluid, it is transmitted throughout the fluid and the same change occurs everywhere. That same pressure is applied outward to the walls of the container. It was discovered by mathematician Blaise Pascal in 1653. This principle has significant implications. Because force is a product of pressure times area, static pressure can be used to exert arbitrarily large (or small) forces by using larger or smaller pistons. This is the principle underlying hydraulics. Also, when under gravity, liquids exert greater pressure at greater depths, but Pascal's law means that the pressure will be the same at any given depth. Consequently, even a narrow column of water, if it's tall enough, will result in high pressure at the bottom of the column. If the bottom of the column spreads out over a huge area, that pressure will remain high, exerting tremendous force.

    Randall muses that, when he first heard of this law, he found it implausible, because it would be able to do things that, on the surface, appear "absurd". He then realized that, not only are these things possible, they either are or have been regularly used for practical purposes.

    The strip shows a classroom in which a character (presumably Randall) is sitting, being shown an image of a simple hydraulic press, which demonstrates the first "absurd" concept. In the image, a large cylinder, fitted with a large piston, is weighed down by a person and a weight labeled "1000 kg". The bottom of this piston is fed by a narrow tube which rises to an opening, into which someone is pouring water, which fills the cylinder, raising the piston. The notion that simply pouring water into a tube, by hand, would be able to lift over a tonne of weight seems absurd, but the pressure exerted by the height of the water, spread over the large area of the piston, is able to exert large amounts of force. The trade-off, of course, is that the cylinder is very large, so a huge amount of water would need to be poured in to lift the piston by an appreciable height. Actual hydraulic presses generally pump the liquid under pressure, rather than relying on gravitational force, but otherwise the same principle applies.

    A second objection Randall raises is that this principle would allow the destruction of entire mountains, with the very low-tech solution of digging tunnels and filling them with water, a technique that would have been available to ancient peoples. By digging even narrow vertical cavities with enough height, the pressure exerted by water at the bottom could become very high. If these channels feed into larger cavities, that pressure would exert across the entire area, creating forces that are potentially enough to shatter the rock face of the mountain. He points out that he later learned of the practice of Ruina montium ("wrecking of mountains" in Latin), which used exactly this principle. This was an ancient Roman mining technique in which small tunnels were dug into the side of a mountain. When the tunnels were filled with water, the rock adjacent to the tunnels would fracture, making it significantly easier to remove.

    In both cases, Randall's attempts to disprove this law by pointing out consequences that he considers implausible fail to work, because those consequences are exactly what are observed (and can be used) in real life.

    Reductio ad absurdum ("reduction to absurdity" in Latin) in the title text is a form of argument in which something is assumed to be true and then the logic is followed until it results in an absurdity or obvious falsehood, which is taken as evidence that the original premise cannot, therefore, be true. In mathematics it's called proof by contradiction. In the title text, however, it is pointed out that some things that happen in the real world are (or at least seem) absurd, so it would be wrong to rely upon this method to conclude that they don't occur.

    Transcript

    [At the left, a teacher is holding a pointer, pointing at a picture on the screen.]
    [The picture shows a hydraulic lift, with a small fluid vessel on the left connected to a tube at the bottom, which connects to a large vessel on the right. On top of the large vessel is a weight labeled 1000 and a Cueball. The fluid in the large vessel is labeled with an upward arrow. Megan's hand is over the small vessel, with a downward arrow indicating that she's pressing on it.]
    [Cueball, Hairbun, and Blondie are sitting at school desks going left to right.]
    Cueball: No, that can't be right.
    Cueball: If hydrostatic pressure worked that way, then you could use it to make machines that exert near-infinite force.
    Cueball: And ancient people could have demolished entire mountains just by drilling small tunnels and filling them with water.
    [Caption below comic:]
    When I first learned about Pascal's law, I tried to disprove it by showing that it would lead to absurd consequences, but it turns out hydraulic presses and ruina montium are both real things.

    Trivia

    In the image, assuming a 2,200 lb weight (1000 kg) and an adult who weighs around 200 lb, both on a circular piston with a 6-foot diameter, the water pressure would need to be about 0.6 psi to lift them. That means the water column into which water is being poured would have to be about 16 inches higher than the bottom of the piston.

    The concept of "runia montium" is similar to a demonstration which was attributed to Pascal himself, in which he supposedly inserted a tall, thin tube into an otherwise sealed barrel full of water. By adding water to the top of the tube, increasing pressure would be exerted inside the barrel, until it burst. This story may be apocryphal (all surviving accounts were written centuries after Pascal's death), but the demonstration works, if you can get enough height.


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    Discussion

    I remember learning about this and thinking it was intuitive, but I didn't really think of these consequences. Maybe everybody is making powerful lifting machines for lifting cars and houses with your bare hands, rather than explaining the article, that there isn't one yet. Pascal's law basically says that if you make one end of a container of fluid X times larger, then any force exerted on the small end is multiplied by X on the large end, so you can make it near-infinite by making the small end very small. But you'll need a little more machinery added (like a gear system) if you want the distance actually moved to be higher. Actually I think that might undo the gains in force entirely. That might be how it happens, it might swap distance for force so the same work is performed.

    I believe they are called bottle jacks, I carry a few with me in the trunk. Seebert (talk) 16:22, 13 May 2025 (UTC)

    Hey, remember that comic where Randall challenged people to fold a paper too small? This hand-makeable device could get farther on that! 172.70.111.110 21:46, 9 May 2025 (UTC)

    yes, that's how it works; the total work is constant and the hydraulic system is converting a small force over a long distance to a large force over a small distance. if you additionally want the force to be over a larger distance, you need to put more energy into the system or else you could push this machine with its own output and get free energy from nothing. really though hydraulics are just smoother, backlashless, equivalents to a gear train in the first place so you generally wouldnt need to use both. - Vaedez (talk) 23:37, 9 May 2025 (UTC)
    Small tube needs to be X times as long to get same displacement. Good for linear force rather than torque. Fluid's own pressure can be the force if tube is long enough. 172.68.55.33 11:41, 10 May 2025 (UTC)

    As someone old enough to remember the slashdot effect, I wonder if XKCD comics generate a similar effect on search engines. Though I doubt they would buckle under the weight these days. 172.69.60.148 22:00, 9 May 2025 (UTC)

    To whoever wrote the initial transcript, remember that we don't include the title text. Barmar (talk) 22:06, 9 May 2025 (UTC)

    I have to wonder whether he has the same disbelief of, say, levers... which allow one to move the Earth. Jordan Brown (talk) 23:34, 9 May 2025 (UTC)

    I'm not sure, but he did make 857: Archimedes Log of n (talk) 10:47, 13 May 2025 (UTC)

    Although some laws of physics are absolute and lead to extreme consequences, others are taught in a simplified form that can lead to wrong conclusions. For example, "Light and heavy objects fall at the same rate" can be used to prove that objects fall at the same rate on the Earth and the Moon - which is far from correct. If the Moon were somehow dropped onto the Earth, it would fall at a certain rate. The Earth dropped on the Moon would necessarily fall at the same rate. So if the Moon falling on the Earth fell at the same rate as a bowling ball, then the bowling ball would have to fall at the same rate on the Moon. When I read Heinlein's _The Rolling Stones_ as a pre-teen, where he describes things falling slower on the Moon, I applied this reasoning and concluded that Heinlein must have made a mistake. The solution to this paradox is that something as big as the Moon will not only accelerate toward the Earth, it will significantly accelerate the Earth toward it, so the Moon does not actually fall at the same rate as a bowling ball. Cphoenix (talk) 01:01, 10 May 2025 (UTC)

    I think if you stand at the shared center of mass of the Earth and Moon, that then you see the Moon falling toward the Earth according to its constant field of gravitational acceleration, as well as the Earth falling toward the Moon according to its constant field of acceleration. It was indeed confusing for me to realize this, involving visit to pages such as https://en.wikipedia.org/wiki/Gravitational_acceleration . F=Gm1m2/r^2 so if m1 is taken out you get a constant F=m1 a2 and vice versa. But I think the page says this only holds if the masses are far enough from each other to be treatable as points. 172.68.55.47 11:57, 10 May 2025 (UTC)

    Besides, isn't it better to just believe in Pascal's Law if it offers a reward of near-infinite force? StapleFreeBatteries (talk) 04:58, 10 May 2025 (UTC)

    I'll wager that you're pleased with that reference... ;) (Whether or not you actually were!) 172.71.26.43 15:58, 10 May 2025 (UTC)

    Could this be referencing or inspired by this recent paper talking about the use of hydraulics to build pyramids? https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0306690 172.68.234.169 08:52, 10 May 2025 (UTC)

    It could be noted that simple machines can multiply forces more or less arbitrarily, but only up to what the machine itself can withstand. Many "why didn't the ancients do that?" can be answered with "bronze kind of sucks". 172.69.246.149 14:19, 10 May 2025 (UTC)

    Right. Archimedes's "Give me a lever long enough..." assumes that the lever is made of a material that won't buckle or snap under the weight of the Earth. If there were a material like that it would make building long bridges much simpler. Barmar (talk) 16:58, 10 May 2025 (UTC)
    It is also quite difficult to efficiently contain a fluid under pressure in a moving system. All considerations of innovation-stiffling aside, the same problem hindered the use of the Stirling engine untill very recently. Force applied to a rigid mediom is much easier to predict / monitor (a solid either holds or doesn't, but it doesnt leak -not in a way that would have been hindering the use in everyday life by "the ancients" anyway). To a certain extent it is also the reason why most common bikes still use cable brakes : sure it't less efficient but it's much cheaper and you can repair them in no time with common garden-shed-grade equipment. 172.71.232.101 (talk) 22:07, 11 May 2025 (UTC) (please sign your comments with ~~~~)
    I think the hydraulic press is different from the lever in that the material only needs to withstand the pressure of the fluid, not the total force exerted by the press. The force is spread over the large area of the pressing end, such that the entire surface area holds the same pressure as the smaller end at every point. 162.158.63.90 23:33, 12 May 2025 (UTC)
    Donald Knuth?

    Is there a name for the teacher character (a cueball with tufts of hair on the sides of his head)? Barmar (talk) 16:58, 10 May 2025 (UTC)

    The teacher appears to be Donald Knuth, though not sure why Randall has used him.--Darth Vader (talk) 09:14, 11 May 2025 (UTC)
    We should add the comic to Category:Comics featuring Donald Knuth then. --FaviFake (talk) 10:09, 11 May 2025 (UTC)

    Here's a reference that claims that water hammer, rather than just depth pressure, was the major effect used in ruina montium: https://blog.ferrovial.com/en/2022/08/ruina-montium-use-water-for-digging-romans/ -- Dtgriscom (talk) 00:04, 11 May 2025 (UTC)

    A few years back I posted a question on Stack Exchange (https://history.stackexchange.com/questions/60189/which-mountain-collapsed-in-france-in-1820-21) about a report, in a 19th-century biblical commentary, of a mountain collapsing due to just this kind of thing - fluid pressure building up. First thought when I saw the reference to ruina montium in the comic was that maybe that's it - but no. So, just curious: anyone have any thoughts on what event is being referred to in that commentary? (The one suggestion that someone made there, about a glacier advancing, doesn't seem to fit.) Thanks! 172.71.23.87 03:38, 11 May 2025 (UTC)

    No idea, but I gotta say, that font or scan is unreadable. Some words are decipherable like הר גבוה and ובמקומו but damn, about half of it is just splotches. No idea how you could read that. --NeatNit (talk) 05:12, 11 May 2025 (UTC)
    I hear. There's a transcript of it (though without the diagram) at https://he.wikisource.org/wiki/%D7%9E%D7%9C%D7%91%D7%99%22%D7%9D_%D7%A2%D7%9C_%D7%90%D7%99%D7%95%D7%91_%D7%9E%D7%91_%D7%99%D7%93. 172.70.175.208 18:50, 11 May 2025 (UTC)
    ruina montium is an ancient Roman gold mining technique in which mountains are crumbled using hydraulic pressure. it was particularly used at Las Medullas in Spain, as detailed in this article https://www.academia.edu/14101438/The_Roman_gold_mine_of_Las_M%C3%A9dulas Orenwatson (talk) 14:37, 13 May 2025 (UTC)

    Should there be a goofs section here, because the white board is drawn with perspective, but the picture on it is not. It will look odd, like the whole contraption is tilted, to the studentsDrkaii (talk) 09:10, 11 May 2025 (UTC)

    I don't know how to write it, but someone should probably write the actual *point* of the comic: Randall coming up with ridiculous scenarios to disprove the theory, only to be informed that those scenarios actually happened. 172.69.70.145 (talk) 23:57, 11 May 2025 (UTC) (please sign your comments with ~~~~)

    Also don't forget that there was a Mythbusters episode to this effect (well, in reverse and with air, but same principle), it's possible to lift a car with a single household vacuum-cleaner just by splitting the hose into a few dozen rather large suction cups. More surface area = more force for the same pressure. 162.158.3.106 (talk) 07:06, 12 May 2025 (UTC) (please sign your comments with ~~~~)

    I'm sure there must be a name for the belief of absurdity (or counterintuitiveness) due to ignorance or failure/refusal to understand the entirety of a circumstance, or overgeneralization of known outcomes that only partially match the circumstance in question. If there is, that should probably be mentioned in the explanation. Does anyone here know what that might be? SammyChips (talk) 16:17, 12 May 2025 (UTC)

    Maybe not entirely congruous with what you describe, but the "Argument from incredulity" is still somewhat similar. 141.101.99.94 22:13, 12 May 2025 (UTC)
    Do you mean forming incorrect inferences by leaving out some of the facts? 162.158.62.109 23:35, 12 May 2025 (UTC)
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