Editing 3087: Pascal's Law

{{comic
| number    = 3087
| date      = May 9, 2025
| title     = Pascal's Law
| image     = pascals_law_2x.png
| imagesize = 400x459px
| noexpand  = true
| titletext = Reductio ad absurdum fails when reality is absurd.
}}

==Explanation==
{{incomplete|Copy-pasted from the talk page, someone who knows this stuff well should take a look:

:''> 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.''

She is not pouring water; she seems to be depressing a small piston. This effects the next section, which claims that the lift would not work as drawn. However, I don't know just how it affects it; would it actually work? --[[User:Calion|Calion]] ([[User talk:Calion|talk]]) 04:40, 8 June 2025 (UTC)}}

{{w|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 {{w|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 {{w|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 {{w|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.

Note that the diagram shown in the comic would not work, however, because the height of the opening of the narrow tube is about the same as the height of the large piston carrying the large weight.  Because the heights are the same, the pressure due to gravity would also be the same, and the 1000kg weight would easily push the water out of the small tube against only the "weight" (pressure) of the air above it.  For this system to work by only pouring water into the tube the narrow tube would need to be much much higher than the piston.  Viewed this way, hydraulics are not all that absurd.  To raise the large piston 1mm would require putting several meters worth of water into the small tube to get the same volume (or in a hydraulic machine, moving the small piston several meters).  This is no different than how a pulley works, where a large weight can be moved 1 meter by pulling several meters of rope out at a much smaller force.  Conversely, if the person pouring the water into the tube started out at the same level as the large weight, the energy expended to raise the water up to the opening to pour it in (by pumping or carrying it up) would be considerable, and equivalent to the energy needed to raise the weight directly, so hydraulics are not some magic way to get around physics.

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 ''{{w|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. 

''{{w|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 {{w|proof by contradiction}}. In the title text, however, it is pointed out that some things that happen in the real world {{tvtropes|RealityIsUnrealistic|''are'' (or at least ''seem'') absurd}}, so it would be wrong to rely upon this method to {{tvtropes|AluminumChristmasTrees|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 [https://www.youtube.com/watch?v=EJHrr21UvY8 the demonstration works], if you can get enough height. 

{{comic discussion}}<noinclude>

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