A Meditation on Buoyancy

This video explores the fundamental nature of buoyancy, and how it arises from molecular collisions and the weight of stationary water. Along the way we will meet some concepts from vector calculus, which will be useful for understanding buoyancy, and which we will also see again in future videos.
Please do not be discouraged if you are seeing some of these ideas for the first time, and are confused by them. That’s totally normal when learning new kinds of math. You’ll find that over time these ideas become very natural.
Three concepts central to this video are:
1. The dot product. This is the simplest way two vectors can be multiplied. Two vectors go in and one scalar number comes out, which relates to the vectors’ size and the extent to which they align.
en.m.wikipedia.org/wiki/Dot_p...
2. The divergence theorem. This is a very versatile theorem in vector calculus that lets you turn flux integrals into volume integrals, and vice versa. Basically it equates the flux through a closed surface to the extent to which the vector field has a source or a sink inside the region bounded by the surface. We used this theorem twice in this video, in two different directions and contexts, so that should tell you something about how widely applicable this theorem is!
en.m.wikipedia.org/wiki/Diver...
3. The normal vector. This is just a vector which is perpendicular (in physics, “normal” is often used to mean “perpendicular”). Typically the vectors are defined to point outward from a surface. Often the normal vectors have length 1, which makes them normalized another sense of the word (vectors with length 1 are often called “normalized”). It’s actually a confusing terminology, some to think of it! But the concept is straightforward.
en.m.wikipedia.org/wiki/Norma...)
Please let me know if you have any questions, comments, or suggestions for future videos, and thanks for watching! :)
#math #maths #physics #science #sciencefacts #buoyancy #calculus #engineering #fluiddynamics #mechanicalengineering #civilengineering #meditation #zen #nature
Thanks to fesliyanstudios.com for the background music! :)

[@shashikantsingh6555]

Most beautiful part is that the creator responded to all the comments.. very kind

[@RichBehiel]

If someone takes the time to leave a comment, I try to respond! :)

[@Charles_Reid]

This is a great video. For the "proof" of the buoyancy law, my teacher just derived the buoyancy force for a cylinder and for a cube and showed that they were the same. I had always wondered about how to prove it for a general shape. Such an elegant proof. You make great content, I will watch your other videos too

[@RichBehiel]

Thanks! :) And that’s funny, my teacher did the same thing. Later on when I saw the general proof, it was really satisfying.

[@JOHNSMITH-ve3rq]

Lmao what an absolute gem this channel is. Just what the heck!?? Dude throw up a patreon I would pay for these

[@RichBehiel]

Thanks! :) I hesitate to make a Patreon because I’m not sure I can commit to posting videos on a regular basis, and I don’t want to let people down. My main priorities in life are family and work, and especially over the next year or two I might not have much time for videos, even though I love making them. I’m working at a startup so if I have to choose between that and making videos, I’ve gotta choose the startup. So I wouldn’t feel right having a bunch of people paying me on a regular basis for this thing I’m not sure I can commit to. Thank you though, I really appreciate your comment and I’m glad you enjoyed the video. But if you’re feeling generous, I do have SuperThanks enabled on my videos, so that’s an option! :) That feels better because it’s for something I already made, rather than something I may or may not do in the future.

[@lordsixth5944]

Thanks to you I am getting back into physics

[@RichBehiel]

That’s great! I’m so glad to hear that :)

[@harrychen3316]

Man this is so GOOD Btw your sound is very clear and euphonious

[@RichBehiel]

Thanks Harry! :)

[@mechwarreir2]

your simulations are incredible insight! Your channel needs more views. - Vi

[@RichBehiel]

Thanks Viraj! :)

[@ivarangquist9184]

Wonderful animations, relaxing audio, and clear explainations. Never have I had such a deep (no pun intended) understanding of buoyancy.

[@gabitheancient7664]

this video made me want to learn vector calculus, amazing

[@DanielKRui]

When you set up the intuition @10:10 with no math symbols, and then gave the simple example of a circle displaying this "cancellation" property this time accompanied by the math symbols, and then changed the animation from a circle to a blob @15:40, I was hit (it really did feel like a physical hit) with the thought "oh! There's perfect cancellation for the blob too!" which made the ensuing mathematics much much much more natural and intuitive. Over the my years of studying math (now at the graduate level), I have come to realize just HOW important it is to have the right picture in mind, or the right core philosophy/slogan. And thanks to your picture, and philosophy of "no shape can face more left than right, no more up than down", this result (and by proxy, the divergence theorem) now feels much more intuitive. Somehow, the most difficult but most crucial component of teaching math is to trick your student into conjecturing the result you want to prove, before you state/prove it! And finally, I must tell you how much I appreciated your philosophical remarks at the end (this is the 2nd video I've watched of yours; the 1st I watched right before this one, was your video on the Hopf fibration and I also appreciated your philosophical remarks there!). People in math sometimes throw around that von Neumann quote "Young man, in mathematics you don't understand things. You just get used to them." There is an element of truth to this quote, but I disagree with the general principle it's pushing for. Many things in math seem miraculous, but once you draw the right pictures/animations, or have the right grounding/core philosophy/slogans, you see that it is "merely" a clever and beautiful combination of "simple" principles. And metamiraculously, these "simple" but paradoxically extremely deep principles feel (at least in my experience) intuitively true (i.e. true in a way that sort of resonates so fiercely that I can feel it in my whole body when I discover them in the foundation of whatever black box I'm currently trying to unravel). And it is the goal of mathematics, or at least mathematics educators, to make what we have to "just get used to" as easy and profoundly simple as possible.

[@RichBehiel]

This is a wonderful comment, thank you! :) And I agree, there are some things in math that are mind blowing and hard to understand (spinors come to mind), but many things can make perfect sense even if at first they seem confusing. I think a lot of that is just due to mathematical notation being so cryptic and parsimonious. It’s efficient and useful once you know it, but hard to learn from.

[@garyscott4094]

A masterful explanation that really makes the internet worthwhile. Thank you so much.

[@RichBehiel]

Thanks for the kind comment! :)

[@atticmuse3749]

This is an absolutely beautiful video, I'm so glad I found your channel. Just finished watching through your series deriving the Klein-Gordon and Dirac equations, and your latest spinor video was fantastic. I wish I had these videos when I was in university!

[@RichBehiel]

Thanks, I’m glad you’re enjoying these videos! :)

[@tedsheridan8725]

Beautiful meditation. Being familiar with vector calc, and having learned buoyancy over 25 yrs ago, you still managed to blow my mind twice (the vector integral proofs). Keep em' coming!

[@logo2462]

Wish we would have done this in my calc 3 class. 10/10

[@victorscarpes]

I think I have the opposite take reductionism in nature. Seeing how amazing things can emerge from such simple principles makes the whole thing even more beautiful to me. Kinda like looking at an ancient sculpture and appreciating the artist that was able to make it with just a block of rock and a chisel. To illustrate my point, I like to quote Leonardo Da Vinci: "Simplicity is the ultimate sophistication"

[@user-lh2hx5xf4e]

Man I'm glad I found your Channel sir

[@marshallsweatherhiking1820]

Seems like this is one of the few derivations that is quick to show without throwing out any details that would make the math inconvenient. As for thing people take for granted that are actually quite remarkable, one is the fact that the solid state of water is less dense than the liquid state. It doesn’t make sense why this would be without knowing that water molecules have a peculiar way of joining into a hexagonal lattices with lots of space when the hydrogen shells are locked, but can squish together more tightly when in motion. Most other solids, like molten metals, are denser than the liquid state. Without knowledge of molecular structure that’s impossible to see with the naked eye, the reason that ice floats is mysterious.

[@arbodox]

Wow! Such an elegant derivation! Your chill voice really does help make me pay attention to the whole thing.

[@ca8064]

Watched the whole thing, don't regret it, thank you :)

[@gabitheancient7664]

I have learnt vector calculus, AMAZING!!!

[@AlexTornado318]

I saw your previous comment and nice job!

[@gabitheancient7664]

@@AlexTornado318 ty

[@colinfischer9183]

This is awesome, thank you for making this :)

[@zacharyshifrel9107]

Not sure if you’re aware of it, but there’s a hilarious idea in the flat earth community that buoyancy can somehow replace gravity and GR. Anyways, this is very well done.

[@RichBehiel]

Oh yes, I’ve seen that idea! 😂 Thanks, glad you enjoyed the video :)

[@skg7531]

I love this channel, you deserve much more attention

[@RichBehiel]

Thanks! :)

[@salsaman4374]

THIS IS AWESOME!!!!!!!!

[@MostlyIC]

Richard, may I suggest you do a follow-on for magnets, one of the things you don't actually understand from introductory physics is why does a bar magnet attract a piece of iron, even though you might have learned about magnetic forces between parallel current carrying wires, and torque on magnetic dipoles, and some other concepts. the answer to this conundrum is also a result of vector analysis, and IMHO worth understanding (and I'm pretty disappointed that no introductory textbooks that I've seen go into it). the video should start with a magnet (dipole) in a uniform magnetic field, IE a situation where by symmetry you can conclude the magnet does not get attracted to either pole of the source of the uniform field. And since you didn't "water down" your buoyancy video, I expect you won't "iron down" your magnets video 🙂 !

[@RichBehiel]

Great suggestion! I’d love to get into electromagnetism, and actually plan to in the not too distant future. Next video will be on spinors, then Dirac eigenstates for free particles in motion, then applying local phase symmetry to the Dirac field to see how all of electromagnetism arises from that elegant principle, then Hydrogen Part 3, but after that I could definitely get into some examples of various electromagnetic phenomena, starting with the “source code” of the Dirac equation and branching out into the specific equations and vector fields and such. The dipole is a great example that’s definitely worth looking at.

[@MostlyIC]

@@RichBehiel looking forward to all of that, but what I'm suggesting is taking a look at a magnet and a piece of iron from the "classical" not "quantum" point of view, where the magnetic field is visualized in a way similar to the hydrostatic pressure field in water.

[@oscaralberto8141]

Thanks to You.

[@DavidLopez-ir3lj]

Woah I’m listening to this while making images in my head then I check the screen later and it turns out it’s so accurate it’s trippy

[@babybeel8787]

24:05 I laughed out loud haha, thank you for your content, really insightful!

[@caisiegel7640]

Amazing !!!

[@GodinciOrg]

Great voice!

[@WookENTP]

Beautiful. The only thing I'm missing in this video for a holistic picture is why pressure is a function of z in the first place? :)

[@RichBehiel]

The deeper you go, the more water on top of you pushing down. Twice the depth, twice the weight of that water. Because water is a liquid and doesn’t sustain shear stress at rest, the hydrostatic pressure is determined only by the weight of the water above any given point.

[@WookENTP]

@@RichBehiel Yes, but why does water push you harder the deeper you go? Why curvature of spacetime creates a force? I mean deeper than that, e.g. Vsauce's "Which way is down" explanation - time goes faster the deeper you go, so the diff in time speed "pushes" things down the same way as, well, buoyancy...

[@WookENTP]

@@RichBehiel I doubt we (humanity) know the answer (yet), I just wanted to give you this small feedback because I can see you try to cover all the levels, because it's important for a complete, intuitive, picture :)

[@nobodythisisstupid4888]

In my calculus II course that i took over the summer last year, depth/pressure was something I struggled with the most. I’m not sure why since it doesn’t seem quite so bad now that I have completed a year of calculus based physics and calculus III, but I wish I had access to this video at that time. Great job!

[@RichBehiel]

Thanks, glad you enjoyed the video! :) Funny you say that, I struggled too with these depth/pressure calculations in calc 2. It’s the kind of thing that’s confusing at first, but then one day it clicks in a way that you’ll never forget.

[@WarrenMooreIII]

Absolutely beautiful presentation. Thank you for the lucid explanation and illustrations

[@RichBehiel]

Thanks, I’m glad you enjoyed it! :)

[@lileloquent]

I was searching for this for over years, from grade 9 and now I'm a student of 12th grade (almost about to finish!) I'm entering college soon. I don't understand vector calculus much but I understand it to some extent, the presentation you have was splendid! I always had my conceptual issues understanding the proper reason of buoyancy but nevertheless you made me a bit relieved through the premise of this video. From 2019 to 2023 a problem, I suffered from the lack of understanding this topic. This was the topic I had least knowledge in. I got some explanations myself, like the displaced volume of fluid was meant to be there, and the fluid geometry around that particular volume insisted on putting a force on that volume in absence of fluid in that specific volume. Thus is how I thought buoyant force worked as I previously presumed from newton's third law, from a mechanical perspective growing from the conservation of some inertia-like quantity (momentum). Though in the end, I found that many people give the same explanation (Though I thought of it for myself before hearing from anyone else). The divergence theorem, I visualized it before, so I kinda know that it's true but I haven't got quite the solid proof yet (didn't search hard enough tbh). So for the time being I'm using it something as I accept with feel. (Not like the buoyant force that I never felt other than seeing practical examples) Thank you again. If you've read the full comment, thank you again, for your time.

[@RichBehiel]

Sounds like you’re on the right track, and are developing a good understanding of vector calculus and buoyancy! :)

[@arthurbehiel4632]

Very uplifting!

[@RichBehiel]

😂

[@StudyEnggFocus]

Hello 👋 Could you make some videos on rotational mechanics and circular dynamics if you have time. Would really appreciate! Thanks for your efforts.

[@GeoffryGifari]

To think that this method also works in the case of the submerged object having non-uniform weight density (like when the object is composed of many parts)...

[@peterc-s6423]

the furthest ive gotten with calculus is some basic path integrals (length of a path, basically), not really touched vector calculus. i have covered a bit of multivariable calculus but thats the extent of my knowledge here. this video somehow made complete sense to me even though i dont feel like i have an intuition for (or even any experience with) surface integrals or volume integrals or flux integrals or whatever. the divergence theorem still makes no sense to me but that is probably just because i dont understand the fundamentals of the flux integrals or volume/area integrals (nor the notation) either way this is a very well put together video and has made me quite interested in learning some vector calculus even though it might not be super applicable in the field ill be studying. thank you!

[@RichBehiel]

Glad to hear that! :) As for the divergence theorem, a good concept that leads up to it is the continuity equation for an incompressible fluid. The velocity of an incompressible fluid will have zero divergence, meaning whatever goes into an infinitesimal volume has to come out (positive divergence can be thought of as a source, negative as a sink, so an incompressible fluid has neither sources nor sinks anywhere). The continuity equation then just says that the flux integral of the velocity field around any closed surface has to be zero, because whatever comes in has to go out, and that’s equivalent to the statement that there are no sources or sinks of fluid since it’s incompressible. Once you’ve imagined that, you can imagine having some kind of field where the divergence isn’t necessarily zero, for example a force field. In that case, there’s still a relationship between how much sourcing and/or sinking is going on inside of a closed surface, and how much is leaking out or falling through the surface. More source = more outflow, so higher divergence = higher flux integral. Same goes in the case of more sink, higher negative flux integral. When there is no source or sink, the divergence theorem reduces to the continuity equation for incompressible flow. The divergence theorem is a mathematically exact way of describing that relationship between source/sink in the volume and outflow/inflow through the surface.

[@peterc-s6423]

@@RichBehiel i still don't have an intuition for anything past a singular integral being essentially an "area under a curve" and a closed path integral being essentially a sum of some value across some path, if that value is just the length then you end up with the length of the path. i was quite impressed with myself for understanding and having a bit of an intuition for the part of the video with the closed path integral of the normal vectors, despite not having done vector calculus! and i found i could sort of extend that to 3d with the blob (though the "vector footprint" being dS in 2d is just an infinitesimally small line segment and i couldn't quite figure out what a 3d one would be) do you have any kind of intuition behind flux integrals, volume integrals, etc. flux is something im not super familiar with, having only learned a little bit about magnetic flux, magnetic flux density, magnetic flux linkage in my physics classes. with regards to the flow of incompressible fluids, it makes sense in terms of divergence, but to clarify do you mean sourcing and sinking as in a source would be essentially increasing the volume (the amount) of a fluid we have, and sinking would be decreasing that amount? so if we can't compress that fluid then the volume isn't going to change and so the volume is being neither sourced nor sunk so the divergence of that fluids volume would be 0? i think if i could wrap my head around that then it'd make it easier to understand the divergence theorem.

[@RichBehiel]

dS for a closed surface in 3D is an infinitesimally small square that’s tangent to the surface. Flux is the flow of the field perpendicular to a surface, so it can be calculated by taking the dot product between the field and the surface normal vector, times dS (to convert flux density to flux), integrated over the surface. Volume integrals convert a density distribution to a chunk of substance in some volume. For example, in quantum mechanics there’s a probability density distribution rho. Integrating rho over some volume gives the probability of finding the particle in that volume. Another example would be, if you have a material with heterogenous density described by some density distribution, then the volume integral of the density gives you the mass of the material enclosed in the volume. What all integrals have in common is that they’re just sums in the continuous limit. Once you see that, then you can do all kinds of integrals. It’s just a matter of asking, what quantity are you adding (the integrand), what’s the thing that goes to zero to keep the sum finite (the differential), and over what domain are you integrating. A vector integral can always be broken up into integrals of scalar quantities, in just the same way that a vector can be written as a set of scalars with respect to some basis. So the tools of calculus work the same, whether finding the area under a curve or calculating a flux integral. It’s all just sums. For sources and sinks, think of a source as a region that magically manifests some water, and a sink as a black hole that magically removes water from reality. Clearly we don’t see such things when dealing with water. But we do see sources and sinks when dealing with electromagnetism, for example, in which charge is a source of the electric field. For incompressible fluid, the divergence is zero so the flux integral is zero, whatever flow comes into the surface has to come out of the surface somewhere else. For electric charge, charge density is the divergence of the E field, so the integral of the charge density (i.e. the charge) is the flux integral of the E field on the surface enclosing the charge, up to some proportionality constant.

[@peterc-s6423]

@@RichBehiel that just blew my mind. i wouldn't think it's a square, as i wouldn't normally think that a square would tile the surface of a sphere, but i guess in the case of an infinitesimally small square it does add up. i think what's giving me the most trouble right now is the notation, and the unknown of how these integrals are actually calculated. integrating over a path makes sense for me as i can represent that path as a set of parametric equations and then use some trig to turn dS into a function of t with dt and integrate from there, for a surface integral of something i don't know how you'd even represent that surface, let alone integrate over it. if you know of any good resources on this that'd be awesome! just for that extra bit of clarity, when speaking of the "divergence" with an incompressible fluid, what is it the divergence of? some arbitrary vector field? a vector field that represents the flow of the fluid? thank you for taking your time to explain some of this, putting it in context has helped and i definitely have a better understanding than i did before (where id see more than one integral sign, say "nope", and just not look into a subject due to that)

[@RichBehiel]

I also was confused by vector calculus notation back in the day. But if you’re already integrating over paths, then you definitely have the conceptual prerequisites to do vector calculus. I forget which textbook I learned vector calculus from, it’s been a while now. There’s some good stuff online though, like 3Blue1Brown videos, and I’m sure if you look up Vector Calculus Textbook you can find a good one. The thing about vector calculus is it just takes practice. The ideas themselves aren’t particularly hard, in that you’ll always think they’re elegant and easy in retrospect. Getting used to the notation and operators can take a little while though. It helps to work with physical examples. Solving Maxwell’s equations is always a good exercise. You might start by calculating the electric field around a point charge, then doing a flux integral on a sphere that encloses it. That’s a nicely symmetric problem, and it’ll offer some insights into the divergence theorem. For the incompressible fluid example, the vector field would be the velocity. For velocity to diverge or converge would require material to be created or destroyed, respectively (or, expanded and contracted). But because it’s incompressible, the divergence is zero, so that’s kind of a special case of the divergence theorem that you can use to prove that whatever goes in through a closed surface has to come out somewhere else. Actually it might make more sense to use the divergence theorem the other way, and show that conservation of in and out implies zero divergence.

[@magnusmotor1364]

"ds" is not an "infinitesimally small quantity" but a variation along a differential line, which is an approximation to a function at a certain point.

[@SuperMaDBrothers]

Great video, but you need to consider who the hell will watch a 35 minute slow video on buoyancy, an extremely basic concept that almost anyone watching math videos on youtube will already fully understand. Would you even watch this video? The proof at 16:31 can be done by just saying "The surface is closed, so if we walk around the surface, the integral must be 0 (nhat is parallel to the direction of the surface). If we rotate all the vectors 90 degrees (nhat is the same as the one in the video), they're all normal to the surface. Easy as that. Please don't overcomplicate with Divergence Theorem. The way we derived this way back in high school was more intuitive in my opinion :P

[@RichBehiel]

Thanks for your thoughtful feedback :) Yeah, the pacing was too slow on this one for sure. Could have been ~25 minutes, probably could crunch it down to 20 without much loss of substance. These days I try to keep the videos shorter. But I do think it’s worth using the divergence theorem in the 16:31 proof, if only as foreshadowing for the way it was used the second time, the other way around. It’s nice to look at it from both sides. And sometimes insight comes from meditating on the simple ideas.

[@SuperMaDBrothers]

@@RichBehiel I see what you mean. This is how you formally prove it with math, but I'm a physicist and I'm okay with knowing something for certain, even without proving it. But keep up the good work :)

[@rdhighlander]

Amazing.. I do like the pound per square foot😂

[@nsxkkxlnmiyo8722]

damn you got me, i was wasting the calcuating power of the modern computer and privileg modern era man onto stupid videos and then i came accorss this. .. .

[@jatatanglobustead3963]

Really cool vid! Just wondering how this, the concept of a continuity equation, and Gauss’s Law (ie for gravity or electricity) relate, since they all seem to involve that double surface integral.

[@RichBehiel]

Thanks, I’m glad you enjoyed the video! :) Double surface integrals appear all over the place, much the same way that volume integrals are very common. Whenever there’s some quantity in flux, or some phenomenon relating to a surface, odds are there’s a surface integral in there somewhere. The continuity equation is just the idea that the rate of change of a substance in some volume is directly related to how much of that substance is flowing out, and coming in, through the boundaries. Gauss’s law is more mysterious, and to really unpack it we’d have to get into quantum electrodynamics. But we can take the law at face value, and see it as a way of describing how the influence of charge drops off as an inverse square. So that’s a statement about the way a certain kind of field is dispersed throughout space, when there are sources and sinks. The buoyancy double surface integral comes from adding up all the microscopic collisions of molecules. These things are all related in that all involve adding up some quantity over a closed surface, but aside from that they’re all pretty different too. Of course, we can apply the same mathematical tools to all these equations. That’s why math is so useful!

[@SplendidKunoichi]

they all make use of the divergence theorem to characterize a conserved quantity in a vector field, here its conserved density or incompressible fluid flow in a velocity potential, others being conserved mass in a gravitational potential and conserved charge in an electric potential the loop binding the double integral together into a surface integral is meant to specify that the surface has to completely enclose the corresponding volume as its boundary for the identity to hold, just like how Green's theorem relates a line integral around a closed boundary curve to the enclosed area calculus III (the one where it gets multivariate) should cover this in pretty good detail, good enough to at least appreciate the full generalization into the Kelvin-Stokes theorem whenever you come across it, very interesting stuff

[@roybelovoskey]

Please tell me he has many of these so I can feel better every day. This made me love life and I am being honest. Thank you and I emphasize those words but recognize the limitations of language neither spoken or heard.

[@RichBehiel]

Thank you for the very kind comment! :)

[@knowscope]

great video

[@RichBehiel]

Thanks! :)

[@BrandonSchmit]

So, if you have two very different shapes that each contain the same volume of air, let's say a very flat and thin rectangle with a total of 5 lbs of carbon fiber, and you have a cylinder that is long and skinny that also uses 5 lbs of carbon fiber to contain the same volume of air as the rectangle: if you place the cylinder in water vertically at a depth where the bouyancy forces are the same as placing the flat rectangle horizontally in the water column, then their difference in how quickly they rise to the surface will be only due to drag forces on the objects as they float upwards? Am I thinking about this correctly?

[@RichBehiel]

Yeah, that’s correct! :) The vertical cylinder has a higher pressure difference between the top and bottom, but a smaller horizontal surface area for that pressure difference to act on. The horizontal cylinder has a smaller pressure difference between top and bottom (there’s an integral involved now due to the curved surface), but a larger horizontal area to act on. The vector calculus in this video shows that when you calculate this integral, regardless of the orientation of the object, the result is always the same.

[@HebaruSan]

Returning to the moving-particle interpretation of 1:04, since water is incompressible, the lower-down molecules must be moving faster than the higher-up ones, in order to exert more force when they collide, right? How do they "know" to do that, since they're just bouncing around hitting the submerged object and each other?

[@GeoffryGifari]

the animations are awesome! how do you make that?

[@RichBehiel]

Thanks! :) I make these with python, using Matplotlib.

[@joshuahannah2070]

Why do you not need to consider the constant in your integration, for example when you integrate dv towards the end of the proof?

[@RichBehiel]

Great question! :) Because the weight density is constant for an incompressible fluid, so it doesn’t vary in the volume. If you’re calculating the buoyant force on a huge blimp due to air, then you could keep the weight density in the integral and its variation would lead to a tiny adjustment on the result, due to the air being ever so slightly thinner at the top of the blimp than the bottom. But for water, the weight density will be constant.

[@clearflow7925]

imagine being a buddhist meditating on buoyancy

[@RichBehiel]

That would be a very *uplifting* meditation 🧘🏻‍♂️

[@aadilansari5997]

Extend it further to stability of floating bodies..

[@shoyu3899]

Yerrrrrrr

[@chrstfer2452]

So uh, greenes theorem?

[@angelamusiemangela]

È questo è un Esempio del punto di incontro delle folate in andamento lineare ,cioe' derivante dalla Prima Folata. E si si dice baci per sempre 💯😘

[@Arthur-so2cd]

my biggest regret is leaving physics uni :(

[@eskidencelalsengor124kg]

Please turkish subtitle

[@Arthur-so2cd]

true misteries don't exist though, do they

[@RichBehiel]

I don’t know! 😉

[@lorenzodiambra5210]

​@@RichBehielso if I'm under 50,000 meters of water it's like I'm under 10 meters? Fantastic! tomorrow I tie myself to a concrete block and throw myself into the gulf of Pozzuoli!!!

[@RichBehiel]

The buoyant force will be the same, to the extent that water is incompressible, but you’ll be under a lot more pressure. But if you want 50 km, you’ll have to go to Europa! :)

[@LasagnaIsGood]

so why assume z=[z,z,z]?

[@RichBehiel]

Sorry, I’m not sure I understand your question. Can you please elaborate?

[@LasagnaIsGood]

@@RichBehiel nvm, i thought z was a vector

[@colleenforrest7936]

I think gravity arises from bouyancy

[@ethanbottomley-mason8447]

Did you watch the video? Buoyancy and gravity are nothing alike. Also, what fluid would gravity arise from? There is no fluid in outer space and yet gravity still acts there. Buoyancy also doesn't depend on depth or distance beneath the fluid, whereas gravity does depend on the distance between two objects. Also, if gravity were buoyancy, then what makes buoyancy happen? There is no reason for pressure to increase with depth without gravity, so buoyancy wouldn't exist. What would cause a buoyant force? Buoyant forces aren't fundamental forces since they ultimately arise from the electromagnetic force, i.e. the force that allows for molecules of water to impart force when they hit something. So no, gravity does not arise from buoyancy. What gravity arises from is the curvature of spacetime and the principle of least action.

[@fg786]

Buoyancy is lame, it's so one dimensional.

[@DavidLopez-ir3lj]

Deadass I came later to finish it it’s till a trip