Latest / Mechanical Engineering Made Simple / (#89) Hydraulics Buoyancy and Ship Stability

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(#89) Hydraulics Buoyancy and Ship Stability · Mechanical Engineering Made Simple

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Welcome back to Mechanical Engineering Made simple.
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Today we are wading into a topic that is deceptively quiet.
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It's a field where if everything is working correctly, absolutely
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nothing moves. But if you get the math wrong,
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you don't just get motion, you get catastrophe.
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We are talking about keeping dams from bursting, ships from
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capsizing, and you know, high pressure pipes from ripping open
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like paper. That's right, we are doing a
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deep dive into fluid statics. Now, statics usually implies the
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the boring part of the curriculum for a lot of
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engineering students. You know, forces in equilibrium,
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no acceleration, no velocity vectors to chase.
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But looking at the source material we've pulled for this,
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a really comprehensive technical text on fluid mechanics, it is
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clear that static does not mean powerless.
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Far from it. Fluid statics is simply the
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study of fluids where there's no relative motion between the
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fluid layers. That's the, you know, the
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technical definition, but in practice it is the study of
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potential energy just waiting to be released.
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Whether that fluid is the ocean pushing against a submarine hall
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at crush depth or hydraulic fluid just sitting in a brake
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line, the physics dictates whether your structure holds up
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or fails. And the stakes are obviously
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very high. So let's unpack this.
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We're going to cover pressure characteristics, how we actually
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measure these invisible forces, the Eureka moment of buoyancy,
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and the geometry that keeps ships upright in a storm.
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And we will finish with a weird edge case fluids that are moving
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but are well, mathematically considered static.
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Which still hurts my brain a little, but we will get there.
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Let's start at the absolute bottom, or rather the
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fundamental definition. We throw the word pressure
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around a lot, but in an engineering context, what are we
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actually saying? It is surprisingly clean.
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The source text defines average pressure with a very simple
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relationship. It's just force divided by area
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P = P / A. Force over area.
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And this distinction is vital because a massive force spread
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over a massive area might result in very low pressure, but a tiny
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force on, say, a needlepoint. Infinite pressure.
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Theoretically, this relationship is the cornerstone of everything
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else we're going to discuss. It explains why a sharp knife
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cuts and a dull 1 doesn't. Same force, smaller area higher.
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Pressure. Exactly.
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And this leads us directly to the first major figure in our
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deep dive, Blaise Pascal. The source material actually
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includes a bio on him, and the guy was a prodigy.
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It says he invented an arithmetic computer at age 19.
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Wow. I think at 19 I was still trying
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to figure out to do my own laundry.
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He was definitely operating on a different level, and because of
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his contributions, the SI unit for pressure is named the Pascal
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1 Newton per square meter. But beyond the unit, his lasting
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legacy here is Pascal's law. OK, lay it on us.
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What is the law? Pascal's law states that the
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intensity of pressure at any point in a fluid at rest is the
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same in all directions. More importantly, if you apply
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pressure to a fluid in a closed vessel, that pressure is
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transmitted equally to every single part of that fluid.
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It doesn't diminish. And this is the cheat code
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behind hydraulic machinery, isn't it?
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It is. The text has a diagram of a
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hydraulic press. It shows how we can lift a car
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with just a human hand. It feels like magic, but it's
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just physics. It comes right back to that
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force divided by area definition.
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Imagine you have two Pistons connected by a fluid filled
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pipe. Piston 1 is tiny, let's say one
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square inch. Piston 2 is massive, 100 square
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inches. OK, I'm visualizing.
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It if I push down on the small piston with 10 lbs of force, I
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create a pressure of 10PSI. Pascal's Law says that 10PSI
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travels instantly to the Big Piston.
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So the Big Piston also sees 10PSI.
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Exactly. But here's the trick.
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Since force equals pressure times area, you take that 10PSI
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and multiply it by the 100 square inches of the big piston
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and that gives you 1000 lbs of lifting force.
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You put in 10 lbs, you get out 1000.
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That is a 100 to one mechanical advantage.
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The math is clear. The second force equals the
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first force multiplied by the ratio of the second area to the
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first area. It is why you can stop a two ton
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car just by twitching your foot on the brake pedal.
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It's all about leveraging the incompressibility of the liquid.
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Now that covers fluids trapped in a box.
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But most of the time engineers deal with fluids out in the
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wild, oceans, reservoirs, the atmosphere.
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The source moves on to pressure of a fluid at rest regarding
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depth. Right, this is where we leave
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Pascal's closed container and look at the vertical column in a
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fluid at rest. Pressure varies directly with
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depth. The text derives this by
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balancing the forces on a tiny theoretical element of fluid.
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The resulting formula is P = P, not plus rho RAGH.
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OK, let's translate that into English.
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Pressure equals the surface pressure plus density times
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gravity times height. Or depth in this case.
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Or depth, right? Rho RAG is also known as the
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specific weight of the fluid. Basically, the deeper you go,
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the more water is sitting on top of your head.
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It's a linear relationship. So if you dive down 10 meters,
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you add a certain amount of pressure.
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Go down 20 meters, you've got to double that amount.
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Precisely This is why submersible holes have to be so
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incredibly thick. They aren't just resisting the
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water around them, they are carrying the weight of the
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entire water column above them. And this brings up an important
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distinction the text makes between absolute pressure and
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gauge pressure. I feel like this trips up a lot
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of students. It does.
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It is all about where you set your zero.
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Absolute pressure sets 0 at a perfect vacuum, empty space, no
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molecules, nothing. Whereas gauge pressure sets 0 at
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whatever the current atmospheric pressure is.
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Right, because we live at the bottom of an ocean of air, we
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are constantly under about 14.7 PSI of pressure.
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But we don't feel it, we are equalized to it.
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So for most engineering applications, like checking your
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tire pressure, we don't care about the atmosphere, we only
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care about the pressure difference.
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So if my tire gauge says 30PSI, that is 30PSI above the outside
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air. Correct.
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If you wanted the absolute pressure inside that tire, you
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would have to add the atmospheric pressure to your
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gauge. Reading the formula is just
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gauge pressure equals absolute pressure minus atmospheric
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pressure. And the text notes that this can
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go negative. Yes, if the pressure is lower
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than the atmosphere, we have a vacuum pressure or negative
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gauge pressure. So we know what pressure is, and
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we know how to calculate it based on depth, but how do we
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physically see it? The source details a few devices
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starting with the manometer. The manometer is the simplest
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application of that row GH formula.
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It's usually U-shaped glass tube.
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You put a heavy liquid in it. Mercury was the standard for a
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long time due to its density that we avoided now.
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You hook one end of the tube to your pressure source, leave the
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other open to the air and just see how far the liquid moves.
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It is elegant because it is undeniable.
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If the liquid is pushed down on one side and rises on the other,
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you just measure the difference in height, let's call it H The
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pressure difference is simply the density of that liquid times
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gravity times H No calibration needed really, it is just
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gravity and mass. But glass tubes are fragile and
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mercury is dangerous, so the text highlights a more robust
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industrial standard, the Borden tube.
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If you walk into any factory or look at a air compressor in a
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garage, that round dial gauge is likely a Borden tube.
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It was invented in the 19th century and hasn't really
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changed because it works so well.
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How does it work internally? It just looks like a clot of
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mechanism from the outside. It is actually a deformation
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mechanic. Inside the casing there is a
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curved metal tube. Imagine a party horn that curls
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up. The key is the cross section of
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that tube. It isn't a perfect circle.
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It is elliptical or flattened. OK, so it is an Oval tube bent
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into AC shape. Right when you pump pressure
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into it, the fluid tries to push the walls of the tube out to
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make the cross section circular. A circle holds more volume than
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an Oval for the same perimeter. As the tube inflates and tries
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to become round, it physically straightens out.
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It uncurls so. Slightly, that tiny uncurling
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motion is linked mechanically to the needle on the face, so you
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are seeing the elastic deformation of the metal
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calibrated to read as pressure. That is a brilliant piece of
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mechanical design. No electricity, just geometry
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and material properties. Speaking of geometry, let's
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scale U. We are moving from gauges to
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civil engineering dams and sluice gates.
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This is where the math gets critical for safety.
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Right. The challenge is calculating the
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force of water pushing against a vertical wall.
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Now my intuition, and I suspect the intuition of many listeners,
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would be to find the area of the wall, find the average pressure,
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and assume that force acts right in the middle, the geometric
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center or the centroid. And if you built your DAM based
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on that assumption, it would likely fail.
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Comforting. So why isn't it in the middle?
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Think back to what we just said. Pressure isn't constant.
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It gets higher the deeper you go.
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So near the surface the push against the wall is gentle.
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At the bottom, the push is massive.
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So the average dot isn't in the middle, it is weighted towards
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the bottom. Correct.
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The dex distinguishes between G, the geometric centroid, and C,
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the center of pressure. The center of pressure is always
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below the centroid. The text actually gives a
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specific formula to find exactly where that point is.
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It involves the moment of inertia.
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It does. It shows that the deeper the
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water, the closer those two points get.
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But they never quite touch. The center of pressure is deeper
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than the centroid by a factor related to the shapes inertia
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divided by the depth times area. We don't need to memorize the
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derivation, but the implication is what matters.
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Why does this offset matter for a generic engineer?
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Torque. If you are designing A hinge for
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a gate or calculating the tipping moment for a concrete
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dam, you need to know exactly where the force is pushing.
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If you assume it's in the middle, you will underestimate
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the leverage the water has to tip your wall over.
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So it's like the water is pushing low, effectively trying
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to sweep the legs out from under the dam.
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That is a vivid image, but yeah, that's exactly it.
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Sweeping the legs. While we are talking about
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structural failure, the text brings up hoop stress.
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This is the force trying to tear a cylinder apart.
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Think of a pipe carrying high pressure steam or even a soda
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can. The pressure is pushing outwards
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in all directions. The question is, how strong does
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the wall need to be? The formula derived is T equals
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PD over 2. OK, so tension equals pressure
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times diameter divided by two, right?
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T being the force per unit length of the pipe wall.
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Notice that D for diameter is in the numerator.
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That means if you double the diameter of your pipe, you
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double the tearing force on the walls, even if the pressure
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stays the same. That seems counterintuitive.
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100PSI is 100 TSI right? True, but in a wider pipe, that
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100 PSI is acting over a much larger internal surface area,
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creating a larger total force that the material has to hold
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together. OK, that is why a small copper
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tube can have thin walls, but a massive municipal water main
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needs thick steel or concrete reinforcement.
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You've got it. So we have covered pushing down
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with depth, pushing sideways on dams and pushing out on pipes.
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Now for the classic pushing up buoyancy.
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Archimedes. The source calls him the
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greatest engineer of ancient Greece.
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Apparently he also invented a planetarium.
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He was a polymath, but his principle of buoyancy is the
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headline here. It states that the buoyant force
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on a submerged body is equal to the weight of the fluid that
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body displaces. The formula is F equals rho ogv.
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Density times gravity times volume.
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This explains why a steel ship floats.
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It isn't about the material, it is about the volume of water it
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shoves out of the way. Exactly.
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As long as the weight of the water you displace is greater
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than the weight of the ship itself, you float.
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But floating is only half the battle.
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You also have to stay right side up.
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This leads us to the section on stability, which I found to be
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the most complex part of the text.
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We have a soup of letters here, GC and M.
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It is complex, but it is visual. Let's break it down.
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You have G, the center of gravity.
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That is the point where the ship's weight pulls down.
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OK, usually somewhere in the middle of the ship.
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Then you see the center of buoyancy.
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That is the geometric center of the underwater part of the hull.
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That is where the water pushes up.
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So when the ship is sitting perfectly flat in calm water,
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G&C are lined up vertically, downward pull upward, push
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balanced. Right now imagine a wave hits
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and tilts the ship to the right. The center of gravity G doesn't
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move relative to the ship, assuming the cargo is strapped
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down properly, but the shape of the underwater hull does change.
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Because the right side dips into the water and the left side
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lifts out. Exactly.
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So the center of that submerged volume shifts to the right.
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The center of buoyancy C moves to a new spot.
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Let's call it C prime. So now the upward push is
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offset. And that offset is everything.
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If you draw a vertical line straight up from that new
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buoyancy Point C prime, it will cross the ship center line at a
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specific point. That intersection point is M,
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the meta center. The meta center.
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It sounds like a comic book location, but I gather it's
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important. It is the most important point
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in naval architecture. The rule is simple.
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For a shift to be stable, M must be higher than G.
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The meta center must be above the center of gravity.
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Why? Because if M is high, the
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buoyancy force in the gravity force create a lever arm, a
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writing moment that twist the ship back to upright.
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And if M is below G. Then the lever arm works the
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opposite way. It twist the ship further over.
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The ship capsizes. So logically you want M to be as
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high as possible, right? You want a massive metacentric
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height the distance GM so the ship is super stable.
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You would think so, and for warships that is true.
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The source mentions warships have a metacentric height of
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around 1m or more. They are very hard to flip, but
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there is a trade off. What is the downside of being
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too stable? It creates A stiff ship.
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If a wave hits, the ship snaps back to upright incredibly fast.
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It is violent. The period of roll is very
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short. That sounds miserable for the.
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Crew. It causes seasickness, fatigue
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and makes it hard to operate equipment.
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Now compare that to a cruise ship.
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The tech says large passenger ships have a much lower
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metacentric height, maybe 0.3 meters.
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That seems risky. It is safe, but just barely.
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The lower GM means the restoring force is gentler.
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When a wave hits, the ship rolls slowly and returns to upright
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slowly. It is a tender ship, much more
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comfortable for the passengers at the buffet.
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That is a fascinating engineering compromise.
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You are tuning the physics for human comfort versus ultimate
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survivability. Engineering is always about
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tradeoffs. You can't have everything.
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All right, we are in the homestretch.
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We have been talking about fluids at rest, but the text
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throws A curveball at the end, moving fluids that are
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technically static. It is referred to as relative
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equilibrium. Which implies the fluid isn't
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moving relative to its container.
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Exactly. Imagine you were in a car
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holding a cup of coffee. The car is moving at 60 mph on
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the highway. As long as the speed is
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constant, the coffee is flat. It behaves exactly like it is
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sitting on a table. But if the driver slams on the
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gas. Linear acceleration.
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Because the coffee sloshes back, the surface of the liquid forms
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an angle. The text gives us the formula
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for that slope. Tan Theta equals a / g.
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The tangent of the angle equals the acceleration divided by
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gravity, so the harder you accelerate, the steeper that
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slope becomes. And this isn't just about
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spilled coffee. If you are designing a fuel tank
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for a jet fighter or a rocket, you need to know where the fuel
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is going to be when you hit the afterburners.
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You don't want your fuel pump sucking air.
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Right, because all the liquid is plastered against the back wall.
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And then there's the rotational version, the spinning bucket.
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Gyrostatics. If you spin a cylinder of
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liquid, the surface curves it, dips in the middle and rises at
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the walls it. Forms a parabola.
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A prolate of revolution, to be exact.
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The math is beautiful here. The height difference Z follows
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the square of the angular velocity Omega.
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The formula is Z equals Omega squared times radius squared
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divided by 2G. That squared term is the
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kitcher. If you spin it twice as fast,
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the height difference gets four times larger.
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Right, this is the principle behind centrifuges.
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You can generate artificial gravity at the outer walls that
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is thousands of times stronger than Earth's gravity.
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It allows us to separate blood cells, enrich uranium or
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separate cream from milk. It is amazing how we started
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with water in a glass and ended up with centrifuges and
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battleship stability. It really reinforces that fluid
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statics is everywhere. It is the invisible
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infrastructure of the world. Before we wrap up, I want to
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touch on the etymology. Note the source included.
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I am a sucker for word origins. We all know Aqua means water.
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Latin. Straightforward.
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But the text traces the route back much further to Sanskrit,
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specifically the word for water offered to Buddha.
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OK, that sounds very spiritual. That route traveled W to become
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Aqua, but it also traveled east along the Silk Road to China and
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then Japan. In 10th century Japanese
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classics, the word AKA appears. Meaning water.
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Meaning bilge water. That lost a little bit of its
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holiness along the way. Just a bit, but it is a reminder
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that whether it is holy water or bilge water, the physics are the
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same. That is the beauty of it.
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The laws of physics don't care about the context, they just
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care about the variables. If you understand the variables
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density, gravity, depth you can predict the outcome.
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And that is the core message here.
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Engineering is the art of prediction.
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We calculate the center of pressure so the dam holds.
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We calculate the meta center so the ship writes itself.
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We do the math, so reality doesn't surprise us.
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Because in fluid mechanics, surprises usually involve
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getting wet or getting crushed. A sober thought to end on that
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brings us to the end of this deep dive into fluid statics.
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If you haven't already, get over to our YouTube channel
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Mechanical Engineering Made Simple, where we post hands on
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how to videos, visual breakdowns, and practical
17:52
engineering walkthroughs you can actually use.
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