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Problem 10 Medium Difficulty

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Answer

Total Force $=\frac{a^{3} \rho g}{\sqrt{2}}$

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Video Transcript

in this problem. We're given that there's a square submerged in water and then where has to find the total force acting on size off this work. Now we're going to send out the total force acting on this will be summation off force acting on the upper house off this word. So this triangle plus force acting on the lower health off the scraps on this triangle. All right, let's start from the upper half. We have our tang again and this is our object. A A Let's say that this is our origin or exposed Alexis pointing, uh, East imposed their wives pointing downwards. So that is pointing south. We know that force is equal to row Tun g Times de temps a sort out his pressure terms area. Let's see that the depth as, uh, measuring from the water surface. And that is why so d is a goto. No, we want to find the area. Oh, let's say that if this distance s why and bless that I were interested in this triangle. As you can see, this angle is equal to this angle. It means that if this is why this should be war as well, and this thing should be what as well. So then the area off less scientific, dismal strip will be this height, which is D y month, spite minus total link. And that is equal to Nana. Why? Why? So the area off this blue shaded regional strip would be to our times D y, because this angle and this angle and fort um, any sub triangle, those angles will be the same. So the area will be to y Tubbs d y weaken, then, right, the total force pants. First, we would need to some forces up acting on those imprint decimal strips. So force would be limit as and go see Infinity summation for one to end you in a row. Tongi times did that, which is why turns to why force I I Tubbs out the wine and we can express that as an integral internal from, um zero to well, now in determine on the limits office internal. All right, if this is our origin, if you asked him not to be the origin, if that has 00 you don't would need to find the level at the bottom of this upper half. It means that we would need to find this total distance if the science A And if we know that if this is an iffy, not this angle is 45 this is 45 this is 90 we could do an easel. See that this distances they were skirting too. So this will be from zero to a oh, skirted of to level out row times g times y times to R D y that is equal to two row g jangles. You're a or skirt of two. Why scrape D y that is equal to two Rogie White Cube or three where why changes from zero to a over a skirt to sort of force acting on the upper Alfa said This is F u physical to row Tongi times a cube or three skirted off too. Now we would like to find the force acting on the board How as well And we're gonna follow basically the same procedure. This is our tank and this is the lover Health. This is the surface. Now we know that this distance here it's a or skirted of two. That is what we found. And we also know that this total sense is equal to eight have skirted two. All right, So basically will have the swill full of the same idea. If you zoom in to this, we know that for anyone, this will be equal to test Big O's. This angle is equal to this angle and this is a right triangle. So the area again off then and the infinite decimal strip with tight T y will be t y time this length where death linked will be what to hover if the total distance from the water surface to the bottom of this object is a skirt off too. Then this distance would be, um, a skirt up to minus why. And since it has symmetric with respect to the this work took a line. And if this is a spirit of two months, while this sign would also be a skirted of two minus y sort of total length in terms of why would be two times a skirt of two minus y and the area of piss poor region would be this total length what bike by the height. So that would be two times its current to minus y times d. Why so what would be the depth again? Less said that f this time something else or done is the lower health is rope. Tunji comes D times a week in a row. We know G. We found a We found an expression for it. Now we need to find an expression for D. Let's say that, um, we assumed this time the origin to be located right at surface. And this is our positive wise that the depth would then be why, from this, the force acting on an infinite decimal element would be Row Tun g Times y Times area, which is two times a scripted of two minus y taps D y. In order to find a total force, you would need to some those areas up. Let's do that. Let's write this one is a human some. So that is, um, limit a Zen ghost. Infinity Summation From one toe end. We have wrote on G times WiFi times to a skirt of two, right? That's why I times Delta y and we can actually write this one as an integral We needed inside on the limits off this inter goal, and we know that here this end off this lower house times from a were skirted up to and it ends at a skirt of two. So the limits office in Tegel will then be a over skirted of two times skirt or two times a day. Grow time, G terms. Why a skirted of two minus y times D y we can rent this one s f f l s. Rogie. This is just accustomed. Independent off. Why a over skirted up to square it up too. Hey, to skirt up to eight times why minus two y scraped arms d y that will be equal to row 10 g 10 skirted to a Y squared minus to like you over three. Where why changes between a or skirt of two and skirt of to a From this, we see that Tandy force acting on the lower house, but skirt up to a cube or rogue e divided by three. Not a total force would be f a few plus f f l. That is a cube. Rocchi, divided by three skirted off too, plus skirt up to a cube Rocchi divided by three. Now, if you most about this skirted off three, we see that then the total force is equal to a cube. Rocchi, divided by skirted off two