 This is gonna be our final video about hydrostatic force and like some other lectures we've done, I wanna end this conversation by revisiting the theorem of PAPAS one more time. But this is gonna be the hydrostatic force version, right? As we see in the past with like surface area and volume, if we know something about the center, the centroid of the region, we can use that to help us avoid integration and find the same thing. Similar things can be done for hydrostatic force and static students really love this observation because they have to do a lot of static problems and they like to avoid integrals when they can. So imagine we have a plate P, which is a flat plate submerged vertically in the water. Then, so we might have something like this. Ooh, you know, just some shape right here, here's our P. Then the hydrostatic force on P is gonna be the product of two things. It should be an area times pressure, right? So the hydrostatic force, it's a pressure times an area, whoops. So what's gonna happen is we're gonna take the pressure, the pressure at the centroid. So the center of mass of our region here. And then we're gonna multiply that by the area of the plate. If you multiply the area of the plate by the pressure at the centroid, this gives you the hydrostatic force. It's pretty slick if you can find it. So in a previous video, we actually saw that we submerged a circle, which has a radius of three, 10 feet deep in the water. And we calculated that the hydrostatic force against that circle will be 7,875 pi over two, the number you see right here. And we did it using an integral. Take a look at that video if you're not sure how the calculation would go. But for this situation, we have our circle right here and it was submerged, the center, the center of this circle was submerged, since the whole circle was submerged 10 feet and it had a radius of three, this means that the depth at the centroid, the center of the circle is seven feet. So if you take the pressure at the centroid, that's 662 and a half times seven, the center right there, which is 437 and a half. What would that be? Pounds per square foot. And then the area, as it's a circle, the area is gonna be pi r squared, that is nine pi. We can see that if you take nine pi times 62.5 times seven, that gives you the value we're looking for. And so that's a very slick way of finding this, the hydrostatic force, if you know the pressure at the centroid, you know the area of the plate, then you know the hydrostatic force by multiplying those two things together. That's really, really, really nice. But as we said many times, that's cautions with the theorem of Papis. In order to get the theorem of Papis to work first, we have to know the area of the plate. Area typically comes about by calculating an integral. I mean, that's actually the original problem that got us thinking about integrals. But also as we will see actually in the next lecture, centroids also come about from calculated integrals. So the thing is, it takes an integral to find an area, it takes an integral to find centroids. So two integrals can be put together to form one integral. That's in general practice, not an effective strategy. But if you can find the area without integration and you can find centroids without integration, then you could actually use this shortcut, which we call the theorem of Papis right here. Like we saw with volume, like we saw with surface area. And also I wanna mention there are theoretical benefits of this centroid formula, even if it's hard to find them. So a problem that shows up in static problems a lot is, what if we have a plate that's submerged? So we have water, we have water right here. But what if we have a plate that's submerged, but it's submerged at some type of angle? So it's not, we're looking at it from a side, right? What if there's an angle to it? How do you determine the hydrostatic force there? Well, it turns out that if you have a vertical plate that's submerged in fluid, such that the hydrostatic force against the plate is say F zero, right? And now that's not supposed to be some reference to like Captain Falcon or anything like that. But if we take our plate and we submerge it vertically, like so, and let's say that the force exerted against this, and again, that's not some reference to like Ray Skywalker or anything like that. If you take the force exerted against this plate when it's completely vertical, we'll call that F zero. If the plate is inclined such that the angle with the surface of the water and the plate is zero, so that is to say if you take this angle right here to be theta, then it turns out the hydrostatic force exerted against the plate, the hydrostatic force exerted against the plate, this is gonna be F, which is just F zero sine of theta. So there's a nice trigonometric argument going on here. And the basic idea behind that is if you take the centroid of this plate, you have the depth of the centroid, that gives you pressure, then you have the area of the plate. What happens if we rotate this thing, right? If you rotate this thing with respect to theta, like here, there's a nice trigonometric argument. Now, as you rotate this thing, the area doesn't change. Of course, the depth does change and that's where this sine of theta comes into play. You're lowering the depth by a percentage, which is this theta. And so you can see very quickly using the theorem, Pappus, that you can calculate the pressure of this inclined plane as some trigonometric relationship. Now, this is if you take theta with respect to the surface of the water. Oftentimes though, the angle you measure isn't going to be with respect to the surface of the water. Oftentimes, the angle you want to measure is actually this one right here. So take like a psi. So if you take its complement, well, if you take the complementary angle, then of course you just have to switch the complementary function. In this case, the force would be F equals F naught cosine of psi. So whichever you prefer, you can do one or the other and it works really nicely. That's the nice consequence of the theorem Pappus. And so that brings us to the end of our lecture, lecture 22 about hydrostatic force. I hope you enjoyed it. Like I already mentioned in the next lecture, we're gonna, in lecture 23, we're gonna talk about centroids and how you can use integration to find those. It's actually a pretty cool topic. Please look into those videos there. If you have any questions whatsoever at any time in our lecture series, feel free to post your questions in the comments. I'd be happy to answer them for you and I will see you next time. Keep on calculating everyone. Bye.