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A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.

563485 $\mathrm{N}$

Applications of Integration

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Campbell University

University of Michigan - Ann Arbor

University of Nottingham

here were given that there's a work to fill them with someone circular gate, and then we're us to find the total hydro started force will invest some lesser gate. We know that the total force is a good to pressure times area and then a single zero times G times that that the pressure's measured at supplied by area. We know the desk TV nog we can easily find or catch a plate or find an expression for that. Now the thing is to find an expression for the area, all right? No, let's assume not. We have and origin located at the center of the semi circle. So that is 00 Let's assume that this is our X axis and this is the Y axis. So this bottom of this gate would be y equals zero level. This is the radius that I would be y equals two level and the upper surface would be y close 10 level. Okay, Now we would like to find the area off the skate, and we know that depending on the depth, the wit off this 10 strip changes and the area of pectin stripped would be doubly y times d y and we would like to find an expression for it with doubly what, as a functional fought Not since this is a, uh, someone circle we not any question of a circle is expected. Plus, y skirt is equal to radius. Christ, our that is to scribe. For most VC, that X is Danica scurried off four months. Why skirt? So this X is actually this part, but we have so that is somatic respected. Wyatt's sort of total what w would be to x would be equal to two times scurried off four months. Why scribed? It means that the area off that tense trip would be equal to two times scurried off four months, one squared, multiplied by the height, which is incontestable fight de wa. All right, if this is our origin, why right here? And let's assume that this is supposed to boy, right? And we know that we measure the pressure taking the, um, surface of order to be the reference meaning that we measured up in this direction. And if this total that office tank, this 10 meters and if this is the post to why it means that as a functional would be 10 minus. What? So let's now, right? Everything that mean And we know that the total force will then be and the forces acting on the stain strips sum up. So that is integral Row 10 g times. Death is a functional. Why was his 10 minus? Why multiplied by area, that is two times skirted off for months West. Great terms D Y. Now we need to determine the limits off this integral. You know that this object changes from wiser to y two. It means that the Levis office integral will be from zero to too. All right, we know we'll throw this row for water is 1000 kilograms per meter cube, and we know that cheesy go to nine point eight meters per seconds. Krystle is splendid and write f s integral from zero to 1000 times 9.8 times to These are the constant. What's plot by 10 minus wire leads by four minus y spirit. You are so the total force that would then be 19,600. That is accustomed. Multiply it by 10 times you tangle from zero to skirted a four minus y spray. Eva minus you take up from zero to wine. Skirted off four months. Weiss buried. Why 40 to go on for a second ago. We're gonna assume not four minus west. Card is equal to you. It means that limits off this new integral, or you will be from 4 to 0. And why do you? I will then be able to negative d over to 40 in Tegel on the first part. First Integral. We see that this skirted a four months. Why Skirt is only this part. Let's use blue is all of this part. And if you some those areas up, you would get the area off the 1/4 of a circle so that the area this would be one forward off the area off a circle. So the total force would then be 19 1006 100 multiplied by 10 times pi times two squared over four minus. If you do that transformation. And if you calculate that interval, we see that that is equal to to over three. You three over to where you changes between four and zero. And as we calculate this in jungle askew, do these calculations, we see that the total force is 56 34 85 so that is 563,485 inches.