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A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Find the hydrostatic force on (a) the top of the cube and (b) one of the sides of the cube.

(a) 314$N$

(b) Approx. 353 $\mathrm{N}$

Applications of Integration

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Missouri State University

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here were given that there's a cube sitting in the aquarium, which one meter, Uh, that is one meter deep, and in part they were has to find a hydrostatic force acting on the top of this tube. Now, since the total height is one meters, and since the one side of this Cuba's 20 centimeters, it means that the distance from the reference or surface off the sec William to the top off this cube is 80 centimeters. You know that force is equal to pressure times area, and that is a cutthroat times G times that times the area. Now let's plug everything that we know, and we know that then still order is 1000 G's 9.8. What is that? But that is the distance from the reference point that is in distaste. Water surface to the top off this cute that is 80 centimeters or 800.8 meters. What's like by area off the top off, this cube will decide 20 centimeters. That would be 1.2 skirt, and if he must fly this, we see that the total force acting on top of this cube is 314. Newton's in part B, whereas to find the force acting on one end off this object. So, Lisetta. Imagine we're looking at this cube from the site. We would have this object, right? So there is this, um that is the aquarium. This is the cube again behind his 20 centimeters. This distance is 80 centimeters and depending on deep depth. So the distance from the reference point we will have and area Let's assume this to B. D. Y. And just to be terribly wise. So the area off dust in strip will be doubly y terms he want. Now what? W what? W pirate does not change with that. And in this case, since it is acute, the what is 20 centimeters. So the area off distance trip will be 0.2 meters times d. What? All right now. What is that? Because we know that again. Forces pressure times area. And that is road times G times that times area. Now we know what areas is a function off this thickness artist in strip. Now let's find an expression for it that, um let's assume not be having origin located right at the center off this less use green color right here. So this is our origin X axis. This is Y axes. Now, if this is a positive, why and if this total height is one ended, that would be one minus one. All right, let's plug that in that You know that then the force will be inter go. So the force will be forces acting on this 10 strips summed up So they're B roll 10 g. What is that? There's a function of why that is one minus y what is area? That is 10.2 times he want now what are the limits office in juggle? Well, we know that this is our origin. So this is y equals zero level and the height is 20 centimeters sort of physical. Otherwise you could 200.2 meters level sort of limits will be done from 2.2. Okay, so died will be then force is equal to the integral from zero point to attack isn't times 9.8 times one minus y times 10.2 d y force will then be 1960 which is 1000 times nine point 8/10 point to interrogate all from zero to point to one minus y do on that is 1960. Why minus y skirt over to where wind changes between Syria and 0.2 force is Donegal to 1960 times point to minus 19600.0.2. Skirt over to there. From this we see that the forces I decided force acting on Sign off. This object is about 353 nutrients.