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A trough is filled with a liquid of density $ 840 kg/m^3 $. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

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$F=526848 N \approx 5.27 \times 10^{5} N$

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 3

Applications to Physics and Engineering

Applications of Integration

Missouri State University

Harvey Mudd College

Boston College

Lectures

03:55

A trough is filled with a …

02:09

The end of the trough in F…

01:54

(a) The trough in Figure 8…

in this problem, we're given that there's an object with triangular cross section Um that has filled what if fluid, What it has still 840 Clark Proprietor Group and then where has to find the total force acting all inside of this object. We know that the force the vehicle to pressure times area and that is equal to density times G times that, Ah, that pressure is measured, multiplied by the area. We know what Roy's were given not we know what she is weakening. You'll find D or find an expression for the did that. Now let's find an expression for the area in order to calculate the total force. Now we would be interested and the area off the sole object. But now the idea is that as the that changes, they went w before I let's say and let's say the height is D y so as D that why changes the world changes. It means that the pressure and area changes, So we would like to find an expression for the depth and the area as a function off, while all right, now let's assume that we have an origin located right here So this is our X axis and this is the Y axis. This is the origin zero zero and we are less interested in the area. Off this object area of distant strip would be since the weather is w fly. And since the height is d Y area would be w y times t want and thing, the first thing we're trying to do is to find an expression for the widow. Please object as a functional. Why? Where? Why we're gonna assume, as the Solis said, that that is why All right, how can we find expression for the width as a function of that? Well, if you were to write any question off this line, this would then give us function for the wet off or one side of the web off this object as a function off step, then less right in the question Off this line, we need the coordinates, office points, accordance off. This one end would be, since the toro length or total length off the one side of state. It means that here it will be four. As census is, the origin for zero will be the coordinates off the first point in accordance up second point will be. Then we would need this height this vertical distance since what side is, uh, eight. And since this angle is 90 and this is 60 we know that then the Hyde will be equal to for over or four skirted off three or it sort of quarters off. This bottom point will then be zero coma. Four skirted three and we're right now trying to ride any question for line. But the following coordinates for common zero and zero come afford scarred three. You know the question of attention will be up for y, minus y dot and we could use either of those points as one at an extent will be. Could slope times X minus accent where slope is the difference between wise divided by difference between exes. So let's use this first point. Why minus zero is equal slope as four started a three month zero divided by zero minus four months black by X months. For from this we see that why is equal to scurried off negative skirted or three times X prize. For since we are trying to write the width off this object as a function of why. Let's leave ex alone and write this one as X is he go to four minus. Why over four Skirted off. Three. All right, this is now the question for 1/2 off the Web and the total with W. Why would be two times texts? That is two times for minus y. Over four skirted off three for the area off the tin stripped. This shaded region would be the with multiplied by the height. So another B two times for minus y over four skirted or three times divine. Now we have everything that we need Total force. Would that be Forces acting on the state's ships summed up We need to determine the limits off the integral and well this object goes from If this is our origin, it goes from y zero level to fly H level, which is four tops. Kurt, police are gonna be 0 to 4. Scurried off free beyond road times g times the that which we assumed it to be. Why initially that'll be raw times g times. Why times the area two times four minus y or were four screwed it up three toes. Do you? Why, we know that row G into or close turns Celeste. Right? This internal and s force is equal to two times wrote OMG integral from 0 to 4. Skirted off three we have for y minus y scribe over four Skirted a three d y. Let's say that we are jingling this area. No. All right. No, we can just take that outside integral and find anti gravity off. This that'll be two times Wrote time G terms to y squared minus y cubed divided by for, uh, sort three Skirted on three where why changes between zero and four skirted off three. So the total force would that be to Rogie Times two times four skirted a three squared minus four skirted off three cube divided by three Skirted off three that is equal to two times wrote of G times 96 minus four cube times three Skirted or three divided by three Skirt three Useful cancel out That'll be than equal to 64 times Work time G that is then we know what joys we know What g is that a 64 times 9.8 times 8 40 Sortie Answers 500 5000 526,848 Nugent's

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