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Problem 18 Hard Difficulty

Suppose that a plate is immersed vertically in a fluid with density $ \rho $ and the width of the plate is $ w(x) $ at a depth of $ x $ meters beneath the surface of the fluid. If the top of the plate is at depth $ a $ and the bottom is at depth $ b $, show that the hydrostatic force on one side of the plate is
$$ F = \int_a^b \rho gxw(x)\ dx $$

Answer

$F=\int_{a}^{b} \rho g x w(x) d x$

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Video Transcript

in this problem Where us to show that total force Acting on an object that is immersed in a fluid with destitute row eyes ro in Tegel A to B road times g times x, w x d x ray is the distance between the surface. Off this fluid to the upper surface of the object on B is the distance between the surface So pitiful er to the bottom surface off this object. All right, we know that force as you go to pressure times area, and that is row 10 g times that times D area. Now let's issue not be have an origin located right here. So 00 Ellis, assume that this is the Y axis and this is the X axis. Now, what we see from this object is that as that that changes so as exchanges. The what changes. It means that if you were to take an area offering to infanticidal strip and if he is to not areas so make X or deadly x times the height G X. So the area What that would change. Okay, Now we know that this is the origin. So this x zero level this is X equals a level and this is X tickles beat level. So the area off distance trip would be w X T X and total area off this whole object than integral w x D eggs. So what is the That's so Roizen material? Poor pretty G is universal. Be found an expression for areas, a function of X here. Now what is that? Well, since we assume this point to be the origin and since we know that that this measured from the reference point and since this is the direction of supposed to backs, we can just used exported that sort of total force would then be in charcoal from, um, well, let's talk about the limits later. Be have density times g times depth. We noticed his ex enemy noted area is WX DX. Let's look about limits as you can see total side of his object This B minus I meaning that, um, the wet will change when X is equal to a and it will be different when X is equal to be, which means that limits after integral will be from a to B. And if you compare this well, what is given, as you can see, we just drive it