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# Finding the horizontal force on the gate if the density of water is w. helpp!?

i posted this twice already but neither of the answers checked with the back of the book. i really need to understand how to do this for my test so please help me and i REALLY REALLY appreciate your time. okay heres the quesiton again.

A control gate in the form of a parabolic segment of base 12 and height 4 is submerged in water so that its base is 2 units below the surface of the water. Find the horizontal force on the gate if the density of the water is w. < thats the end of the question.

more details: the parabolic gate has the equation 9y=x^2. and there are points (-6,4) and (6,4) on the parabolic gate in the text theres these formulas. p=F/D=wh F=w h A how would i set this up? and what would be my limits ?

### 4 Answers

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So first find the intersection points with the water surface:

"height 4 is submerged in water so that its base is 2 units below the surface of the water."

So y=1/9 x^2 shifted down by 2 becomes y=1/9 x^2 - 2

And the intersections with the surface are at y=0:

0=1/9 x^2 - 2

x^2 = 18

x = +/- 3√2

F = Pressure x Area

But Pressure is also a function of depth y (or d, or h, whatever):

Pressure = w * g * y ("w" = rho, usually)

So the incremental force at depth y, due to an infinitesimal strip dy deep and x-(-x)=2x wide, is:

dF = dP * dA

= (w * g * y) * (2x dy)

It's easier to parameterize for y than x.

So if we know y, x is given by:

y = 1/9 x^2 - 2

x = 3 √(y+2)

So

dF = (w * g * y) * (6 √(y+2) dy)

dF = (6wg) y sqrt(y+2) dy

Then integrating between y=-2 and y=0 :

F = (6wg) ∫y √(y+2) dy between y=-2 and 0

Make a substitution z = y+2, thus dz = dy

F = (6wg) ∫((z-2) √z dz) between z=0 and +2

= (6wg) ∫ z^(3/2) -2z^(1/2) dz)

= (6wg) [-2/5 z^(5/2) + 2(2/3) z^(3/2)]

= (wg) [-12/5 z^(5/2) + 8z^(3/2)]

= (wg) [-12/5 2^(5/2) + 8*2^(3/2)]

= (wg2√2) [8 - 12/5 * 2]

= (wg2√2) [8 - 4.8]

= (wg) 6.4√2

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sahsjing, you should say your approach is the same as mine, because I posted the approach first :)

• math55,

My approach might contain some errors. The thing is you didn't post the picture. I assume the gate has parabola shape, opens up and most part of the gate is above the water surface .Is this right? But I don't understand why " there are points (-6,4) and (6,4) on the parabolic gate"? These two points should be above the water surface, aren't they?

If you can't post the picture, would you just post the answer? I am 100% sure I can figure it out.

If p = wh, then

F = â«PdA

= â«w(2-y) 2xdy [y: 0...2], (dA = 2xdy)

= â«w(2-y) 6ây dy [y: 0...2], (x = 3ây)

= 9.0501 w

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bictor717,

The density is a function of height. Therefore, your idea doesn't work. By the way, what do you think the parabola opens up or down?

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smci,

Your approach is the same as mine. You missed a sign only. Check it. You should get the same answer as I showed here.

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smci,

I am very glad that finally you got the same answer as I had showed here. :)

By the way, you really don't need to modify the equation, which makes the integral more complicated.

• All you need to do is integrate over the pressure of the water over the submerged area of the gate. I will do everything algebraically and put numbers in last. Things are much clearer that way and, if I misunderstood some point, it is easy for you to correct it.

The gate has the form: y = a*x^2. Here a = 1/9

The base (origin) is at a depth of b. Here b = 2

the density of the water is k. k is not specified but for dimensions in feet it is 62.4 lb/ft^3 for meters it is 1000 kg/m^3

The water pressure is proportional to the density so it is a straight line fnction with a value of b*k at y = 0. It increases with increasing depth (decreasing y) at a rate of k. These two point give the slope and intercept of the pressure function. The pressure as a function of y is:

p = b*k - k*y = k(b - y)

The force on a differential area is the product of the pressure and the area:

dF = p dA = k(b - y) dy dx

All that is left is to set up the limits and integrate. y ranges from 0 to b and x ranges the width of the gate, from -sqrt(y/a) to +sqrt(y/a). Since the x limits are a function of y, you will integrate on x first. the first integration yields: k(b - y) x dy

Applying the limits gives: 2 k (b - y) sqrt(y/a) dy

Integrating this on y (and simplifying) gives: 4ky(b/3 - y/5) sqrt(y/a)

Applying the limits of 0-b:

F = 4kb(b/3 - b/5)sqrt(b/a) = 8kb^2 sqrt(b/a)/15

Putting in the numbers: F = 8 k 2^2 sqrt(18)/15 = 9.051 k

The final result depends on your selection of units to choose k.

• Wouldn't you just find the area of the gate and multiply that by the density?

The area is int (4 - x^2/12) dx, from -6 to 6.

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