 A pool of water is contained on one side by a hinged gate. The gate is pinned in place with a stop, which is able to withstand 9,200 pounds of force before failing. The gate is 4 feet tall, 5 feet wide, and hinged at the top. How tall is the pool of water able to be before the stop fails? I'm going to approach this a little bit backwards. I'm going to consider the moment acting on the gate. If the gate is hinged at the top and the stop is providing 9,200 pounds of force at the point of failure, located 4 feet away from the fulcrum, then the force of the water pressure, which I will call FW, must counteract that moment at whatever distance away from the fulcrum it is. I'm going to abbreviate that with a dummy variable. It doesn't matter what we call it. I'm not going to call it x because I don't want to confuse it with the x-axis. So I'm just going to call it epsilon. And then we're saying the moment around point A, the sum of moments around A, must equal 0. So 9,200 pounds of force multiplied by 4 feet minus FW times epsilon must equal 0. In our approach here, we are going to be trying to write FW and epsilon in terms of h, the height of the water. Once we can write them in terms of h, we can plug them into this equation and solve for h. I will begin by trying to calculate FW. I know F is going to be gamma multiplied by the height from the surface to the center of gravity of the gate, multiplied by the area of effect of the water. Remember that gamma times hcg here is just density times gravity because that's gamma multiplied by height, which is our Pa equation that we know and love. We're saying force is equal to pressure times area. That's where this equation comes from. It's nothing new. So I can write FW as gamma times hcg times area. And then I will substitute in density times gravity in place of specific weight, since we happen to have more convenient access to density values for water. For hcg, I will recognize that I'm talking about this distance here. The distance from the surface to the center of gravity of the gate, which is going to be the exact middle of the gate. Now, I don't know hcg because h is my unknown. So what I'm going to want to do is try to write hcg in terms of h so that I have something to solve for. I will recognize that the relationship between h and hcg is that h is equal to hcg plus half the height of the door, which is 2 feet. Therefore, h is hcg plus 2 feet and hcg is h minus 2 feet. So instead of writing hcg, I will write h minus 2 feet. Then the area of effect of this door is 4 feet by 5 feet because it is 4 feet tall and 5 feet wide. So the area of effect is 4 feet times 5 feet. I know gravity, or rather I can assume gravity. I'm assuming standard gravitational acceleration, which is going to be 32.2 feet per second squared. And the density of water I can determine if I make some basic assumptions about the water. Since I don't know enough to treat otherwise, I'm going to assume that the water is at standard temperature and pressure and that it is regular water. It is not sea water. It is not salt water. It is just water. So for that density, I will go to table A1 in my textbook, which is the properties of water at a variety of temperatures. So in the appendix, if I mosey over to table A1, I can see density values for water as a function of temperature. Standard temperature and pressure would be one atmosphere and about 20 degrees Celsius. So I could grab 998 kilograms per cubic meter. However, note that everything else in this problem is imperial. So it's probably going to be more convenient for me to use an imperial density. So I will just scroll to the right and instead of grabbing 20 degrees Celsius, I will use 68 degrees Fahrenheit, which is 1.937 slugs per cubic foot. Remember that in this table, each row is supposed to be at equivalent temperature. So you should be able to just read across. You don't have to convert the unit. So 1.937 slugs per cubic foot is close enough for our purposes. 1.937 slugs per cubic foot. And at this point, I have FW as a function of H. And for convenience here, I will plug in everything else and try to just write it as a function of H with the units already handled. That'll make my sum of moments calculation a little bit easier. So I'm going to write 1.937 slugs per cubic foot, multiplied by 32.2 feet per second squared, multiplied by H minus 2 quantity feet. Note that by doing this, H is in feet. So I'm building the unit into our equation so that it's easier to plug into the sum of moments later on. But this only works if H is in feet. And then 4 feet times 5 feet. And I will calculate a quantity in pounds of force because the force required to overcome this top block is already given in pounds of force, so that will be fewer units to have to keep track of in the moment calculation. Cubic feet cancels three of these feet. I recognize that a pound of force is a slug multiplied by a feet per second squared. So slug cancels slug, feet cancels feet, second squared cancels second squared. I'm left with pounds of force. So if I pop up my calculator here, I can write this as 1.937 times 32.2 times the quantity H minus 2 times 4 times 5. And I get 1,247.43 times H minus 2. Yeah, I know I could have just calculated everything else except for H minus 2. That would have been just fine and then written out H minus 2, but apparently I felt the need to write it in my calculator. I could also write this as 1,247.43 times H minus 2 times whatever that value is, which would be 2,494.86. That would be perfectly valid. One might be more convenient to plug into my moment equation, but maybe not. So that's FW. For epsilon, let's remember that on the gate, the center of gravity is two feet down. And we calculate the position of the center of applied pressure, the CP value, relative to CG. And it is relative with a variable called YCP. And remember, YCP is defined in the upward direction. It is towards the surface. So if I write epsilon is equal to 2 feet minus YCP, that will work regardless of where CG actually is. Even if it was lower than CG, YCP would be a negative number, which means that I would be calculating 2 feet minus a negative quantity, which would yield a number greater than 2 feet. So by defining it this way, we don't have to worry about the actual direction. I will point out that in almost all cases, we consider CG is actually higher than CP. And YCP is going to be negative the moment of inertia around the centroid in the x-axis direction times the sign of the angle between the surface of the water and the gate, which in this case is 90 degrees. Sign of 90 degrees is 1, so we can effectively ignore that. Divided by the height from the surface to the center of gravity of the door, which again, we are going to write in terms of H multiplied by area. So I'm going to say YCP is negative Ixx times sign of 90 degrees divided by HCG times area. And for the moment of inertia, I have this convenient chart of the moments of inertia of some common geometry that we're going to be considering. We have a rectangle, our base is 5 feet, our length is 4 feet. Therefore our moment of inertia is 5 feet times 4 feet cubed divided by 12. Sign of 90 degrees because again, the surface is at a 90 degree angle from the gate and we are dividing by HCG, which we are writing as H minus 2 feet. And we are multiplying by 4 feet times 5 feet. And our goal here is going to be to write the quantity in feet because we want to be able to plug this in to our epsilon equation and then plug that into our moment equation. And it already is in feet because cubic feet cancels feet, feet and feet. Again, asterisks here, H is in feet for this calculation. This equation only works because we built in the feet dimension for H. Otherwise we'd have to handle that unit in our calculation in the moment. Literally in the moment while we're calculating the moment. Okay, so I will write this as negative 5 times 4. That should be 4 cubed feet cubed by the way. That's a 6 calculator. Come on, not a carrot. Times the sign of 90 degrees, which is 1, but I hate just to be thorough. I'll actually write it properly. 90 degrees. Okay. And then we are dividing that entire quantity by 12 times H minus 2 times 4 times 5. That's 12 times, not 12 minus. Calculator, what are you doing? Just delete that so I'm not reminded. So we're left with negative 4 thirds divided by H minus 2. If I represent that as an approximate quantity, come on calculator. It's negative 1.3333333 divided by H minus 2. And that again is in feet. So we got a, well, I guess assuming for the moment that H is greater than 2, then I'm left with a quantity that is less than 0, which means that I'm going to be talking about an epsilon value that is greater than 2. Anyway, epsilon then is 2 feet minus negative 1.333 feet divided by H minus 2, which I will write in feet. And for convenience, instead of writing minus negative, I'll write that as a positive. So 2 plus 1.3333 divided by H minus 2 times feet. The next step is plugging this and this into our moment calculation. So instead of writing FW, I'm going to write this quantity here. And instead of writing epsilon, I'm going to write, oops, need to throw in my units. This is pounds of force. And then instead of writing epsilon, I'm writing this quantity. Pounds of force, cancels pounds of force, feet cancels feet. And I'm left with one equation and one unknown, and that unknown will yield a quantity that it is in feet. And that is the answer to the question. So to continue, we could do the algebra, but that's a little bit outside the scope of this class. This is fluids class, not algebra class. So I will just let my calculator solve it. And I guess I will solve the algebra by hand after this. That way, once you see the answer, if you really want to see the algebra, you can stick and watch the rest of the video, but otherwise you can move on. By the way, if you are working to this on an exam in one of my classes, you are perfectly welcome to use your calculator or Wolfram Alpha or MATLAB to solve the math. I am not expecting you to have to do the algebra. So we are solving this equation for x, calculator yields, too few arguments. Ah, because I have one too many parentheses. Okay, now I know that's not right. Seems to be how it's written. Oh, I see. It thinks we're multiplying. We're not subtracting. The calculator has a negative operator and a minus operator, and I typed negative when I should have typed subtract. Okay, it yields 16 feet. See, when you do things on a calculator, it definitely works out better. 16.08 feet. And that is the answer to the question. But if you're curious about the algebra, let's go into the algebra dimension. All the units cancel and I will write this. Instead of taking 9,200 times 4 minus this quantity, I will write them on opposite sides of the equation. So I'll have 9,200 times 4 is equal to 12,000, excuse me, 1247.43 times h minus 2 times 2 plus 1.333 divided by h minus 2. We'll distribute, so 9,200 times 4 is equal to 1247.43 times h minus 2 times 1247.43 times 2 plus 1.333 divided by h minus 2. And then I will foil these. That would be first, which is 2 times 1247.43 times h. And then outside, which would be plus 1247.43 times 1.333 times h over h minus 2. And then inside, which would be minus 2 times 2 times 1247.43. And then last, which would be minus 2 times 1247.43 times 1.333 times 1 over h minus 2. Okay, now let's compute all the multiplication so that we don't have to keep track of all of those dots because I'm surely going to miss those. So calculator, come back. 9,200 times 4 is 36,800. 36,800 is equal to 2 times 1247.43. 2 times 1247.43. Let's hope that that wasn't a rounded quantity because we are going to incur some losses. 2,494.86. And then 1247.43 times 1.333. And naturally, it would only be right for me to write this as 4 thirds so that I don't have to write an infinite number of 3's. Plus 1663.24 times h over h minus 2, which I will deal with in a second. And then minus 2 times 2 times 1247.43, which I'm going to write as 4 times. So minus 4989.72 minus this garbage 2 times. I guess let's just grab that and let's throw a 2 in there. 3, 3, 2, 6.48 times 1 over h minus 2. Aha, okay. Then I'll group together the constants. So adding 4989.72 to both sides, 36,800 plus 4989.72 yields 41,789.7 is equal to. And then I have 2,494.86 times h plus 1663.24 times h over h minus 2, minus 3, 3, 2, 6.48 times 1 over h minus 2. And there are, you know, a number of ways that we could approach this, but the way that I'm going to approach this is by multiplying everything by h minus 2 at this point because that will get rid of that denominator. So times h minus 2 is equal to 2,494.86 times h times h minus 2 quantity plus 1663.24 times h minus 3, 3, 2, 6.48. And then I will distribute so 41,789.7 times h minus 2 times that number, which is 8,35,79.4 is equal to 2,494.86 times h squared minus 2 times that number. 2,494.86 times 2, which is 4989.72 times h plus 1663.24 times h minus that constant at the end minus 3, 3, 2, 6.48. Go write that as, okay, let's go, 0 is equal to 2,494.86 times h squared and then plus 1663.24 times h minus 2,989.72 times h. It's 2,989, John, 72 times h minus 41,889.7 times h. And that's all the h to the first power values, right? Yeah, and then minus 3,326.48 plus 8,3579.4. Okay, and then 2,494.86 is the odds that I'm going to remember that that's a 4, not a 9. Let's fix that, 2,494.86 times h squared plus all three of those coefficients. So, 1663.24 minus 2,989.72 minus 41,789.7 yields negative quantities. I'll write that as plus negative 43,116.2. Excellent handwriting times h and then we have 8,3579.4. Terrible handwriting. Man, what's happening? I need to zoom in occasionally, John. Plus 80,252.9. And look at that. We have a quantity that is 0 is equal to a, let's say, times h squared plus b times h plus z. So the answer to that is going to be, of course, x is equal to negative b plus or minus a square, but b squared minus 4ac all divided by 2a. So, x is equal to negative b plus or minus the square root of b squared minus 4 times a times c all divided by 2a, where a is 2,494. Okay, John, you can zoom in. It's fine. I try not to zoom in because I don't want to discombobulate people, but no one's watching this part anyway. So, who cares? a is equal to 2,494.86. b is equal to negative 43,116.2. And c is equal to 80,252, I think? Yeah, 0.9. Okay. So we will have two answers, one that's possible, one that's not. So, let's write this as negative sign, not to be confused with the subtract sign, negative, negative, 43,116.2 plus a calculator plus the square root of quantity, negative sign, 43,116.2 squared minus 4 times 2,494.86 times ad 252.9 close front c. Did I add a leading front c? Let's hope so, divided by 2 times 2,494.86. And then I will jump back, double check. Yeah, I did. Look at me. Okay, that answer is 15.1602, 15.1602. And then if I change that plus to a minus, oh, you can't see what I'm doing. That was a shame. That's a negative sign, not a minus. There we go. We get 2.2, 2.12, 2. Okay. So one of these is possible, one of these is not. The 2.122 form isn't going to be possible, so I'm calling 15.16 feet our answer. Now, 15.16 versus 16.08. Which is the better answer? Definitely the 16.08. Did you see how many times I multiplied garbage by two or three or four? Any rounding error that would have appeared up here is going to just be magnified over and over and over again. So I would definitely recommend doing as little arithmetic in your actual calculation as possible. We don't want intermediate steps that are multiplication or division. I mean, gosh, where did that 1247.43 come from anyway? That was probably around a number in and of itself. The best thing to do under these circumstances would be to just calculate the quantity with a single calculation step that is keeping track of everything symbolically, like plugging this in to your moment calculation instead of this. Anyway.