 A pool of water is contained on one side by another hinged gate. The gate is hinged at the bottom, which we're calling B, and held in place by a horizontal force at point A. That force is indicated as P in our diagram. What force would be required for equilibrium? By that I mean how much does P have to be to overcome the force of water? Well, again we have ourselves a moment problem. The gate is hinged at the bottom. That's a fulcrum designation, not the letter A, by the way. And P is pushing up here, and the force of water is pushing somewhere down here. Before this to be in equilibrium, the forces must be balanced, by which I actually mean the moments must be balanced. So the sum of forces around point B must be zero. I'm calling positive in the clockwise direction just for fun. Therefore, zero is equal to fw times the distance from B to the center of pressure, which I'm going to call epsilon, just for fun for now, minus P multiplied by the distance from P to B, which is 3. P to B is 3. You see? And then like with the previous example, we end up in a situation where our center of applied force is defined relative to Cg. And the distance between them is always defined up in Ycp. Therefore, I can say epsilon is going to be the distance from the bottom of the gate to the center of gravity plus Ycp. And for that information, let's refer back to our table. I know if I have a hemispherical gate, the distance to the centroid from the bottom is 4 times the radius divided by 3 times pi. So 4 times the radius, that radius here is 3 meters. So I can take my calculator and type 4 times 3 meters divided by 3 times pi, which is 4 divided by pi. Thank you, always calculator. So it's 1.27324 meters. So epsilon then is going to be 1.27324 meters plus Ycp. And just to be accurate here, I'm going to leave that as it was. I'm going to write that as 4 times or 4 divided by pi meters plus Ycp. As I want to limit the opportunity for rounding errors to creep, I'm going to draw that pi a little bit better, much more better. Anyway, so Fw times 4 meters divided by pi plus Ycp minus P times 3. We, our goal is going to be to write this out only in terms of P so that we can use this equation to solve for P. So I'm going to want to try to figure out Fw and Ycp separately. For Ycp, we have the same calculation as we did in the previous example problem, which comes from this slide. We have the force of water is equal to gamma, the specific weight, which is density times gravity, times the height from the surface of the fluid down to the center of gravity times the area of effect. So Ycp then would be negative Ixx times sine times theta divided by Hcg times area. This is Fw is equal to gamma times Hcg times area and Ycp is negative Ixx times sine 90 degrees, which I'll write this theta for now, divided by Hcg times area. And the big bear of a problem in the previous example was the fact that Hcg was written in terms of H. This time around though, we know the height of the water. The height of the water is given, it's 8 meters. Therefore, Hcg is 8 meters minus 4 over pi meters, or 8 meters minus 1.27324 meters. So I know all of those quantities. That way I can just calculate Fw and Ycp as actual numbers and just plug them in and solve hopefully a much simpler algebra problem. So let's just calculate those separately for now. Again, my favorite thing to do in these circumstances would be to plug everything in symbolically and I think that that approach will serve you better. But for the purposes of illustrating this example, Fw is going to be written out separately, calculated separately, and plugged in as probably a rounded number. So density time gravity times Hcg times the area of effect. Rho, the density of water, is going to come from table A1. We are assuming standard temperature and pressure here. So we'll grab the density of water at standard temperature and pressure from table A1. Remember that for our purposes, standard pressure is one atmosphere. Standard temperature is 20 degrees Celsius. Therefore, the density of regular water at standard temperature and pressure is 998 kilograms per cubic meter, probably a longer horizontal line, much more better. Kilograms per cubic meter. And then we're assuming standard gravity. It's a 9.81 meters per second squared. And then we have eight meters minus four thirds, excuse me, four pi meters. And then we are multiplying by the area, which is half the area of a circle that has a radius of three meters. So area of a circle is pi r squared. Therefore, half the area would be pi over two times r squared. So pi over two times three meters, which I can write as three squared, meters squared. Now it doesn't specify a unit for its answer, but if we're solving this equation, it's probably going to be easiest if we work it in newtons and meters. So therefore, I'm going to calculate my force here in newtons so that I can get an answer in newtons a little bit more conveniently. I'll move the decimal place over. That's helpful, much more better. So a newton is defined as a kilogram meter per second squared. And kilogram cancels kilogram, meters squared, meters, meters, that's four meters because these are subtracted. So really, I should have written that as eight minus four for pi quantity meters, but you guys, I'm sure know what I'm talking about. Meter squared times meters times meters, four meters, three plus one is also four. Then second squared cancels second squared, leaving me with newtons. So calculator, if you would deign to perform a little bit of labor for us here, come on. You can do it. Nine hundred ninety eight times nine point eight one times the quantity eight minus quantity four divided by the pi symbol times the pi symbol times three squared divided by two. We get nine hundred and thirty one thousand thirty nine newtons. So I'm going to change my mind on the fly. I'm going to say that we probably want an answer in kilonewtons. So let's calculate this for now in kilonewtons. Nine hundred and thirty one point zero three nine kilonewtons. Cool. Halfway there. Then epsilon, which again remember is four meters over pi plus YCP. Actually, let's just write this as YCP. Let's not make things more confusing. YCP is what we're calculating now. That was negative Ixx times the sign of the angle between the surface and the gate divided by hcg times area. And if we refer back to our table full of moments of inertia, Ixx is zero point one zero nine seven six times r to the fourth power. Let's see if I can remember that. One zero nine seven six one zero nine seven six one zero nine zero point one zero nine seven six times the radius squared times sign of 90 degrees because again our gate is perpendicular to the surface of the water divided by eight minus four meters divided by pi times the area of this half circle is pi over two times radius squared. So let's not scroll up iPad. I don't know what you're doing. Let's draw a big horizontal line again and get to work negative. Then I'm going to draw, okay, how do I do that? No, I think it'll be okay. That's r to the fourth John r squared. He was so distracted by the coefficient. By the way, I'll double check one zero nine seven six one zero nine seven six. Look, we did it. Zero point one zero nine seven six times the radius, which is three meters. So three to the fourth power meters to the fourth power times the sign of 90 degrees, which is one divided by eight minus four over pi quantity meters times pi over two times three squared meters squared. And we presumably want an answer in meters. So I will double check that these cancel appropriately. Four meters in the numerator, three meters in the denominator, and leaves me with meters as my answer. So here we go, zero point. Excuse me, negative zero point one zero nine seven six times three carat. Four times one times two divided by quantity eight minus quantity four divided by pi I don't know why I was such a stickler about that. Instead of running six point whatever seven three could have had way fewer math steps here. But you know, we're being arbitrarily precise. That's not the pi symbol. Come on calculator, you can do this. Look at that pi symbol. And we're going to have to do it again calculator. So hang on to that times three squared. Could have canceled a couple of the exponents on the threes, but it's fine. What are you mad at calculator? Come on, it's fine. Leading parentheses I see. So we get negative zero point zero nine three four eight nine meters. So we get negative zero point zero nine three four eight nine meters. And with that, we have enough to calculate P. I will take this equation, plug it in down here and solve for P. Zero is equal to P would be equal to FW times four meters over pi plus YCP divided by three meters. So I will write that as 931.039 kilonewtons times the quantity four over pi meters plus negative zero point zero nine three four eight nine divided by three meters. So meters cancels meters. I'm left with a proportion that we are multiplying by our force. And that proportion is four over pi minus zero point zero. I'll just grab that number plus this thing. Aha, divided by three. So that number is a little over a third. So we are benefiting by the lever arm that this gate is acting as assuming for the moment that we want to minimize the required force to hold the gate shut. Anyway, 931.039 times this proportion, we get 366.131.