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Expansion of a Hemispherical Dome

Patrick Ford
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Alright guys, so hopefully try this on your own. Let's go ahead and work this out together. So we have a geodesic hemispherical dome made out of aluminum. So what does that mean? Hemispherical dome is something like you might see in a greenhouse or something like that. Um So it's kind of just imagine if you took a sphere and you kind of just cut it exactly in half. Right? So this is sort of like this hemisphere goes down like this. Now, what happens is it's made of aluminum. And on a certain day where you have a temperature of negative 10, you measure the radius of this hemispherical dome to be m. This r is equal to 25, it's half the distance across the entire thing. Then what happens is you're gonna measure it again on a warmer day where it's 30°C now because we're working with a metal like aluminum, it basically is going to expand but it doesn't just expand in one dimension, expands in all three dimensions because we have a three dimensional objects. So basically what's happening here is that on a warmer day? Everything has kind of swollen up and expanded like this. So now you can kind of imagine that the dome is going to be shaped like this. Right? I'm obviously just exaggerating this, but basically the radius of this has increased a little bit. And now what happens is that this geodesic hemispherical dome has a little bit more interior space inside of it. So basically that's what we're looking for in this problem, how much more interior space? And what that means here is we're trying to figure out, well, how much volume did you add to the dome just by increasing the temperature? So that's gonna be delta v. What is the change in the volume? That's going to represent how much more space you have inside that dome. So now that we know what variable we were looking for, let's go ahead and get started with our volumetric thermal expansion equations. We just have one for delta v. So this is gonna be delta v is equal to beta times the initial volume times the change in the temperature. Now, if we're looking for delta V, we just have to figure out everything else on the right side of the equation. Now, beta is just going to be our aluminum of volumetric expansion coefficients. The initial volume. We actually don't have that. That wasn't given to us in the problem. What about the change in the temperature? Well, let's see the we're going from negative 10°C and then we went to 30°C. So, that just means here that the change in the temperature delta T. It is just equal to 30 minus negative 10. This is the difference which is just 40 degrees Celsius now, because we're working delta. Again, it doesn't matter for you Celsius or kelvin, you could convert this to kelvin, which is what you're gonna find is it still just works out to 40. All right. So, we have what the change in the temperature is. Now what we have to do is just find the initial volume of this hemisphere. So let's go ahead and work that out. Right, So the initial volume of this hemispherical dome, how do we do that? Well, you may remember from geometry or trigonometry or whatever, that if you have a sphere like this, I'm gonna draw this out real quick. The volume of a sphere is going to equal four thirds pi r cubed. But we don't have a sphere that we're working with. We're working with a hemisphere. So the volume of a hemisphere is just going to be half of that. Right? It's half of one sphere. So it's one half of v sphere. So we can do is we can just basically cut this fraction of four thirds in half. So we're just gonna do two thirds pi r cubed. So that is the volume of the hemisphere. So that just means that your v not is just gonna equal two thirds pi times the radius. What's the initial radius at negative 10 degrees Celsius? We're told that it's exactly 25 m. So we're just gonna plug in 25. I don't have any conversions. You cube that what you'll get is three points 27 times 10 to the fourth and that's in meters cubed Alright, so now we just take this number and we plug it back into this equation over here. So, your delta v is just gonna equal beta, which is 7.2 times 10 to the minus five times the initial volume, which is just the 3.27 times 10 to the fourth. And then we're gonna multiply by the change in the temperature, which is 40°C or Kelvin doesn't matter. And then we're doing what you're going to get here is the change in volume is 94.2 m cubed. So that's actually a pretty substantial difference, basically what happens is that again, the aluminum is kind of expanded on a hot summer day and therefore you have a little bit more space, a little bit more volume inside of that hemispherical dome. Alright, so that's it for this one. Guys, let me know if you have any questions.