Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $ a $ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse.
$(2 a / 5,2 a / 5)$ if vertex is (0,0) and sides are along positive axes
right. This is a problem. Number fifteen from section for chapter fifteen of early transcendence. A petition by James Stewart prompts us finds in your mass of Obama in the shape of the necessities, right? Triangle with equal sides of likely intensity in any form is proportional to the square of the distance from the Vertex opposite. I got news. It's the first thing we want to do is just strolling picture. So we're told that we have a right triangle, and each of the legs has, like, a all right. And so we're looking for the center of mass and the center mass is the point. Somewhere in this triangle, where is Jai? Balanced it on the end of the pen. It would not fall. It was stay perfectly balanced. So, in some sense, it's the average of me, Uh, X coordinate where all the masses, okay, and the white court, where all the mass is looking. So the first thing we're going to need to do is actually just compute the mass and so good. And so if we put in some sort of court system, so just put the X according exact sis along one bag in the play access number. We're gonna let this verdicts. The origin in there is our mill. It's protects me the origin because you know that the density is proportional to the distance from this pool and the origin is a great point to compute the distance from any of the nice distance formula from the origins. If I have this point right here, looks why and then the distance here, he's going to be square. Thanks for squaring because squared, that's the spirit sort of scream. Basically, just And so we're told the density of this triangle is proportional to this distance Where and so now if we wanted to find them ass of this triangle, the mass in general, it's from basic physical science is going to be the density times the area. And so if the dense New York constantly would just take that constant density vilified by the area is trying on your help, Max. Okay, but that's not kicks. If we take it's a little square here, Mom. Now, this small little differential square is goingto have area. Yeah, which is gonna be dx times. Why and then the small little square We can actually assume that the density is crossed. Okay? And so the density here, we're told, is proportional to the distance squared from the work. I talked to several of Dean. The intensity that any cortex, why it's proportional, which means there's some constant such that the densities that constant times square y squared and now knows that we don't know this number's seek. So either we don't have enough information. So the problem or is going to turn out to not matter and that's going to be the case. The center mass is going to be independent. That's proportionality constant. So in our square over here, we can assume that the density is constantly equal to this value given by the ex Michael. And so we have the small, little different mass cam. This is just Sydney the point only sometimes. Thanks. It's just good, Erica. And now we want to know the total mass. You can just add up all these little squares. And now, of course, these air differentially small squares. So we're gonna have to use a vinaigrette. And because we're looking for two dimensions ex, um, I never actually he's a double emigrant. And so what? We have is a mass the girl for X and y Well, I am actually could end that you're sort of you're gonna have to get X and y involved so we'Ll have to put in our limits of integration. Why? But the grand if something see x squared square Yeah Excuse me Now the way I have written I've written the ex first So we'LL do this eccentric girl first actually doesn't matter by fifteen East from this anagram is kicking a differential so we can actually embrace more want But if we look at X first we're going to fix a why a white coat. So here and now, where is X going excess starting at zero? Go away until you get this line here. And what is this long line when you're right? This is a function of why? Well, it's not too hard to see that it's going to be okay. Us What? There's a couple ways to see that It's a slope. It's a line of slope negative one as X intercept a year. But that's what it's going to be. So X is going to go from zero Tuesday. Why and why is gonna be going from Europe more? No, you actually compute. Yes. And I know it seems that it doesn't have anything to do with Senator Vast. But we really do need to know the master calculated the center of mass because, well, the similar mass is going to be basically an average over this mess, Senator, but versus just compute this shit. Like I said mass giving about this on your girl going from you, eh? Thanks, guys. Hey, why in your hand? So we have this constant. Just get affected that out of the thunder girls, then we have square by square. Why now? We just gotta work. So we're doing x first. So we need to find an anti guerrilla of dysfunction, huh? Just a polynomial. So anti drug X squared X cubed over three. Why squared? It's constant from the excitement, which is sex by squaring. They're evaluated zero minus. Why? It's playing in a mine. Why? Yeah, it's over. Hey. Hey, Mike. Why? You know, So I really don't need to plug in the lower limit because it's your functions here. Okay, Now, this is just a polynomial. Consider a couple ways to do it. You do. U substitution No, or you can just expanded out. I prefer just to expect it out. Um, they really doesn't matter. I mean, this first one, if you let new a Miles, why, you? Well, we're just going to get a negative sign here. Why? And so this first, then a girl. I was just going to He's my everything. So I can just look at this first time right here. Do this substitution. It's just going to me, the inner girl. Okay, So why is equal to zero use from a twice a zero, You three, near where we have this negative sign, King and bless this one. I'm just going to expand out. Okay, Clyde. Great minds, human. Bye. Okay, so one thing we can do right away as we can just get rid of this negative sign if we just switch and zero. And at this point, this is really just a count. One question. So this this integral here. Well, when I take the anti Drew that you can't, you're going to get you to the fourth of four, which is going to give me a U to the fourth of twelve. And then I'm gonna value zero and about you today. That's going to give me minus a to the force over twelve minus zero. But that whole thing zero so begin. I'm just kidding. Four acto from this guy and then plus take Nancy Drew here. Hey, why? Yeah, And miss line Where? Four. I went from zero to you. Okay, but and I'm just going to get subtraction. Look at together. What ain't it forth? Twelve? His term comes with a painter before over three minus into the forest for, you know, I'm almost done. So now we just have to add. Okay, so here's one You know, the fourth number twelve. We had a common denominator. This's going to be for need to the fourth of it all, Linus Missile. Me three needed the fourth number twelve. So all in all we get to you. Still other cops in there, too. And, of course, twelve Cher's C times. No, over six. And so that'LL be our mass. And now notice our masked depends on this constant C and it depends on basically the size of a triangle. We're not that That's just the mask. So what is the mass have to do with the senator's glass. Well, if we look back at this triangle, we found the mass. You know what? The X cordon of December Mass that's gonna be here long. This leg, What I could D'Oh, they just cut up the rectangle in strips and I just want to take and each one of these Canada and infinitely, uh, thin strips. Say us one. Well, I want to take the mass and waited of this Valued X in the god of all those those massive trying his X coordinates Well, then divided by the total mass. And that's just going to give me an average of for the masses along the X axis. And I want to take it of course room You're a over the hole triangle. So basically, what I'm saying is I want to integrate, add up all of the mass, but I want await it with this X, and that's gonna give me kind of expected value. You know where the mass is and then and just getting by. It's the same thing. Where what? I just want to add up all the mass. It's raining. It's why, says Wise is telling me where alone it's war horizontal strips. Kind of what? The masses that each one of those strips. And then that'LL just give me an average. That I will. Then what mask? This is very much analogous if you're confused too, you know, finding an average grade. What am I doing in the numerator? I'm just adding up, uh, kind of, you know, the x coordinate. This is like a throne of students in the class, the total mass and each one of those strips for your students that their grade and then I just dividing by the total number of students, something like it really is just an average. And again, wherever this point is, that's going to be the balancing point. Also called the Essentially Hey. Okay, So I really need to do is just compute news to dinner girls here and we'LL start with ex. So if I look at this girl Thanks. Okay. Well, actually, your rights to recall what d mm. Is this the sea experience wise? Great. Why? And then of course, the same integration will be the same. So it's more there. So your call. We're going from zero to aimless. Why? What is going on here? A and I'm waiting with X here, X woman tm it's going to be okay with this this constant understand? Good impactor out because you've been extra. Why? And I have this sort of Vincent me escort y squared in the meantime study area. Thanks, Buck. Well, actually, I think this one is always a little bit easier, at least initially. So we do. The X and a girl first actually have this X here, which is going to allow me to do a nice the substitution of Yuki X squared plus y squared. Yeah, thanks. Yes. No. So then I can replace this x Times GX in here. You So get this factor in half that from California's just good practice to put all the constants out of the immigrants. Defector. Thank you. No, Mark it. So I have sex. The ex gets dissolved into you, and I just have a new here still monkeying while here. Nice sniffing out where my unit is going, OK? Selling why is equal to that's where X is equal to zero square. Yeah, right. And then what? X is equal to a mine? Why? Well, I have a minus y squared. So here in excellence. Maybe minus y. Here. What? Scream? Great. Hey, hey. Why? Why? Good, Great. And so just actually do that. CEO Yoo, No, to a right now. This is going to become you squared. Yeah. Square. Never again about getting this from black squared, Tio. It's Mars plus screaming. Okay. All right. So you can factor out this factor of from what to see for now. Okay, So, integrating a right? Yeah. Big jacket. First term is news. A month squared, squared, squared minus. All right. Good for you. Oh, right. So it's not the nicest thing, but it's doable. All right. So again, we could expand this out. Uh, that's probably the best thing to do with this point. So if we have thank you. No, I didn't mean to do that, Thank God. Okay, so I have to live back, so All right, See, Four. It's good. Right this way. And, Grant, I was ever going to square this. Are you a A minus y squared? It's wise, friend. So nice way to just distribute about some squares. Four. Can you really have two times squared? A minus y squared spider and minus what course? So you and something good happens, they're wise to the force will cancel. Yeah, tens over it is like this. And so this first guy, we definitely just wanting the substitution, so just let me be. That's why before you can even do it. So we're just going to pick up a negative sign? Come on, go ahead and skip a couple steps. Essences Kwan. So there's a Miles one of the fourth is is if I just do the substitution, I'm going to get a negative, But it's just gonna be a minus one. But of course, with these negative, so eh Okay, well, then, next turn. Well, let's just expanded out, and then we're gonna have some scratch. Very quite scared, eh? No. Hey, s So we should probably go ahead and expand this out. Ms. Was going to be what? Squared Squared? Minus two. Why? Cute. What? Or all times? This too. That's what we're immigrating and a great first term. Just get why schemed a squared over three minus Teo. Why? Over. Okay, and that's it. Soon our kids see over for and now when I go out with a this first chairman, get zero on the negative skin cancer with the negative, I subtract five. Oh, right. Plus Okay, Teo, And look a here. They're going to get in a three. Hey, keeper for. But we also have this factor to move to a little bit more in class. Who now? It's just more had infractions are here. Reform. Five common nominator. Uh, well, it looks like we're gonna have to get over yet, too. Snakes. Hey, and go on. So here we have twenty. And of the fifteen or sixteen, and now here, you really have to service Will be Ernie End of the Fist sixteen. And here we have what that's going to be. Well, well and then a questionable too. So it looks like we're getting too. So I totally get it for over six. Look inside now. Eight of the fifth over five. Again. That's twelve. And the fifth over sixteen. Plus for here. Twelve others, of course. Yeah, for see, That's going to be sixteen. So get sixteen. Can I see you for a course? Sixteen. Sixty More than a simple prize, too. Sync the fifth over. All right, Snow Done. Kelly's with the X coordinate. The X coordinate mass call is this double and a girl. Thanks. Um, a mass, actually. Hey, team sing. Hey, six, You call and this is going to simplify to a over five. No notice seized. You canceled to the senator. Masses of people see. But of course, it does depend on the size of the triangle. And now we could do exactly the same thing For the Y center of mass, you can get in the exact same wonder girl. It would work exactly the same. But something really nice to notice is because of the symmetry of the density. If isis switch at X and why nothing's going to change. I mean, you can also see that when we actually do this Computer wasn't a girl. If I just replace this with why I'm getting exactly the same, you know, grand, just relabeled with X and y said the Y center mass. It's going to be the same today. But if you don't believe me, of course you can just damage all. It's very similar