A wire of density $\delta(x, y, z)=15 \sqrt{y+2}$ lies along the curve $\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+$ $2 t \mathbf{k},-1 \leq t \leq 1 .$ Find its center of mass. Then sketch the curve and center of mass together.

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Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Okay, folks. So inducing video. We're gonna take a look at problem number 34 where were given a density function, and we're going to be looking for I'm the center of mass of a wire lying long occurred and then we're going to sketch the curve and dissenter roughness. Okay, so first of all, let's let's just go ahead and sketch out the curve here. The curve, This is the Z axis. Um, this is the Y axis, and you'll see why I'm drawing in this way later. Um, so you know, along the curve were given to function for why at parameter rise by t and were also given a function for Z, which is also parameter rised by t. And why is T squared minus one and disease to t? That means I can write why, as Z over two squared minus one because he over twos t t squared is just super to sward. And that is 1/4 Z squared minus one. So now you see why I'm drawing this wine z axis like this. And that's because I want to write why as a function of Z, because that's really easy to draw because we have a problem. We have 1/4 and Z squared minus one. And the Vertex of that problem lies here. This is Nick. This is negative one. And you have, uh, a problem that looks like this. And because of the fact that that our range is limited to negative one and one for T. So T is in the range of negative one and one. That means that first of all, well, let's just plug it in here when tees negative one. Why is zero okay? So that means or why can I get Can I get greater than zero and same as when teeth equal to positive one. That's also when Why is zero all right? But we have graphed out the curve. Now we can go ahead and look for the center of mass. And as you recall, the there isn't a form. There is a formula for Senator. If Mayes and that formula is simply given by this, it is a fraction fraction. Um, which is integral. The M. That's for the denominator and numerator is going to be are gm. Okay, that's that's really just the weighted sum of all of the positions of the particles. Um, that constitutes the the object whose center of mass we're looking for, anyway. Um, because of the fact that the curve r T does not have the next component, I can rewrite this center of mass vector in in this way. So it's really just one over m began stands for the total mass, um, multiplied by some of why come. But I had multiplied by, um let's see why the M plus Z had multiplied by Z Do yet. Okay, So basically, we're gonna have to do this integral and in distant to crawl, and then we're gonna add them up and divide by the total mess. All right. Anyway, the total mass. Let's let's look well. There's really a lot of into girls to do here. We have three of them. The total Maris. Let's do that. First total masses, religious density multiplied by yes, okay. And that we know how to do it because we're given the density function, which is really just 15 y plus two multiplied by d years. Um, between first of all, this is even a square root of why plus two square root of one plus why prime of Z squared, multiplied by deeds E and between. Negative too. And positive, too. Okay, um, this is equal to 30 from 0 to 2. Um, Skerritt of why plus two, one plus c squared over four. Dean Z. Okay. And this when you crank all of the numbers out, this is gonna give you 30 um, zero to z squared over four plus one. Did you see? And this is going to give you 80 when you crank out all of the trivial algebra stuff. So that the total Maris, Um, I you know, I skipped over a few intermediate steps year. Um, for example, here, I really should have plucked in. Why the function of Z, which is 1/4 c squared, minus one. And then when you multiply the two scrubbers together and you do some algebra and you you're gonna end up with a anywhere fast Italo Mass, let's now go ahead and, um, and do disinter crawl. So for one, we have why d m it? But as you remember, de me's Delta DS okay. And that's going to give you Z squared over four minus one because that's what why is multiplied by 15. Why plus two do? Yes. We, uh, plucked in what we got previously, Kate. And this is gonna be 15 negative too, to to the's great over four in minus one, multiplied by Z squared over four plus one multiplied by Deasy. And this is equal to 15. Negative, too, to to Z to the fourth over. 16 minus one DZ. And I'm going to skip through a couple of a couple lines of algebra here and just directly arrived at Negative 48. That's the first integral the segment Integral. I want to do the same thing for the second. Integral is see in Delta DS because that's what the M is that is equal to 30 actually. Excuse me, that's not true. That is equal to let's see 15 Z. Um, why? Plus two one plus why Prime of Z squared, u Z and this is equal to 15 z Do you squirt over for last one, daisy? And this is between negative to you and to and when you Ah, when you go ahead and plug in the numbers here, you're going to arrive at this just 1/4 c Q DZ plus zzz between negative two and two negative two and two, and this is gonna give you zero. Okay. And so this is the second integral. And apparently our second integral gives us zero. And that's a good thing. Let's let's go ahead and try to figure out what the center of Mass is. So the center of mass, which is a vector, is just gonna be won over mass z had Z d m plus why had why the m This is zero, as we just calculated. And this is equal to negative 48. Okay, so we have negative. 48/80 multiplied by white hat, and that is going to give you negative. 3/5 multiplied by white hat. And let me ah, just copy down this graph that I drew in the beginning of the video and let me ah, draw out this point. Negative. 3/5 is gonna lie and little bit below the center. You're all right. So this is negative. 3/5. That is the location of this center of mass of this wire. And thank you for watching. I know that this is a lot of algebra lot of math, but But you made it You You made it through the video, so thank you for that. I will see in the next video by

University of California, Berkeley