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Numerade Educator

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Problem 36 Hard Difficulty

Find the center of mass of a thin wire lying along the curve $\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+$ $(2 / 3) t^{3 / 2} \mathbf{k}, 0 \leq t \leq 2,$ if the density is $\delta=3 \sqrt{5+t}$.

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Okay, folks. So let's take a look at problem number 36. We're going to be looking forward to Senator after my ass of a thin wire lying along a curve and were given the density of the of the of the wire cats. You know, there's there's a few preliminary things that I think you should know. First of all, senator of masses of Vector. Um, and it's got, you know, ah, formula for it, which I'm assuming that you already know Senator Mass is really just a weighted some of the positions of all the particles that make up the object that you're looking for this under of a mass for, um, divided by the total mass. Of course. So what I mean by that is we have integral are gm, okay? And we're integrating over the entire volume or area of of this mass. Okay. And because ours a vector, um, are are is just, you know, x hat plus x at X plus. Why had why places he had the I'm gonna So I'm going to write this thing as a some over I, um, multiplied by a fraction x i d m multiplied by EI where e. I is the unit vectors like X had Why had he had an X I means X, y or Z, depending on I while So when I is equal the one I have X, I equals X. And when Isaac goes to X, I equals why and when I equals three x, I equals Z. But anyway, that's just a notation. So now let's 1st 1st fall. Let's go ahead and start looking for M, and we're done looking for em. We're gonna do three intervals, so we have I equals one and two and three. But anyway, that's that's good. Let's get started Mass of the wire. It's just Delta. Yes, and were given a function for Delta. And Delta is parameter prized by the by the new variable T. So we have before plugging in Delta. Let's just go and rewrite yes, as ex dot squared plus y dot squared z dot squared, multiplied by DT, where I used the notation where dot on top of a letter means the derivative of that letter with respect ity. All right. Anyway, we have between zero and two three times Route five plus T desk Delta multiplied by the square root of one plus four plus t d t. And this is really trivial now because, um um okay, lets see five plus t for it into grand d t zero to. And when you crank this out, you get 36. Okay, so that's the That's the mass. And so we're done for for this part, the nominator. And now we're going to to be We're gonna be doing three intervals. All of them look like this. The 1st 1 is when I've won. The 2nd 1 is when I was two and the 3rd 1 is went Eyes three. Okay, so now I have one. We have x multiplied by the M, integrated along the wire. OK, so X is just eggs. Um, well, let's not this Let's, um Let's not start with one. Let's start with why, which is when I equals two. So we have Why d m um that's equal to two times xdm. So if we have this integral week, we also have this. Anyway, um, we have two times tee times three five plus t day T because that's, um, this is X X is as a function of tea is just tea because that's given in the problem. And the M is d m is just dealt with the s. And if you look at this, you get five plus t DT. So I'm just, you know, I'm just reusing the results that we have previously obtained. This thing is his Stelter DS in Delta TSZ with the M. And so this is because this is three five plus t d t. I took it and I plucked it here. That's all I'm doing anyway, So we have Let's see, we have 60 the to five t place T squared DT. And when you crank those numbers, how you get 76. Okay, So as for xdm, that's simply 76 over to which is 30. Let's see, 38. Yeah, because this integral is twice this integral. So So to get this one, you just divide this by two anyway. All right, so that's the second integral. Um, we have one last to go. Z d n is going to be equal to 2/3 t to the power of 3/2 because that's what C is. Z is 2/3 t to the power three halfs that's given in the problem is a given multiplied by Delta DS. Um, but I'm not gonna write all that. Yes, I'm just gonna reuse the results that we have previously obtained because Delta ds is just three five plus t DT. Okay, now, this could be simplified into twice the integral 3/2 five plus t DT. And this thing can be a well, first of all, let me plug in the integration limits first when you Ah, go ahead and crank this how you get a really nasty number. Well, not that nasty, but a kind of nasty here because it can't be simplified. Further Group two. All right, so that's third. And do grow now. We have all of our components. We're gonna put them together and and and get the center of mass factor. So the center of mass is limit. Copy this down here, center mass is this thing. And we have this and we have all three of these intervals. So, um, let's go ahead and read it out. We have masses 36. Of course, the X component is 38. That's this one and the y component in 76 and the Z component. If is this huge chunk of number and I'm gonna go ahead and simplify this a little bit. Maurin 19/18. 19/9. Four. Route to over seven. Fast the center of mass Vector. And we're done for this video. Thank you for watching.