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Numerade Educator

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Problem 42 Hard Difficulty

Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve
$$\mathbf{r}(t)=t \mathbf{i}+\frac{2 \sqrt{2}}{3} t^{3 / 2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}, \quad 0 \leq t \leq 2$$
if the density is $\delta=1 /(t+1)$.

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Video Transcript

Okay, folks. So now we're going to be talking about problem number 42. But this is problem number 42. Um, this problem, we're gonna be looking for the center of mass and the moment of inertia, of everything wire lying along a very complicated looking curve. And we're also giving a density function, which is premature ized by the variable t. Okay, so first of all, you know, there's gonna be a lot of computations, and because of the fact that we don't have a lot of time, I'm just gonna skip over a lot of the trivial and time consuming computations and leave those to you. And all I'm doing here for this video is giving you a rough idea of the strategy or, um, or an overall picture of what you should have in your mind when you're doing this problem. Anyway, for this problem, first of all, we're gonna be looking for the center of mass. And as you know, Senator Mass is a vector is not just the numbers of vector of three numbers. Um, so we're going to be doing basically three inter girls for this honor of mass and moment of inertia. as well. Moment of inertia. Um, you're gonna have to find three numbers for the moment of inertia. And that corresponds to, um, Mom. Inertia about X. About why and about the Z axis. Okay, so that's that's where we're going with this. Let's go ahead and start writing out some steps we have. First of all, a center of mass is going to be written as one over the total mass multiplied by integral are D M. Now are is obviously the vector of X, x, y and z. Okay, so this integral is really three into girls, okay? And I'm just running at you in a very compact way. Um, and the total total total mass. Let's go ahead and try to find the total mass here. Okay. Um, so very the total mass. It's obviously is actually very simple. It's Delta. Yes. Um, because this is D m. And if you do that injured, if you do the integral you get the total mass anywhere. I'm going to write Delta as one over t plus one multiplied by the S by. There's another way to write DS, which is to be right. As an expression in terms of tea. So, um so how do you write DS in terms of tea? Well, as you remember, DS can be written as the X squared plus d y squared plus thes e squared. And this could be written as ex dot squared plus y dot squared plus z dot squared d t. Where I used the notation dot on top of a letter to represent time derivative and and were given the curve X as a function of tea is just tea And why the function of T is something something and Z as a function of t it another function. It's a proble teas guard over to. But anyway, if you do the time and derivative if you take you to differentiate with respect to time although all three of these functions and you plant them back in here, you're going to you're gonna have one, plus I to t plus t squared DT. All right, now, this is gonna be written as one over t plus one square root of T plus one squared d t from let's see from 0 to 2. And if you crank out this integral, you're gonna get to Okay, so that's the total mass. Um, we have found the total mass of the wire. Let's go ahead and look for the three Integral three inter girls for this for this thing. Um rdm is really x had xdm Plus why hat? Why d m blesses He had z d n. Okay, so that's what we mean. That's what I mean when I say three into girls. So let's go ahead and crank them all out. We have first of all, we have xdm that's simply written as X adult A D. S and this is gonna be written as while x as a function of tea is just tea that's given in the problem. Um, multiplied by adult A d s. But we have dealt a DS here. We have an expression for Delta DS. This is Delta. Yes, this is Delta. Yes. Um, Delta DS is simply one over t plus one plus one d t thies to cancel. So you get DT So Delta DS is that the same thing as DT? So we have tgt all right? And from zero to to have you crank this out, you get obviously to all right, so that's the first integral the second integral. This is going to be to route to over three. Do you? The power of 3/2 D T from 0 to 2. And if you crank this out, you get 32/15 and that's the second integral. What about Z D M? That's that's equal to integral of integral of teeth. Guard over to halted. Yes. And you crank this out, you get for over three. So, the center of mass, we have finally completed all three integral Well, four under girls, actually, because they're still the total mass, which is also been integral. Um, so the center of mass can be written as one over M um, integral xdm integral Y d m into Rosie G. M. And we have all three numbers. So we're just going to, um, divide up and divide them out by the total mass. So you get one and 16/15 and 3/3. All right, so that's the center of mass. It's a vector. Um, and we're done for for the center of mass. Let's start looking for the moments of inertia. So I x So that's go ahead and start looking for the X component First I axed can be written as integral Delta Our scored minus X squared DS, which is well, Delta DS, as we said, is just DT So we have y squared plus Z squared 80 from 0 to 2. Um, we have an expression for for why end for it? See, in terms of tea, why is simply to route to you over three t to the power of 3/2 Z ISS employees t squared over two. And if you square them, you plug them in here and each we do the integral you're going to get 2 30 to over 45. Okay, so that's I X. Why? Why is very similar? We have delta R squared minus y squared. Notice the similarity here instead of instead of executed we have y skirt. Anyway, we have multiplied by DS Delta. The yes is the same thing as DT. And so we have X squared plus C spurred E, t and X as a function of t is t and a few square it and you put them in and you do the integral from 0 to 2, you're going to get a number 64 over 15 there's I Why and now that's too easy. I z is very similar. They're all very similar. Delta R squared, minus disease word. Yes, and I want you to know that the difference here anyway so dealt with es again is just DT. So now R squared minus C squared is simply expert plus y squared. We have an expression for X here in terms of tea and we have an expression for why here in terms of t Now, if you square them, you add them up and you plant them in here multiplied by D t. And you do the integral from zero to you're going to get 56 overnight. You know, it's very it's actually very simple. This is not a This is not a competent. This is not a conceptually difficult problem. It just requires a lot of computations, which I have skipped over because of time constraints for the video. Anyway, I hope enjoy the video. I will see you in the next video