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Problem 36 Easy Difficulty

If a wire with linear density $\rho(x, y, z)$ lies along a space curve $C,$ its moments of inertia about the $x-y-$ and $z$ -axes are defined as
$$I_{x}=\int_{C}\left(y^{2}+z^{2}\right) \rho(x, y, z) d s$$
$$I_{y}=\int_{C}\left(x^{2}+z^{2}\right) \rho(x, y, z) d s$$
$$I_{z}=\int_{C}\left(x^{2}+y^{2}\right) \rho(x, y, z) d s$$
Find the moments of inertia for the wire in Exercise 33.

Answer

$I_{x}=I_{y}=4 \sqrt{13} k \pi\left(1+6 \pi^{2}\right)$
$I_{z}=8 \sqrt{13} k \pi$

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Video Transcript

Okay. So we're trying to find the moments of inertia Puerto Vallarta in exercise 35. Okay. So um from exercise 35 it's given that the density is some constant K. And cough is given by the part metric equation. L. T. Equals two sovereignty. And then to Carson ot three T. From 202 T. Calls to pipe. Okay, so now the the equation. Okay, let's mark this as one. I'm going to find the X. Component of the moments of inertia. So I said Becks this is given by the integral along the curve, Y squared plus Z square times the density. I saw it. We know that the density it's just a constant K. And then parametric equation L. T. Equals to sign of the to co sign of T. Three T. So this means that y square plus the square equals four times casinos quality plus nine T. Squad. Okay, we just need one more thing. Ds. This equals the link of our priority D. T. So from this we can find the value of the moments of inertia. Okay So let's find our primary tea. Which he calls two times. Of course I don? T and then negative too sovereignty. I'm three. So the length of our prime duty. This he calls four times course sign of square T. Plus four times science quality plus nine which he calls squared 13. Okay. So now we're just plugging these expressions into the formula of the nasha. The low elements of the integral. It's given us zero and the upper limit That is two pi. So I We go from 0 to 2 pi and then Y squared plus Z squared. This is four times. Of course I know square T plus 90 square. And then the density function this is given by K. And Ds. Is the length of our priority. DT soy square 13 D. T. Okay. So we're just gonna fuck her out. The constant here and then use the seniority of the integral plus nine 10 to Pi T square DT. Now for the integral of course china's quality. We need to use a technique called the reduction formula. So the reduction formula for the call sign for the integration of the call sign of call sign to N. O. T. From A to B. This is given by of course in a penalty D. T. Because And -10. Brien integral from A. To B. Call sign of em minus two T. D. T. Plus one over N. Times. Co sign of N -1. Opti times sine of T. From A. To B. Mhm. Okay so now we have N. S. Too. So we're just gonna plug in these um these numbers there and then we get ice affects because K. Times Square Bridge. 13 times for Times 1/2 From 0 to Pi won t t Plus 1/2 course. I know t sign of T from 0- two Pi. And then we have the second term as this. Sorry we're just gonna talk that even as well. You see cause three times nine times T square T. To the third three From 0 to 2 pi. Which he calls, he says four times One of the two. I'm stoop I plus zero plus three times two pi to the fed. So The final value is K. Square three times four pi plus three times 2x2/3. Yeah. Okay. So that's for the X. Component. And then we need to find the Y. Component which is given by integral along the cops. See X squared plus Z square times the density. Yes. Okay. So now we're just gonna plug that the expression in and then we get 0-2 pi for signs square of T. Plus. Now in T square times K square 13 D. T. Which he calls K. Times Square 13 times four times The integral from 0 to Pi science square of T. D. T. Plus nine times seem to go from 0 to 2 pi T square D. T. Okay, so for the first time of the integral uh we're again going to use a reduction formula for saint and Okay, so in the world from A to B. Or sign to N. O. P. D. T. Yeah, This can be found by the formula and -1. Urban integral from A to B spine, N minus two. Opti DT -1 over. In times co sign of T times sine To end -1 lT from A to B. Okay so we're just gonna plug the formula in and then we get I said y equals K. time squared 13 times four times 1 over. two Integral from 0 to Pi one DT -1/2 time is of course I know t sign of T. From zero to pi Plus nine times T to the 3rd over three. From 0- two Pi. Now now we're just gonna integrate the simplified expression and then get four times one of two times two pi minus zero plus three times soup. I to the third Which he calls K. Times Square 13 times four pi plus three times two Pi to the 3rd. And then that will be our what components of the moments of inertia. So the last one, the last component um The formula for the Z component is X square plus y square times roll X Y. Z. Yes. So we're just gonna again plug the expression in And then get 0 to buy. Times full time scary. Times square. It's 13 D. T. Which he goes 4K. Times square 13 times to buy. She calls eight times school 13 times K pi.