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