7.( point) Consider a wire in the shape of a helix rlt) = 4costi + 4sintj + 7tk,0 < t < 21 with constant density function Plx,%z) = 1 A, Determine the mass of...

Find the mass and center of mass of a wire in the shape of the helix $ x = t $, $ y = \cos t $, $ z = \sin t $, $ 0 \leqslant t \leqslant 2\pi $, if the density at any point is equal to the square of the distance from the origin.

In this problem of vector calculus we have to find the mass of a wire in the space. And we have given that the thin white is in the space of a helix. See Such that X. is equal to costing. Why is equals 2-70 and That is equal to 40. No Where T. is the value from 0 to 2 pi. Who's very well mass density is proportional to the square of distance from the origin. That means density is proportional to the square of distance which is X squared plus y squared plus 30 square. From here we say that density is K times X squared plus y squared plus Z squared. Moss is given us density. It's a function of X. Y. Z multiplied with Diaz allowing the currency. Now we have to find the value of the so dss is differentiation of two Because it equals to two signs. This is -2. Sinus squaring this term. So this way they will be for a sinus square tape plus definition of this to 70 is equal to to cost is squaring. So this is Four courses quality different section of 40 is full so this is 40 square which is 16 and This is five to not set 40. So this is five T. So this way they will be 25. Now this is DT So when we added so sinus squared plus courses square T. Is equal to one. So four plus 25 equals 29. So the S. A. G equals two under rule 29 80. Moss is here. Rule of exquisite escape multiply X squared plus Y squared plus B squared. So X squared plus Y. Is squared is equal to food And that is where it equals to five T Is square which is 25 T sq nowadays equals two. Underwrote 29 80. From here we can right constant out of this integration which is K under 29 integration. Now we have to put the limit Of integration that is from 0-2 pi. So we have to put 0-2 pi here. 0 to to buy here. Integration of four plus 25 day square DT. So this is okay and the route 29 Now integration is 40 plus 25 PQ divided with three. Now Upper LTD to buy lower limited zero when you put upper limit. So they said okay and the road 29 No four multiplied with two pies on the stage 85 plus 25 multiplied with this is two pi cube which is eight by This is to buy whole cube which is eight by cubes. 25 multiply eight days 200 by Cube divided with three which is the right answer from us. Yeah

In this problem of vector calculus we have given that the thin wire is in the shape of a helix. See Such that X. is equal to two costing And why is equals 2-70 and no 30 is equals 25 T. Now we have to also we have given that is wearing from 0 to 50 0 to 2 by actually 0 to 2 by we have to find the mass of a wire in space whose mass density rule of X. Y. And zed is given by K. Which is a constant. So most of a 10 1 in a species given way dissident city, a function of X. Y. Zed multiplied with days along the currency. First we have to find the will of DS. So DS is calculated as dx divided with duty. Holy square plus D. Y divided with DT. Holy square plus dessert divided with DT whole square multiplied with DT so the extra But we did it. So this way we will be Two scientists squaring this term. So this really would be under root of four. Science is quality with minus sign. So this will become a positive sign. Definition of sinus costs. So squaring this term. So this time will be full courses square T. Now plus differentiation of five. T. Is five. So this value is square. Again when we solve it, this is equals to 25 plus four which is equal to 29 Because Sinus squared plus cosine squared is equal to one. So food is common, four plus 25 equals 29. So this is 29. DT Role of exercise. There is a constant, so moss is equals two. Okay, the S equals to 29 29 80 and he's wearing from 0 to 2 pi so this is 0 to 2 pi. Integration of this term. Music close to Okay under route 2090, so this is D. And the winner is from 0 to 2 pi putting the par value. So this is really weird way to pick Under root of 29 when he put zero. So this term is zero. So we have the right answer is to pick a and the route 29 is the right answer.

Job or has the co ordinates t to swerve through over three t three halfs and half of STI square 40 from zero to chew. Oh, it calculates the module off the derivative. We're gonna have one plus team. So there's the mess. Is Dona the S L C which is for the valley in there too? One d t. So, which is to explore is and was he over M So 1/2 terms x don't B s. We'll see which is equals to one. And why Boer is why x z over and 1/2 times Why did he s c when she's too thers? Z bore is, um that's why over now equals to one have tons Zito, the S and C, which is 230 to 45. So that's the sense of mass. And for the movements of inertia, we have ice of X equals two x squared. Sorry. Westward plus thesis Weird, don't it? Yes. Which is given by, um, the answers, but you're was three and, uh, ice. The why is extorted plus thes where DVDs which is you close to 60 64. 15. And I said xe is ex worthless voice word. The other the s to the sixth Woman night recall over Sealift Mass is one succeed or team and to over three. That's it.

In this problem of vector calculus we have given that the thin wire is in the shape of a hell X. Such that X. is equal to three costing And why is equal to 370 And that is equals to 40. Where T. is wearing from 0 to 2 pi. We have to find the mass of 10 wives whose variable master state age. The role of X. Y. Z equals two. two plus it so much is given us integration of rule X. Y. Said. Ds along the car, lucius. See first we have to find a way of this. So this is calculated as dx divided with DT Holy Square plus. Dy divided with DT Holy square plus dessert divided with DT Holy square multiplied with DT. So D. S. DX When we differentiate. So this way we will be -3 signs. So squaring this term. So that's where they would be nine. So many square T. When we different state three signs of this while you were three costs is squaring this term. So this is nine crosses square D plus. The vision of 40 is four is squaring this time is 16 and Gt sinus squared plus cosine squared is equal to one. So we have nine plus 16 is equal to 25. Under 25 equals +25. So this is five DT No he is wearing from 0 to 2 pi. So we have to put the limit of T from 0 to 2 pi row of X. Y. Z. Two plus said and that is equal to 40. So this is two plus 40. The S E. Equals 25, multiplied with DT. Now we have to integrate this term so M is equal to Integration of 10 is equal to 20. So this can be written as here Message equals to 20. Putting the limits from 0 to 2 pi plus. Integration of 20 T is equal to 20, multiplied with the square divide with two, putting the upper limits and lower limit. No, when we solve it, the city equals two. This way it would be 25. So this is 25 0. And here this really would be putting the value to so that's where they were to square. That is four pi squared divide with two is equals two. He had 40. This is 40 by square and this is the right answer. Mhm. Mhm