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4 (3 marks). Consider the force F(I v) field F(r,y) = rli+y8j. Find the moving particle along work done by any path from the point (1,0) to the point (2,...

Question

4 (3 marks). Consider the force F(I v) field F(r,y) = rli+y8j. Find the moving particle along work done by any path from the point (1,0) to the point (2,

4 (3 marks). Consider the force F(I v) field F(r,y) = rli+y8j. Find the moving particle along work done by any path from the point (1,0) to the point (2,



Answers

A particle moves along line segments from the origin to the
points $(1,0,0),(1,2,1),(0,2,1),$ and back to the origin
under the influence of the force field
$$\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+2 x y \mathbf{j}+4 y^{2} \mathbf{k}$$
Find the work done.

Our problem in 77 using green serum. So using green serum get the following in a girl simplifying, then searching over the polar gonna have three double integral far square D A. I'm plugging in our limits. We get three Well from Georgia Pie there too, if our cubed DRD theater and this comes out to 12 pie.

In this problem of line integral. We have to find the work done by force field F on a particle moving along the given path. So here is part I'm showing you. So this is the path we are moving from say this age .102. This is 01. Now force F. Force field F is given as F of X. Y is equal to X. Is where I minus X. Y. G. And we have given that the currency is such that X is equal to cause Q. T. And why is equals two. Sign Q. T. And we are moving from 1-0. No from here we say that artie artie would be equal to the stage because Q. T. I plus find cube T. J. Now we have to calculate DDR. So this is the art would be differentiation of course cube is three course is quality and the position of course is minus signs that say minus three courses square T. Scientific I this is scientific. I'd and deposition of science Cube is equal to three sign square T. And cost T. J. From here we have X is equal to Costco and Y is equal to send you so function of X. Y will become a function of T. If we put X equals to cost you. So that's where they will be Cost to the Power 60 I minus cause cube and signing cube. So that's it cause QP multiplied with sign QP and this world they would be multiplied with G. Now we have to evaluate F dot dear for finding we work. So sign This is your cost to the power 60 I. Is multiplied with this term. So this is equal to minus three. Close to the power aid. So this is Close to the power eight and Scientific I. And now this is multiplied with sign. There's a cost you. So this would become -3 Cost to the power 40 And signed to the Power five T. So this assigned to the Power five T. And here we have eliminated iron data. So this is multiplied with DT And now we can take comments say and value of T. T. Is wearing from zero to PI divided with two. So from here we see that value of T. Is wearing from zero to pi divided with two. So this is from 0 to By the way with two. Now we are taking -3 close to the powerful message T. And scientists as common. So we would have here Cost to the power 40 plus signed to the Power 40 as common. Now when we again take some calculation we are making this term signed to the power 40. So this term can we return as this is three costs to the power 40 sci fi And This is caused to the Power 40. And sign this can be written as sign is quality science security is one minus causes square T. So this age plus 1- Courses Square The Holy Square. And here this value is DT multiplied with the city. and then again integration from 0 to Pi divided with two Here also integration from 0 to Pi divided too. So this is work no minus three cost to the powerful T. Scientific when we opened this square. So that's where there would be two courses square to cost to the powerful actually. So this is to cost to the powerful d minus two cause a square T plus one. And here DT now when we simplify it. So they said equals two Work done as equals to integration from limit 0 to Pi divide with two. This is -3 multiplied with two. So this is six. Cause to the power eight city and Scientific now three calls to the powerful multiplied with minus two. So which is equals two plus six Cost to the Power 60 70 plus listen one multiplied with ministry. So this term it's -3 cost to the Power 40. Science baby. And multiplied with DT. Now we have to integrate it simply. So integration of six cost to the power eight design. Tea is simply here cost to the minus cost to the power nine divided with nine. So this it equals to two divided with three Cost to the Power 90. And hear this term is with minor science. So we have to write it with a minus sign. This is -6. No Integration of six cost to the powers six multiple. Every science. So they say -6 divided with seven. Cost to the power 70 plus integration of these terminals. Three to the power five cost to the power five t. Now this is equals to work. We have to put the limit zero to pi the head with two. Now when we put the upper limit so caused by is equal to one and corr zero is equal to zero here. Caused by developing two is equal to zero and cause zero is equal to one. So we have When you put up our values of this term is 000 and minus when we evaluate this term so This is equals two. to divide with three -6, divide with seven Plus three, divide with five is equals 2-. That's where they would be 43 -43 divided with 105 years. The right answer.

So in this video to the work come by it no field on the particle so that I could movement from 20 Teoh negative to zero along the curve. Why is he was this for four minus and from negative to a spirited to zero along the line six. So the first thing we know is that the work done by the forced her long before the bullet. Well, to, um, green stare, which is the double integral over the domain T dick you, by Jack's finest people devoid. Okay, No, um, market has stalled this in full accordance. So first thing we notice is that the radius schools to phone zero to and the angle goes from Syria. And now D Q by D. X just derivative execute plus three x voice square Perspective X, Just three Expert bus three Life sweat and the people I devised a secure native of X with respect. Why, which is just cereal? Because there is no time Sparky or di Sita. We can pull it three out as a common factor. And then we're gonna be left with X squared, plus y stubborn. But we know that X script was spared sparse First was just three x squared times are the rd Veda. So that's just gonna be re are cute. Your times NATO The interval of that is three part of power for divided by four. And the remit of integration is from 0 to 2. So we're gonna looking to win zero week it 12 minus zero, just 12 and now that's a constant. So the weaken politically outside any interval of the haters just stated and that, uh, the lives of integration It's poems you wrote a play for export those your right, If I know again, we're going to subtract 12 times by minus zero, which is simply 12 pie. If we go back to our graph, we know what is a business counterclockwise. So there is no need for the orientation its counterpart by So there's no need to change the sign. So the work done by the field in the room, the particle things to

In this problem of line integral. We have to find the work done by force field F of X. Y is equals two minus light. I minus X. City for the given figure I'm showing you. So this is a required figure. And we are moving from the point this is 20. So 202. We are moving to -2 and zero via this curve. And the cover is such that They said why is equals two. We can say Why is equal to under root of four -X square. and we are moving from 20 to -20. So first we have to parameter is this term and we are moving from 20: -2 and zero. Now we have two Parameters. So arty would be such that we can make this term as X squared plus Y. Strategy goes to fold this term is can be written like this. So artie would be to cost T. I Plus two scientific plus to sign D. J. Where value of T. Is from 0 to 0 to pi. So this is from zero to pi. Thank you when we differentiate it. So this is our Dashti different vision of course is minus signs. So they said minus two scientific I. And differentiation of sinus calls. So this is to costea. Yeah and now we have F for finding the work. We can say F. F. T. In terms of death X is equal to -2 signs here -X. Is equal to cost actually. And why is Equals to two scientists. So from here we say that this is minus to sign so minus to sign T. I and minus took off so this value is -2 cost e.g we have F and we have our so for finding the work we want F dot dear. From the limits zero to pi we have the limit of T. So doing the dot product F dot dear. So From here we see that this is equals 2 -2 and multiply this -2 sons. So this really would be for Sinus Square. So this is four Sinus quality and minus two causes multiple ways to cost so which is minus four. Close the square. So this is minus for courses square T and we have integration from the limit zero to buy and this is with the T. Now when you solve it for Sinus Square T -4 courses where it is simply equals to four can be written out of this integration and we have Sinus square t minus clauses square T DT An integration limit from 0 to Pi. Sinus waiting minus crosses square is simply cost to T. So this is simply yeah, This way we will be minus of course to T. So this can return -4. Integration from 0 to Pi and cause to T DT integration of course to duty is simply minus is here. So Integration of course is signed. So this is -4, multiplied with Scientific and divided with two. Also We have to put the limit from 0 to Pi. Now when we solve it this is minus sign duty And they say multiply but to also so this is -2 signed duty. And we have limit from 0 to Pi. When we put upper limit sign this really would be to pie, so sign two by zero and signed by 70 is also zero, so this value is equal to you zero. So we have the right answer at zero.


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