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Question (4 Points)Given F(x y) = Zxyi 2y)j. The work done by the force in moving particle from (0.0) to ( 1. 2) along the straight line joining the tWo points iS:b...

Question

Question (4 Points)Given F(x y) = Zxyi 2y)j. The work done by the force in moving particle from (0.0) to ( 1. 2) along the straight line joining the tWo points iS:b) %d) ,0 ;

Question (4 Points) Given F(x y) = Zxyi 2y)j. The work done by the force in moving particle from (0.0) to ( 1. 2) along the straight line joining the tWo points iS: b) % d) , 0 ;



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 $$ \textbf{F}(x, y, z) = z^2 \, \textbf{i} + 2xy \, \textbf{j} + 4y^2 \, \textbf{k} $$ Find the work done.

Here we fuss. Plot the four sources Displacement forces on y axis and displacement on X axis. Here we have. If Lord, this is zero. And here we have minus F. Lord, this is explored and this is two times off. Ex Lord, if you plot then this goes The line goes from picks mod and meet If Lord here given a one dimensional force therefore fix effects which is a function of X The world is simply equal to the area under the curve. So what will be called to integral off Ex Initial two x file force dx for about a The port is shown here we take explored to be a positive and the world done. He's inactive. There's the object moves from ex physical 202 x Is it going to explode? And the world is a positive as it moves from X physical Do it smart two x is equal to two x amount So this is negative here and here. It's a positive since the 80 off a triangle we know any offer trying will be No, he's the one or two times a based I'm altitude. Then the work done from X more acceptable to 02 X is equal to X. Marred is doubly. One is equal to minus X Lord times if no divided by two and the world done from X p Simple to fix Nod two X is equal to to explode is given by W to that easy Cool too well, um two ex lord minus X terms if f more divided by two. This gives us explored F Lord divided by two. The total work is then some off to So not who is one plus w two on this gives us minus one over to f more explored plus one or two explored f mort to be zero but be the integral for the work Done. Who can be written is a w integration from 0 to 2 Ex Lord, This is f into the function we have If more x divided by explored minus one d x are integrating this we get if more times x squared, divided by two ex Lord minus X integrating from zero to to explode this user stance her to be zero

Everyone today we're going to solve problems off 14 months. You're functioning X. Come on by equal explain. Come up. It's a square. See which is executive? I square from zero comma zero Want a long one are equal to okay Learn minus T which is less than zero. Why equal to the which is execute o T square zero Less than a record today Less than record the one via dice equal to one X dash equal toe pick So w vehicle to, if not tr equal to in the wealthy, wrote one The Cube coma theorist Before don't toe the Ghanem are in the rose 0 to 1 to the rest of four, plus the rest of four Duty which is equal to integral 0 to 1, three tiers to four duty three by four Terrorist if I from 0 to 1 which is equal to three by five Thank you

Whereas to find the work done on a particle that moves along line segments from the origin the points 100 1 to 1 0 to 1 and back to the origin under the influence of a force field f where f is the force field Z squared I plus two x y j plus four y squared. OK, so we'll use Stokes is here, Um, rather than computing work directly. That's why this is in this section. So we'll let s the the planer region, which is enclosed by the path of the particle. Now this is a plainer region because there's only four different points here and do all lie in a plane. So, in fact, they lined the plane Z equals one half. Why, Therefore s is the portion of the plane c equals one half why for X between zero and one because that's the largest tax ever gets. And then also we have y lies between zero and two because it's the smallest and largest y ever gets. And see that moves from the origin 2100 2121 20 to 1 and back to the origin from above that's a counterclockwise orientation. And so our surface this induces an upward orientation for our surface. Now the curl of our force field F this is eight. Why I plus two z j plus to why, okay. And therefore by Stokes theorem the line integral to the work done Which is the line integral over the curve C of f This is the same as the surface integral over s of the curl of f and this is the same as the double integral over the region D in the X Y plane of And because our plane is a function of X and y, you can use a shortcut. So we have the opposite, uh, the first component of our force field of the curl of our force field. Just negative eight y times the partial derivative of Z are playing with respect to X, which is zero minus thes. Second components of the curl of our force field to Z times partial derivative Z are playing with respect to why which is one half plus the third component of the curl, which is to y d a. So simplifying and plugging in Z in terms of X and y we get iterated into growth from 01 integral from zero to, um becomes two y minus. And this is one half. Why de y dx? It's really this is just three halves. Why eso taking the anti derivative with respect toe? Why beget three hats time integral from 0 to 1 of one half y squared from 0 to 2 DX evaluating We get integral from 01 times three d x, which is just three So using Stokes there and we found that the work done was just three.

So I guess in reality a power you have to use a computer or calculator to you and I'LL compute the first part. So once around the circle culture class wise so kind of standard Parametric ation Mexico's to co sign T Boy hose to society and the tea goes wrong. The other two pi So what is F or far off? T? We're simply replaced x by to society And why by two scientists Second power should be fourth scientific all scientists in our prime iss negative to sigh Inti to co sign t no so integral integral from zero to two pints a dot product of these two, it's negative. Eight sci fi coz I scurti plus a sai inti Cose I squirt ti So you find if I didn't make any mistake their teacher this who term looks like they're canceling. Therefore we should get zero Well, let me tell you actually, if I compute anything wrong ex Chris for cause I score thirteen x wise for scientific society The ribbed him is Neck deep Sigh and cose I That's right. Then we takes Ah, we take a shot for that So we're negative a psycho's I square eight cycles I sparkle. It looks like they're indeed can sew, sew, sew This this inter cross here


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