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Problem 47 Hard Difficulty

(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle $x^{2}+y^{2}=1 .$
(b) Is this also true for a force field $\mathbf{F}(\mathbf{x})=k \mathbf{x},$ where $k$ is a constant and $\mathbf{x}=\langle x, y\rangle ?$

Answer

a. $W=0$
b. $W=0$

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Video Transcript

this'll questions were asked to show that the work done by a constant force field wants people to see on the path that we were given. Is that a circle of radius one? So we have X squared. Uh, that's why I swear it is well to one. And we know to pure amateur eyes circle. So we just said x equal to co sign T like will sign t and he goes from 0 to 5 and then our are It's just society off site tea and then our prime It's negative sign t common cause so we could stick the derivative off CO sign T. All right now And now what? We're told that West units that are forced feel this cost. So we're gonna set our force field after two equal income and just constants were eman n are constants of CN STS, right. So now we know that the work is the integral along the curve of f dot are. And then now what we're gonna do is we're gonna write everything in terms of cheap. So that's where the work will simply be the integral from 0 to 5. Well, I m call much of the vector and public end, uh, about product with the vector Negative Sign Club coastline. So this is just the product, and what we get is the integral for you're a to buy a negative M Sigh Inti Bus and Co sci fi. The M ISS constant the MND and are constants so we can pull them to the outside. And the integral scientist is simply negative co sign teeth. But remember, there's a negative in front of the magic schools away, and the Integral co signed you've just signed. All right, so now we're gonna plug in to buy and then subtract, Subtract from that the value at zero. So this is an who sign to Pai minus co signs you're also co cited by is just one no sign zeros. Also one bus and sign of two pi zero sign of 00 So we get em. Times one minus one. She just zero an an time zero minus. Here with this is against you. So the work done by a constant force field is zero. So we showed that now part B were given of the force field. This k Times X were X is a vector field given by Ex cover life. So now what we do is we just, uh, multiply like a So our vector field becomes Horsfield. I'm sorry becomes k X, come off the line. And then what we get is we know from our privatisation about that X is just close I inti and why you signed. All right, now we're asked to find the work, and we have to show that it's is you. So you know that the work is just the integral along the curb See off at stock d are another way to write that is the integral from 0 to 2. Yes, I'm 0 to 2 pi at which is x co sign D comma. That's our case goes I e chronic A sci fi Don't our prime tea that they are crime. We also found that above have been part is just negative Sign t close I, Inti and then DT So now we're gonna find adult Frana. So that's just negative, K. So we're just gonna multiply this first term this first turn and then after that, this second term times this second. So we get negative co scientist negative Keiko site. He's fine teeth was scarce. Scientific o society. So these to cancel we're just left with zero. So again now we showed that the work for this new force field, which is Ah, Kate, Times X, where X is just, uh the vector X cover line is also equal to tears of the work in Syria.

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