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Problem 37 Hard Difficulty

Work by a constant force Show that the work done by a constant force field $\mathbf{F}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$ in moving a particle along any path from $A$ to $B$ is $W=\mathbf{F} \cdot \overrightarrow{A B} .$

Answer

$f_{B}-f_{A}=F \cdot \overrightarrow{A B}$

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Video Transcript

Okay. This is from 37 answers to show that the work done by a constant force for U F, which is a B a from moving apart. Okay, is job You work after dark? A B So the star of zealous named Victor wasn't a is X y z in that and point B let's name it is X B. Why be Z a p? All right now, first off the f ah a I plus B j plus c k Ah, you're m and your Emmys A. And when you do partial, these would any respect X y or z doesn't matter because it doesn't contain in that they own a partial is actually zero because there's not contain any X one z in it. Therefore, they must be equal. So you're f is is that which means that you have a conservative view and wouldn't you know what you need if I find the potential? Yeah, potential little f all right. Because you're f the in the grove have that we are the work of this is that a to B is going to be a little f a B minus half obey. All right, where f little efforts, The potential bigger. All right. So let's go ahead and find the little effort so well, because this section partial f would respect to X going to be a therefore f was going to be a X plus some Z uh, Y and Z, and then you do partial f would respect that, too. Why you get this is their older plus partial G with respecting why and then you compare it to get that sexually bees. Therefore, you're g going to B B y plus some h Z and we're doing partial f would respect to Z. That's just ish Problem Z and we know that it's C by P compared. So Trixie going to be C Z uh, plus a constant, of course. So what's your little elf? Little half going to be a X plus B y We'll see. Easy plus, um therefore effort be minus f a Just work. Just going to be a oh pees on X B. Close be Tim. Why be we'll see them zero b minus a tim, except a plus Be tim. Why, eh, we'll see Tim Z o a. And ah, constancy. Cancel away. In fact, you can uh, factor. L A. So next be minus X A plus B Thames. Why? B minus y es close. See tam's E A B minus zero A and which is a little, uh, this is exactly the same. Yes. Um, Graddy in of as, uh, sorry. No grand in here. Grounded. Yeah, grabbing their little f dot with the vector a b in the crowded and a little fsp caf that a b because a b his what vector Where you write it in that's gonna expiate my sex a white B minus way and Z b minor z a And then the big average Just gonna be a B c. When you've got these factors you get that's product and that's it. See you in the next video.

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