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Consider a particle on which a force acts that depends on the position of the particle. This force is given by $\overrightarrow{\mathbf{F}}_{1}=(2 y) \hat{\mathbf{i}}+(3 x) \hat{\mathbf{j}} .$ Find the work done by this force

when the particle moves from the origin to a point 5 meters to the right on the $x$ -axis.

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Alrighty. So there's a particle here at my origin of this system, and it's gonna be under the influence of ah, the's forces here that, interestingly enough, depend on the position of the particle. What does this mean? Well, it means that Sarah Particle was at same position for three on our corner Naxi system. You're gonna wanna plug in four here for the Actually you're 12 Jay, you know, a plug free, and for the way, you're gonna have six. I see if six sites will jay and so the force basically changes depending on your position. It's pretty interesting. Now, the trick question here are tricky. Part of the question I should say is that you move from you move five meters to the right on the X axis. So we go here when we go here five years to the right. So we're displacement. Vector is going to be equal to five meters in the eye direction and there's zero meters in the direction we don't go anywhere on the Y. Axis is all strictly on the X direction. So we were defined. The work that this force accomplishes, we can go ahead usar work dot product equation and roll it this year dotted with our five meters in the eye direction, plus our zero meters in the J direction. Close this down. Did you the dot product number You take your to wine multiplied by the five meters. I should also mention that really didn't leave that Newton's sitting next to these guys. Just see you guys know anyway, nonetheless, five times to why? So we get 10. Why in the eye direction is equal to work and then you're gonna do you're three x times your tons your zero three x time zero yet plus zero uh, for the J direction plus zero. Yes, based lead with you. Only if you include that there. So the work and remember actually actually mentioned one of the really important thing You're vectors. You know, veterans go away when you're talking about, um, work works, not a vector. Cuz you never actually news publication. All that's left is to 10. Why? Here's the tricky part. We don't go on to the Y axis at all. Why is equal to zero? We go from 0 to 5 in a straight line and we never touch anything on the Y axis Because of that. This is 10 times zero and the work done is equal to zero jewels

University of Alabama at Birmingham