To extricate an SUV stuck in the mud, workmen use three horizonta...

Question

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E4.2. (a) Find the x- and y-components of each of the three pulls.

Accepted Answer

everyone in this problem. We have three archaeologists who are pulling an old relic buried in the ground using three horizontal ropes. Okay so we have an image below that shows the applied vector forces and we want to determine the X. And Y. Components for each of the three poles. Okay so we have put a. Okay here in red. We have pulled B. I'm gonna circle in blue and we have pull see I'm gonna circle here in green. So starting with pull a. Let's go ahead and draw this out. Mhm. So we have pull a here which is 685 newtons applied. Okay then we have the components. Okay, so this is going to be the Y. Component of A. Along the vertical and then along the horizontal is the X. Component of A. Okay now you'll notice the way we drew this triangle was in relation to the X axis. Okay so our um angle that we were given is relation to the Y axis. So this is going to be 90 minus the angle we were getting. So this is going to be 36 degrees. And you could draw it the other way. You don't have to draw it this way. You could draw it the other way. But I like I'm going to keep everything in relation to the X. Axis when we're drawing these just so you can see um the relationship between. Alright so that is our pull A. Okay now let's go ahead and find those components A. X. And A. Y. And you'll notice here A is in the first quadrant. Okay, so the X and the Y components are both positive. Okay, so when we're looking at A X. Well this is going to be equal to A.X. is on the adjacent side to our angle. So we're gonna be related through coast. We're gonna have cosign of our angle 36° eight times the hypotenuse which is 685 newtons. And this is going to give us an X. A. X. Component of 554. newtons. Similarly for why we get the Y component and why is on the opposite side of our angle. So this is going to be related through sign. It's a sign of 36 degrees times our hypotenuse. 685 newtons is going to give us a Y component of A of 402 0. Nunes. Okay. Alright. So we're done with A. Now we can go ahead and do B and C. Let's move down. So we have some more room to work for both B and C. So we're gonna move down and I'm just going to redraw B for us. And what color did we make? Baby is going to be blue. Alright, so let's re drop B And B was a vector given like this. Okay. In quadrant two it has some white component and some X component with a magnitude of 784 newtons. And the angle to the negative X. Access we were given is 44 degrees. So again, the horizontal, it's going to be the X. Component of B. And the vertical is going to be the Y component of B. Alright, so let's do the same thing again. We're looking for B X. Well, the X component is on the adjacent side so we're going to relate their coasts. So we're gonna have cosign of 44 degrees times the hypotenuse. 784 newtons. Okay. And let's think about this. We're in quadrant two. Okay, so our X component is going to the left so it's actually going to be negative. Alright, great. This is going to give us working this out an x component of B of negative. 564.0 Newtons. Okay. And doing the same for the Y component, B. Y. This is on the opposite side. So we get instead of Kassian were relating through sign sign of 44 degrees times the hypotenuse. 784 newtons. Okay. And I left the space for the negative. Do we need the negative? Let's think about this. We're talking about the Y component in this case. We're in quadrant two. So why is positive. Okay. So are Y component is upwards? We don't need the negative. Alright, I'm working this out. We're gonna have 544.6 newtons. Alright, we're done A We're done be all we have left is C. Okay. And see again. Let's draw this in relation to the X axis. So we have the vector C. Like this With a magnitude of newtons. We're giving the angle in relation to the Y axis. Okay, Well If we take it that and subtract it from 90, we're going to get the angle here related to the X-axis. So we're gonna have 58°. Then we have C. X. Along the horizontal and see why along the vertical, the X and Y components respectively of our pulse. E. All right. So let's consider C. X. Let's talk about sign. First C is in the third quadrant. So both X and Y are negative here. Okay, So our X component is going to the left. So we're gonna have negative, Hey, we're on the adjacent side. So we're relating through coasts degrees Times The Hypotenuse. 469 Newtons. Okay. This is gonna work out to -248. Newtons. Alright. And for see why again, third quadrant Y is going to be negative as well. The Y component of our poll is going down. So we have negative were on the opposite side. So we're relating through sign a 58 degrees times the hypotenuse the magnitude of that pole. 469 newtons. And this is going to give us a Y. Component of C of negative. 397.7 newtons. And so you'll see here the way we've set it up every time we're talking about an X. Component, we're using coast and every time we're talking about a Y component, we're using sign. And that's why I made all of those triangles. So that were related to the X axis when we're talking about the angle so that we were consistent. So you could see that pattern but you don't need to do that. Like I said, okay, you could draw them in relation to the Y axis. You just have to be careful with which one you're doing and paying attention to whether you should be using sign or coast there. Okay, Alright, So let's go back up now that we have everything and we'll see that the final answer that we have found is going to be F. Okay, Force of A. In the X direction is going to be 554.2 newtons. And in the Y direction it's 402.6 Newtons recall it. The I in the J I had is the unit vector in the X direction, and J hat is the unit vector in the Y direction for B. We have negative 564 newtons in the X direction and 544.6 newtons in the Y direction and four C. We have negative 248.5 newtons in the X direction And negative. 397.7 Newtons in the Y. Direction. Thanks everyone for watching. I hope this video helped see you in the next one.

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E4.2. (a) Find the x- and y-components of each of the three pulls.

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