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$\bullet$ Three horizontal ropes are attached to a boulder and produce the pulls shown in Figure $1.25 .$ (a) Find the $x$ and $y$ components of each pull. (b) Find the components of the resultant of the three pulls. (c) Find the magnitude and direction (the counterclockwise angle with the $+x$ axis ) of the resultant pull. (d) Sketch a clear graphical sum to check your answer in part (c).

a) $A_{x}=86.6 \mathrm{N}$$A_{y}=50 \mathrm{N}$$B_{x}=-40 \mathrm{N}$$B_{y}=69.28 \mathrm{N}$$C_{x}=-24.08 \mathrm{N}$$C_{y}=-31.96 \mathrm{N}$b) $R_{x}=22.52 \mathrm{N}$$R_{y}=87.32 \mathrm{N}$

Physics 101 Mechanics

Chapter 1

Models, Measurements, and Vectors

Physics Basics

University of Michigan - Ann Arbor

Hope College

University of Sheffield

University of Winnipeg

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this's Chapter one problem. 48. What So in this problem, were given three vectors, each with a magnitude and direction, and were asked to find the X and Y components. Look into these vectors, essentially, so to do that, since you got magnitude and direction, we're going to use these equations. Ex component equals I'm al itude times the co sign of the angle, and this is where the angle is measured. Clock counterclockwise from the X axis, the positive it's Texas and our why the wife opponent is our sign. Data worth data is again the same angle measured counterclockwise from the positive. That's access. So let's first look. And Dr A, which is drawn and red. Here are magnitude is 100 Newton's and the angle were already given measured appropriately counterclockwise from the X axis is 30 degrees. So for the ex component, we do 100. Newton's the magnitude times the co sign of 30 degrees, and you end up getting 87 mutants and for the wise opponent, same thing, but with a sign and you get 15 units for the other defectors, you're going to follow exactly the same procedure you've got your magnitude and direction, and you plug it into the formal is in the top left. The only tricky part is for the other rector's. We don't necessarily have the angle. In fact, we don't have the angle given as measured counterclockwise from the X axis. So we've got to figure that out for the greenbacks hereafter be. We want this angle. I've just marked in green, and we know that this angle is 30 degrees. The whole thing marked in green is 30 degrees plus this 90 degrees between the X and y axis. So the fate of that we want to use protector be in. Our formula is 120 degrees. That's the 30 degrees plus 90 degrees, so you'll plug that into the formula. You take your magnitude 80 Newton's times the coastline of 120 degrees for the ex component, and you'll get minus 40. Newtons, which is good, should be negative. You've been seeing that it's pointing in the negative extraction for the Y component. You have 80. Newton's time signed 120 degrees and you get 69 mittens and again we construct on the diagram. It should be larger than the ex component magnitude and positive. So that's good. Finally protector. See, we are given that this angle here is 53 degrees. But what we want for our seita is this angle measured all the way from the positive that plexus counterclockwise around. So that would be this 180 degrees plus 53 degrees. So you ended with data equals 233 degrees. And again, we're given our magnitude of 40 mutants. Just label that there and data 233 degrees. You plug it into the equations on the top left and you end up finding that your components the ex component is minus 24 Newtons and the Y component is minus 32 unions. And these check out they're both smaller than 40 Newton's and magnitude. They're both negative. And since this angle's greater than 45 degrees, this thing is a little more vertical than horizontal, so the Y component should be a little larger in magnitude in the ex component. Great. So that's the end of party for Part B. It wants the components of the resultant of the three pools, so the result is what you get when you add these vectors altogether, and you can add vectors by adding their components separately. So add up their ex component and add up all these ex opponents to get the X component of the resultant and add up all the wife opponents to get the wife opponent of the resultant. So for the ex component, let's go around and collect the ones we've got. We thought 87 Newton's from the Red Vector minus 40 Newtons from the Green Vector, minus 20 for Nunes from the Blue Vector. And since these all have the same units, I'm just going to draw two units once for them, and you end up with 23 meetings for the why. It's the same thing. We've got our 50 Newton's plus 69 unions, minus 32 unions. You can write that in yourself and end up with 87 mutants. All right, you're part See, it wants the magnitude and direction of this result doctor. So just like we originally were able to find X and Y components because when you were given man with your direction, you go the other way, too. You confined magnitude and direction when you're given X and y components. So, uh, let's jump up top to do that. The magnitude of a vector from its external components could be gotten basically from the Pythagorean theorem. So it's the square root of our X squared. Plus are why squared and so are our ex. And our why are 23 Mittens and 87 Newtons. So you put them in and you get 90 Newtons, which seems reasonable, just a little larger than the larger of those our angle again, This li is measured counterclockwise from the X axis is the inverse tangent of our Y Over rx. And so, in this case, that is going to be the inverse tangent of 87 Newtons divided by 23 mutants, and we get 75 degrees. So all three of these pools ad together to be equivalent to a single pull of 90. Newton's in the direction 75 degrees counterclockwise from the X axis, so we can try to draw this on here after party. It's going to be a little crowded on this board. If you're doing this at home on your paper, you're gonna wanna have, ideally, a ruler and a protractor to get all this perfectly. We'll do the best we can here, so to add apart vectors, u want to point. Put them all tip to tail. So I'm going to start with my red vector as pictured. I'm going to translate the green vector. Better be over. I'm going to keep it. Same magnitude. Keep its direction. I've just translated it so that it's tip is at the other one's tail, and it so happens that this should be a right angle here because each the green records tilted 30 degrees counterclockwise from the Y axis and the red vectors tilted 30 degrees counterclockwise from the X axis. So there's still 90 degrees from each other, and now the blue vector. Let's translate that. Keep it the same magnitude. Same direction comes to about here, and so will draw the resultant thes air all tip to tail. So we're going to go from the tail the 1st 1 all the way to the tip of the last one, and this is our result in Vector, and let's see if that makes sense. We said its magnitude was 90 Newton's, so from the magnitude of the diagram, it should be about its length to be about halfway between that if the red and green vectors and it looks like it is and it should be 75 degrees clockwise from the X axis. And that looks about after it too. So I think we're all set.

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