Two tugboats are pulling a disabled speedboat into port with f...

Question

Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between these forces is 28.2°. Find the direction and magnitude of the equilibrant.

Accepted Answer

Welcome back. I am so glad you're here. We're told that two bicycles are pulling a bicycle that has a broken chain, start our bicycle with a broken chain. It can be sad. The forces exerted by the two bicycles are £12.08 pounds. And the angle between these forces is 35.6 degrees. Ok. So let's draw to the right of this bicycle. The two bicycles that are pulling it, one has a force of £12. So that's its magnitude and that's going to be a little bit longer than the other line pulling it, which is £8. So we can label these, the magnitude of the longer line is 12, the magnitude of the shoreline is eight and then the angle between these forces is 35.6 degrees. Were asked to determine the magnitude of the quirt and the angle it makes with the £12 force. Our answer traces are answer trace a £19.1 at an angle of 165.9 degrees with the £12 force. Answer choice B £20.2 with an angle of 168.1 degrees with the £12 force answer choice. C £17.3 at an angle of 167.4 degrees. With the £12 force answer choice. D 18.8 £18.8 at an angle of 172.2 degrees with the £12 force. All right. So we're talking about the equilibrate, which is going to be heading the opposite direction of where the bikes are being pulled. So what we're trying to find is heading behind that bike, but we need the magnitude of it and the angle that it makes with the £12 force. So we can label that theta for our angle and our magnitude of that equilibrium vector can be X. All right. So first, we've got to figure out the magnitude of our resultant and we can imagine that we've got a parallelogram here. So we can take our £8 of force and add that from the terminal side of our £12 force vector and that'll be parallel with it. So we'll have another eight there. And then just to visualize the parallelogram, we can draw the £12 force from the terminal point from our £8 force. And now that's heading to the terminal point of our other £8 force. So that's another 12 and there we have a parallelogram or you can imagine that I drew it as perfectly as a parallelogram and the um magnitude from our broken bike to where our 12 and £8 forces meet is going to be our resultant. So we'll label that r we know that the magnitude of our resultants is going to be the same as the magnitude magnitude of our equilibrium. So we want to start by finding that magnitude. How do we do that? Well, we are going to be using the law of cosines to find that. Now, first of all, we want to figure out that angle that's across from our resultant and we can do that because we recall from previous lessons that adjacent angles of a parallelogram are supplementary, which means that this 35.6 degrees that were given if we add it to the angle that is across from our resultant. So we'll say we can call that angle B if we took 35.6 degrees plus our angle B, that's going to be because they're supplementary, it's going to be 180 degrees. And to solve for our angle B, we'll just subtract 35.6 degrees from both sides. And we know that our angle B is going to be 144.4 degrees. So now we're all set to use our, let me rewrite that. So it doesn't bleed into the other part. So that's 144.4 degrees. And so our law of cosines, we recall from previous lessons that our resultant our R is going to equal the square rut of. And then on the inside, it's going to be our, our two side links that are adjacent to our angle that we're taking the cosine of. So that'll be 12 and eight, we're going to square 12 and add that to eight squared. And then we're going to subtract two, multiplied by again, those two adjacent sides 12 multiplied by eight, multiplied by. Now, the cosine of our angle, which is 144.4 degrees. Now, you can plug all of that into your calculator, make sure that you're in degrees mode and when you have done so you will get an R value of 19.08 or as all of our answer choices are to one decimal place, that will be 19.1 and we're in pounds. So there is our magnitude for R 19.1. And so we know for our equilibrim, it's going to be the same as that resultant magnitude. So our X is going to be £19.1 great. We know our magnitude. Now, we just need to figure out the angle that it makes with the £12 force. So if we go back here, we need to first figure out what the angle is from our broken bike and heading between our resultant and our £12 force bike. So we need to figure that out, we can label that angle A and in order to do that. We're going to use the law of signs, we recall from previous lessons and we'll use that alternate form if we've got the sign of that angle, a divided by the opposite side. So that's going to be eight side length that will be equal to. Here. We use one that we know we'll use the sign of 144.4 degrees divided by the side opposite of that, which is our resultant that's going to be 19 point. We can use if you keep what was on the calculator or you can use more decimal 0818. Now, it's just a matter of plugging this in to the calculator, you can multiply both sides by eight before you do. So, so that on the left, you just have the sign of A equals. And on the right, you plug in the sign of 144.4 degrees multiplied by eight divide that all by 19.0818. And you'll get the sign of A equals something like 0.244053693. And then you can use the inverse sine function on your calculator inverse sign of that 0.244053693 and you hit enter and you will get approximately 14.1 degrees. So that's our angle A in here it's 14.1 degrees. But we're trying to figure out theta. But thankfully, we know because it's going to be a supplementary angle between angle A and angle theta. We take 100 80 degrees minus our 14.1 degrees and youll get our angle theta at 165.9 degrees. And there we have both the magnitude of the equilibrium and the angle that it makes with the £12 force. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between these forces is 28.2°. Find the direction and magnitude of the equilibrant.

Key Concepts

Vector Addition and Resultant Force

Equilibrant Force

Law of Cosines and Law of Sines

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