Two rescue vessels are pulling a broken-down motorboat toward ...

Question

Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.

Accepted Answer

Welcome back. I am so glad you're here. We're told that two Siberian Huskies are pulling a sled on a snowy terrain. Let's draw the sled. This will be, I don't know, we can imagine that's a sled. The forces exerted by these dogs on the sled are £16.05 pounds. And the angle between these forces is 18.3 degrees. All right. So we got one dog is really significantly pulling its weight here with this, the magnitude of its line is £16. And I don't know, we can just imagine that this is a, a strong dog here and then pulling away from the sled just 18.3 degrees in the other way, label this angle 18.3 degrees is another little puppy and this one is not quite so strong. It's only pulling with £5. So the magnitude of that line is much shorter. The magnitude of that vector determine what we're asked to do here is determine the magnitude of the equilibrium and the angle that it makes with the £16 force. Our answer choices are answer choice, a £21.9 at an angle of 177.3 degrees with the £16 force answer choice B £24.5 at an angle of 174.1 degrees with the £16 force answer choice C £20.8 at an angle of 175.7 degrees with the £16 force and answer trace D £22.4 at an angle of 176.8 degrees with the £16 force. All right. So we're looking for the equilibrim, which is going to be going in the opposite direction of our sled. So that's going to be the equilibrim. We need the magnitude of it. So we can, we can label that e the magnitude of our equilibrim and the angle that it makes with the £16 force. So we can, we can label that theta well, in order to find the magnitude of our equilibrium, we're going to first need to figure out our resultant because the magnitude of our equilibrium is going to be the same as the magnitude of our resultant. It's just in the opposite direction. So if we want to think of our doggies and we can get a parallelogram going here, mine's not going to be perfectly drawn. You can use your imagination of it. But from the terminal point for that £5 force, I'm going to draw a 16 uh vector with 16 magnitude parallel to our other 16 and then from the terminal point of the £16 force connecting to the other terminal point of the other £16 force, we'll have a £5 force vector. So now we have a parallelogram and then connecting those creating a triangle connecting the places where the sixteens and the fives meet, we have our resultant. So I'm going to label that one as our and to figure out the magnitude of that resultant, we are going to use the law of cosines. And in order to do that first, we need to figure out the measure of this angle that is opposite of our resultant. And we can use either triangle. I'm going to use the top triangle. So that angle is helpful because we have parallelograms. And we recall from previous lessons that adjacent angles of a parallelogram are supplementary. And so that means our sleds angle at 18.3 degrees plus our angle. In question, we can label this one angle C is going to equal because they're supplementary, they'll equal 180 degrees. So to solve for angle C, we'll subtract 18.3 degrees from both sides and we'll get a measure for our angle C of 161.7 degrees. All right. So we have got that angle and now we can use the law of cosines. So we recall from previous lessons, the law of cosines tells us that for our angle C that's going to be equal to the square root of. And now all of this inside the square root radical, we've got A squared plus B squared minus two, multiplied by a multiplied by B multiplied by the cosine of angle C. So see here, what that's talking about is the side length for our R that's going to be R equals the square root of. What's our A A can be our first side length at 16 and B can be our second side length. That magnitude of our vector at five. So A can be 16 squared plus B is five squared minus two, multiplied by A is 16, multiplied by B is five, multiplied by the cosine of our angle that we know the one that's opposite of the magnitude, the sight length that we're finding. So see here is 1 61.7 degrees. Now make sure that your calculator is in degrees mode, you can plug all of that in and you will get um the magnitude of our resultant is going to be £20.8. So the R length is 20.8. That's the magnitude, which means that our equilibrium length is also £20.8. So there is what we're looking for the, for the magnitude. Now, we just need to find the angle that it makes with the £16 force. So in order to do that, first, we are going to figure out the angle length inside of our parallelogram that is between our resultant vector and then that £16 vector. So I'm going to label that one angle B. And so to solve for angle B, we're going to use the law of signs. We recall from previous lessons that the sign of that angle B divided by the magnitude that is directly across from it, that's five is equal to. And then we'll use another pair that we know. We know the sign of our 161.7 degrees divided by that our vector, which is the 20 point. And if you go back on your calculator and use that whole long thing 20.8064 then you can multiply both sides of the equation by fives. So that on the left, it's just that sine of that angle B equals on the right plug into your calculator sine of 161.7 degrees. Multiply by five, all divided by 20.8064 will give something like 0.075455739. Then use the inverse sign function of that long number, 0.075455739. And you will get something like 4.3 degrees. And now that's our angle B 4.3 degrees, but we're looking for theta there, but we know that our theta and our angle B are supplementary. So again, we're going to take 180 degrees, subtract our 4.3 degrees. And that will give us our theta's value of 175.7 degrees. So there we have the magnitude of our equilibrium and the angle that it makes with the £16 force, we look at our answer choices for £20.8 and an angle of 175.7 degrees. And that matches with answer choice c well done. We'll catch you on the next one.

Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.

Key Concepts

Vector Addition of Forces

Law of Cosines

Equilibrant Force

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