Two forces act on a point in the plane. The angle between the ...

Question

Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.

forces of 116 and 139 lb, forming an angle of 140° 50′

Accepted Answer

Welcome back. I am so glad you're here. We're told that two forces of 108 and £128 act on a point in the passenger train, calculate the magnitude of the resultant force. If the angle between them is 155 degrees, 40 minutes. Our answer choices are answer choice. A £231 can trace be £53. Answer choice C £202 and answer choice. D £124. All right, let's try to draw this. So we've got two forces acting on a little passenger train. I think this is like the little engine that could maybe I don't, you know, we'll pretend that's a passenger train and two forces acting on it. The angle between them is pretty large angle. It's 155 degrees 40 minutes. So one of them will one vector will draw one pretty much, almost straight up. We'll label that 108 and the other one almost going straight down a little bit longer. That one will label 128. And we know that the angle between them is 155 degrees, 40 minutes, let's convert that simply to degrees. So that it's a little bit easier to put into the calculator. So we know that if we're converting 40 minutes, two degrees that there are, there is one degree in 60 minutes. So our minutes units will cancel out 40 multiplied by one is 40 divided by 60. And that simplifies to be two thirds degrees if we take 155 degrees plus two thirds degrees. And we keep that as an exact value in a fraction, that will be 467 divided by three degrees. So we can label our angle 467 divided by three degrees. Now, what are we looking for here? We're looking for the magnitude of the resultant. So let's create a parallelogram parallel to that 108 vector uh magnitude vector I'll draw from the terminal point of the 128 magnitude vector parallel to that 108 and about 108 of a magnitude. And then connecting the two terminal points of the two vectors with the magnitude of 108 will draw another line connecting those. And that will have a magnitude of 128. And so the resultant is connecting the two initial points of our two initial factors with the terminal points of the two vectors that we drew in the parallelogram. And we just need to find the magnitude of that. We can label that C. And in order to find the magnitude, we can use the law of cosines as soon as we figure out the measure of the angle directly across from it, which we can also label as an angle measure. C because we recall from previous lessons that adjacent angles of a parallelogram are supplementary. We know that taking our angle that we know the 467 divided by three degrees and adding that to the one that we're looking for. Angle C, they're supplementary. So that will equal 180 degrees. And if we subtract 467 divided by three degrees from the 180 degrees, we'll get our measurement of C which is equal to 73 divided by three degrees. So now we can use that law of cosines. We recall from previous lessons, the law of cosines if we're solving for C will have C equal to the square root of and then it will be A squared plus B squared minus two, multiplied by a multiplied by B multiplied by the cosine of angle C, all inside of the radical. We know what our angle C is. What are our A and our B? Well, A and B can be either of the sides adjacent to the angle that we know. And so we can say that A is 108 and we can say that B is 128. So now let's plug everything in. So all underneath the radical, we're taking the square root of A squared is 108 squared plus B squared is 128 squared minus two, multiplied by A, is 108 multiplied by B is 128 multiplied by the cosine of C is 73 divided by three degrees. And you can plug all of that straight into a calculator. Be sure you're in degrees mode before you hit, enter, you'll get an answer of 53. That's our resultant force magnitude and everything here is measured in pounds. Well, all of our magnitudes are measured in pounds. So we look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.
forces of 116 and 139 lb, forming an angle of 140° 50′

Key Concepts

Law of Cosines

Vector Addition

Angle Measurement

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