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 250 and 450 newtons, forming an angle of 85°

Accepted Answer

Welcome back. I am so glad you're here. We're told two forces of 183 150 Newtons act upon a point in the passenger train. Calculate the magnitude of the resultant force. If the angle between them is 78 degrees. Our answer choices are answer choice. A 426 newtons. Answer choice B 329 newtons. Answer choice C 377 newtons and answer choice. D 359 Newtons. All right. So we've got this little passenger train, I guess this is like uh more of an old style locomotive or something. Yeah, I think that's a steam engine. Ok. Anyway, it's a passenger train of the past and we have two forces acting upon it. One is 180 newtons, one is 350 newtons and the angle between them is 78 degrees. So from our train, we can draw one vector heading up. That can be 180 newtons. That's its magnitude and another with 78 degrees between them, that one is longer, that's 350 newtons and we can label 78 degrees. We need to calculate the magnitude of the resultant force. So what we can do here is create a parallelogram. So from the terminal point of the 350 or the 350 magnitude vector, we can draw a parallel vector to our 180 magnitude vector. And from the two terminal points of the 2, 180 magnitude vectors, we can connect those drawing a parallel line to our 350 magnitude vector. And so the resultant is going to be connecting those two initial points to those two terminal points of the vectors that we added. And we can label that C to determine the magnitude of that, we can use the law of cosines if we first figure out what the angle measure is of the angle across from our resultant. So we can do that because we recall from previous lessons that adjacent angles of a parallelogram are supplementary. So if our angle, if our unknown angle is angle C, we can say that our 78 degrees plus our angle C equals 180 degrees because they're supplementary. So if we subtract 78 de degrees from both sides, we'll have our angle C equals 102 degrees. So now we can use that law of cosines. We recall from previous lessons. The law of cosines tells us that for solving for CC squared would be equal to A squared plus B squared minus to a multiplied by B multiplied by the cosine of angle C. So what are our A and our B A and B can be either of those adjacent angles to our 102 degrees or adjacent sides, not angles. But let's say that A is the 180 B is the 350. So now we can just plug things in, we're solving for C, so we're going to be taking the square root of both sides. So our C will be equal to the square root of. And now A squared A is 180 squared plus B squared B is 350 minus two, multiplied by A A is 180 multiplied by BB is 350 multiplied by the cosine of angle C is 102 degrees. Now, you can plug all of that into a calculator, make sure that you're in degrees mode. And when you plug all that in, you're going to get something like 425.5. And we round that up for our answer to match our answer choices. And that will be 426 newtons. And that is our resultant force 426 newtons. We look at our answer choices and that matches with answer choice. A 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 250 and 450 newtons, forming an angle of 85°

Key Concepts

Vector Addition of Forces

Law of Cosines

Angle Between Vectors

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