Two forces of 692 newtons and 423 newtons act on a point. The ...

Question

Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.

Accepted Answer

Welcome back. I am so glad you're here. We're told two forces, 898 newtons and 356 newtons are concurrent, their lines of action pass through a common point. If the resultant force is 1044 newtons, what is the angle between the two forces? Our answer choices are answer choice. A 104.2 degrees. Answer choice, B 75.8 degrees. Answer choice, C 112.9 degrees and answer choice. D 67.1 degrees. All right. So let's draw what we know. We've got those two forces that there are lines of action pass through a common point. Let's make that common point be the initial point for each of our two vectors. So we've got one at 898 newtons. That is a, that's its magnitude and it's a longer magnitude than the other one with the same initial point of 356 newtons. That's its magnitude. We don't know what the angle in between is. That's what we're looking for, but we can just draw it with an estimate. Now, we know what the resultant force is that's kind of the line that passes in between them. So if we make this into a parallelogram, so from the terminal point of the 356 magnitude vector, I'm going to draw a parallel line that is also a vector with 898 magnitude. And then connecting the two terminal points between the two vectors with 898 magnitudes will draw a parallel to the 356 vector magnitude. And there we have, you can imagine it's a great well drawn parallelogram. And so connecting from the two initial points of our initial vectors to the two terminal points of the vectors we added for the parallelogram that is our resultant. And we can label that with a magnitude of 1044. So as I mentioned a moment ago, we're looking for the angle in between our two initial vectors, we can label that angle theta and we can't figure that out yet. But we recall from previous lessons that adjacent angles of a parallelogram are supplementary. So its adjacent angle is the one right across from our resultant, we can label that angle C. And so if we took our theta that we're looking for plus our angle C, they will be equal to 180 degrees because they're supplementary and we can figure out the measure of our angle C because of the law of cosines. So we recall from previous lessons, the law of cosines is that because we're dealing with here, our angle C is what we're looking for. We'll say that C squared equals A squared plus B squared minus two multiplied by a multiplied by B multiplied by the cosine of angle C. What we're looking for is angle C. But what's our A and our B and our C, those are our side lengths or our vector magnitudes C is the one directly across from our angle C. So C here is the 1044 A and B could be either of the two remaining, but let's say A is 898 and B is 356. Now let's plug things in. So C is 1044 we'll be squaring that plus A is 898. We'll be squaring that plus or excuse me, 1044 squared is equal to, we're not adding equal to a 898 squared plus B is 356 squared minus two, multiplied by A is 898 multiplied by B is 356 multiplied by the cosine of C which we don't know. Now, the rest of this is mostly algebra and it gets to be big numbers. So we've got 1044 squared is 1,089,936 equals 898 squared is 806,404 plus 356. Squared is 126,736 minus two, multiplied by 898 multiplied by 356 is going to be 639,376 which is still multiplied by the cosine of c. Now, we can combine like terms 806,404 plus 126,000. 736 is 933,140. But we can subtract that from both sides. And on the left 1,089,936 minus 933,000, 140 is 156,796 equals on the right. We've just got that negative 639,376 multiplied by the cosine of C. So we can divide both sides by negative 639,376. And that will equal our cosine of C. Now we're going to use inverse cosine to solve for this. So it into your calculator inverse cosine of and I would just keep it as this fraction negative 156,796 divided by 639,376 before you hit enter, make sure you're in degrees mode and you'll get an answer of approximately 104.2 degrees. But remember that's just our angle C and we're looking for theta. So we've got to plug that back into our equation where theta plus angle C 104.2 degrees equals 180 degrees. Now, you can subtract 104.2 degrees from both sides and will solve for theta, theta equals 75.8 degrees. And there we have it, the angle between our two forces. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.

Key Concepts

Resultant Force

Law of Cosines

Trigonometric Functions

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