Forces in 2D Legacy Problem #5 Guided Solution

Here we have a problem in which we're asked to determine the resultant force or the sum of two forces. Now, whenever we're adding vectors of any kind, such as these force vectors, we always appreciate if the force vectors are at right angles to one another, because the act of adding such vectors involves the use of the Pythagorean Theorem, something that most of us are very, very comfortable with. But the bad news here is that the two forces that we're given are not at right angles to one another. You might want to draw the two forces just to get the feel for the direction that they go. Put a dot on your page, and then there's a force that goes at 53 degrees, which indicates that relative to east, we're 53 degrees north of east. You can go ahead and draw that force vector, and then there's a second force vector up on the dot or the half back from the other team. And that second force is a force which is at 107 degrees. That's about 7 degrees to the west of north. So you can kind of draw a vector that's going mostly north, but a little bit to the west, and that's the second vector. The bad news, as you see, is they're not at right angles to one another. The good news is that we can make them be at right angles to one another. That is, we can take the two forces and we can resolve them or break them up into their parts, a part going horizontally and a part going vertically. And we can use the parts of the vector to create one horizontal vector and one vertical vector by adding the parts together. Then once we've done that, we have now created a situation in which we have just two forces, one of them going horizontally and one of them going vertically. Now we use Pythagorean theorem and SOH CAH TOA to determine the answer. That essentially is the strategy that I will employ in this problem. Now somehow you're going to need to organize yourself. So I recommend that you use a table similar to the one that's provided here on this help page in order to organize the data. In fact, you might want to just use the same table. Now I would begin by calculating the horizontal and the vertical components of Jerome's force. And getting the horizontal is as easy as going 1230 Newtons times the cosine of 53 degrees. And getting the vertical is as easy as going 1230 times the sine of 53 degrees. This always works whenever you have the magnitude and the direction expressed as the counterclockwise angle of rotation from east. So go ahead and calculate those and enter them in the table. Avoid rounding until you get to the final answer. Now I had to do the same thing for Michael's force, 1450 Newtons at 107 degrees. So for horizontal I just go 1450 times cosine of 107. You'll notice you get a negative number. Whenever you get negative using this process it means something. It means something about the direction of the force. Here if you get a negative for a horizontal component that means it's headed west. So I would write down either negative or west where the previous horizontal component for Jerome was east. And then repeat the process with the sine of 107 degrees to find the vertical component for Michael's force. So you go 1450 times the sine of 107 degrees. Now you have two horizontal components and two vertical components and you'll note in the last row of the table we have to add them up. We have to take the horizontal and the vertical, the horizontal component for Jerome and for Michael and add them together remembering that Michael's is negative or to the west. Do the same thing for the vertical component. Then when you're done you should have 316.29 Newtons positive for the horizontal component and for the sum of the two vertical components you should have a whopping 2368.96 Newtons. Now you have two forces and they are at right angles to one another and you have to determine the resultant force. This is probably a more comfortable business as usual situation for you. Simply draw out the two forces, one of them going very, very far to the north, the other going a little bit to the east and add them together and as you do you'll be using Pythagorean Theorem and you should get 2389.98 Newtons and you can round that to the proper number of significant digits, that would be 2390 Newtons. Now finally you can determine the direction of this force. It's not enough to say that it's going northeast, you have to give more detail about how far north of east or east of north it is. So depending on how you've drawn your triangle here, I've drawn mine with a northerly component going up first until the arrowhead of that I've added my easterly component. So I'm going to find the angle between the resultant, which is the diagonal of this triangle and the vertical line and when I do that I get a value of 7.6 degrees. So I can express the direction as 7.6 degrees east of the north direction, 7.6 degrees east of north. Now alternatively you might have drawn the triangle a different way and calculated a different angle and when you did you may have gotten something like 82.4 degrees north of east.