Forces & Load: Force Addition Graphic Math

So in the spirit of the ARE, I would like to continue the previous video and solve this problem with numbers. So here's the problem: we had two cables, A and B, with a thousand pound force pulling on them, and now we have four answers to choose from: how much is the force in member A, how much is the force in member B. And I'm realizing that I should have put tension compression in front of these answers, but they're all tension, so let's not go there. So looking at the answers, I'm not looking for the correct answer. I'm looking for the wrong answers because I'd like to eliminate. So answer number one is incorrect because the two forces in A and B are not equal. So that one is eliminated. We're left with three answers. Looking at the three answers, answer two and four say A is larger than B. Answer three says B is larger than A. Looking at the diagram, the force of 1,000 is vertical. B is shallower than A. A is steeper than B. A is more in with the 1,000 pound force, I should eliminate this one. Because A should end up with a larger resistance than B. Very good. We're left with two answers, two and four. Answer two is incorrect. Because they should not add up to 1,000 pounds. Any answer that adds up to 1,000 is incorrect. A square plus B square should equal to 1,000 square, not A plus B is 1,000. And if you look at the answers I've given, they all add up to 1,000. Sorry, not they all, but the first two add up to 1,000. That is immediately eliminate. So the correct answer is A is 866 and B is 500 pounds. Both are in tension. Now, how did we get those answers? That's the important thing, because if it's fill in the blank, you've got to do the work. And it's a little bit involved, so you might elect to skip the question, whatever. That's fine. Let me just solve this graphically. I have a thousand pound force going downward. And I know that A plus B, vectorially, at 30 degrees, B at 30 and A at 60, should add up to a resultant that falls on this line to cancel. That should be the resultant of A plus B. It should be 1,000 up to cancel 1,000 down because the system is in equilibrium. Very good. So let me draw A and B, and let me draw them based on the correct angle that they subtend. So if A is this much, sorry, let me draw it a little bit better. If this is 45, this is a little bit steeper. Here is A. How much? How long is it? I have no idea. It's that much. Now let me come to the head of A, and let me put the tail of B. Now, B is doing not 45, but a little bit less, 30 degrees. So, there is B. Now, the only problem is I passed that red line, and I should not pass that red line. Let me fix this and say B must end up being that length. In order to end up with a resultant on the vertical, that is that much, that cancels the 1,000 down. So it's important to end up on that vertical. So A, I don't know how much A is. It could have been that much. Could have been that much A. Well, then B has to be that much to end up on the vertical. If A is only this much, B has to be that much. We have to end up on the vertical to cancel the 1,000 pounds. So let me redraw my diagram a little bit better, because I have to calculate now. If this is the 1,000, and that looks like a 60 degree, and there's my vertical, and there is B. So let me work out the angles. This is 60 degrees. This is 1,000 pounds. Well, then this is 30 degrees, and I know that the angle between A and B is 90. The angle between A and B is 90. So this is a 90 degree, although it doesn't look it. Sorry about that. And this must be 60 degrees because that angle is 30. Very good. So now I messed up my drawing. Let me draw it one more time. Sorry about that. Where is it? Ik ga het zo naar ben So redrawing this one, let's just clean up a little bit. It looks like A plus B needs to fall on the vertical. So let me extend B a little bit. Here is B. A is doing 60 degrees with the horizontal, which means this is 30. This one is 90, as we said earlier. And this one is 30. and this one is 60. So I have a right-angled triangle right here. Let's shade it maybe for clarity. So here's this triangle, and this triangle has a 30 degree, 60 degree, and 90 degree, and the sine of 30 degrees is opposite over hypotenuse. I could use the 60 degrees and say sine of 60 is opposite over hypotenuse. It doesn't matter which angle you take. I know from a calculator that the sine of 30 degrees is 0.5. That is set. It doesn't change. And looking at the 30 degree, the side opposite of 30 is B. The hypotenuse is this resultant of 1,000 that has to cancel the other 1,000. So, the hypotenuse is 1,000 pounds, which tells me, I cross-multiply here, b is equal to 500 pounds, or 0.5 times 1,000. Looking at the cosine of 30 degrees, the cosine of 30 degrees is equal to the side adjacent divided by the hypotenuse. Again, from a calculator, the cosine of 30 degrees is 0.866. And looking at this, the hypotenuse, looking at this orange triangle, the hypotenuse is 1,000 pounds. The side adjacent to 30 degrees is A. The side opposite is B. So, side adjacent is A, which tells me that A is cross multiply 866 pounds. And sure enough, A is larger than B. And even graphically, this is longer than this one. And if this one... Okay.