Forces & Load: Force Addition Algebraic Math 1

In this video, I'd like to talk about force addition using the algebraic method. We have covered, hopefully you've looked at the video that discusses components, and the other video that discusses the graphic method. So, given a multiple choice problem, I would always like to see if I can eliminate some answers. That is critical. So, versus fill in the blank, I'd have to do the work and might decide to skip it because it takes some time. Very good. So, looking at this question, calculate the resultant for the forces shown. And it gives me four answers. And I'm really not interested in solving this problem. I would like to try to eliminate. So, let me see what I can do. 150 pounds at 30 degrees, 100 pounds at 70 degrees. I think the resultant, using the graphic method, I think the resultant is that diagonal of that parallelogram. And visually, it looks like it's longer than 150 pounds, and longer than 100, obviously. So I think this answer is out. It can't be smaller than 150. Now, the other suspicious answer is D. I don't think 100 plus 150, 100 at a certain angle and 150 at a different angle, you can't just add them algebraically to get 250. Now, if they were in the same direction, yes, I could add them. But right now, they're in different directions. I think this answer does not make sense. I'm left with two answers, both of them in the correct quadrant. I mean, I have no idea how much this angle is. It could be 60, it could be 70. The magnitude of the remaining two answers is more than 150. I have no idea. I'm left with two answers, A and C. Either one could be correct. Looks like I need to buckle down and do the math. Very good. So that, I'm going to explain the algebraic method because I couldn't choose between these two answers. Very good. So using the algebraic method is quite simple. We're going to go in like we did with components. We're going to go in and get the sine of 30 degrees, the cosine of 30 degrees. We're going to get the sine of 70 degrees, the cosine of 70 degrees. These are fractions. We're going to break down our separate forces based on the inclination of each. So sine of 30 from a calculator is 0.5. This one is 0.866. And then for 70 degrees, the sine of 70 degrees looks like it is 0.94, and the cosine is 0.342. So let me go ahead and multiply by 150 pounds for the first one. I'm breaking down the 150 pounds based on 30 degrees, and my two components turn out to be 75 pounds, And the other component is how much? 129.9. Very good. So those are the two components of 150. Which is which? I don't know. Which is horizontal, which is vertical? I'll check later. Right now I'm just doing the math. I'm going to break down the 100-pound force based on sine and cosine of 70 degrees. And I'm going to end up with two components. One of them is 94 pounds, and the other one is 34.2. Very good. So now it's time to figure out which component is horizontal, which is vertical. Looking at the 150 pounds, it goes up and to the right. It looks like that much of it is horizontal, and that much of it is vertical. and it looks like the angle is less than 45, therefore the run is greater than the rise. I have two components. You must be the horizontal. You're the smaller of the two. You're the vertical. Looking at the 100-pound force, it goes to the left and up, and it's pretty steep. The vertical is larger than the horizontal. If these are my two components, the vertical is larger than the horizontal, well then this is the vertical, this is the horizontal. So 94 pounds on the vertical and 34.2 on the horizontal. So the algebraic method says break down each force into a horizontal and vertical component and then add the components separately. So for 150 pounds, I have two components. The horizontal is the larger, so this is 129.9. and the vertical is the smaller of the two at 75 pounds. So the vertical is the smaller, that's the 75 pounds, and the horizontal is the larger of the two. Now I need to check the arrowheads. It looks like it's going up and to the right. Here's up, positive. Here's to the right, also positive. Now the 100-pound force looks like it's going up and to the left. Up and to the left is negative. Let's look at the components. It looks like the larger is the 94. Make sure you don't switch these. The larger is the vertical, and the smaller is the horizontal. So with the algebraic method, you just very simply add the horizontals and the verticals, and you end up with a resultant that has, I'm going to add these two, that has a horizontal component of plus 95.7, and it has a vertical component of plus 169 pounds. So here's my resultant. So let me draw this resultant on this diagram. It goes from here to here. Its horizontal projection is plus, to the right, 95.7, and up 169 pounds. And so the resultant square is 95.7 square plus 169 square. A square plus B square equals C square, the Pythagoras theorem for that red triangle. So the resultant turns out to be, how much is that answer? 194.22, which is answer C. Now, how do we get the 60 degrees? That one is straightforward also. You have a diagonal resultant whose magnitude is 194.22 and whose rise is 169 pounds and whose run is 95.7. and we would like to figure out this angle right here. So how much is this angle? Well, this angle, let's call it theta. The tangent of that angle is its rise over its run, or rise is 169, and run is 95.7. And so the tangent of that angle is equal to how much is the math? 1.765. So there's some angle out there whose tangent is 1.765. So you go to your calculator and you say tangent inverse. So that's the key on the calculator above tangent. So you might have to do second or something like that to get tangent inverse. So we're asking the calculator to give us an angle whose tangent is 1.765, and it returns 60 degrees. So the tangent of 60 degrees seems to be 1.765, or rather 1.765 is the tangent of 60 degrees. So my answer here is 194.22 at an angle of 60 degrees with the horizontal.