Forces & Load: Force Addition Algebraic Math 2

Let's go to the next problem. Problem number two says calculate the equilibrium for the forces shown. So this one is the equilibrium versus the resultant. So let me think about this for a minute and let me pull out my graphic method again and let me say here's a parallel to the other line of action. parallel the first line of action and I should expect a resultant in this quadrant and an equilibrium which would be opposite of that one. So I should expect an equilibrium in this quadrant and a resultant in this quadrant. Very good. So let me look at my answers. Something at minus 10 degrees, let's just agree on one thing, that is positive angles, that is negative angles. That's just convention. So I need my answer to be in quadrant, should be between 90 and 180 degrees. So I'm looking at these angles, trying to eliminate some answers. This one says that 200 degrees, you must be out. 200 degrees is somewhere here, so that's not reasonable. At 60 degrees, that's 45, that's 60. That's not reasonable. And then at 169, could be, I don't know, it's somewhere here, maybe. But minus 10 is incorrect. So we got the answer without even working, just eliminating three of the answers by looking at the quadrant by just graphically putting them head to tail, head to tail, and figuring out where the resultant is. So let me do the math now to come up with these answers that I have up there. So let's do the algebraic method again, and it's the same numbers as the previous problem, So I'm just going to write down the components of 150 pounds based on sine and cosine of 30 degrees. And we end up with what we had previously, which is, let me just pull out my paper. Those are 75 and 129.9. Those are my two components for the 150, and I'll remind you, looking at this graphically, the run is larger than the rise, and it's up and to the right. This must be 75 pounds, and the horizontal component must be 129.9. Excellent. As for the 100 pounds, previously it was pointing up and to the right, sorry, up and to the left. Now it's pointing down and to the left. The angle previously was given with the horizontal. Now it's given with the vertical. It doesn't matter much because the sine of 20 degrees is the cosine of 70, and the cosine of 20 degrees is the sine of 70 because 20 plus 70 equals 90 degrees. Those are complementary angles. So the sine of 20 degrees is 0.342, and the cosine of 20 degrees is 0.94 times 100 pounds. And I get two components, same as before, 34.2 and 94 pounds. I just need to look at the diagram and recognize that this time, we're in that quadrant, we're pointing downward, we have two components, 34.2 and 94. It still looks like this is pretty steep. You must be 94. And, sorry, 94, not 94.2. And the other component here is 34.2. So let me set up my algebraic table, and let me say the following: I have H and V, horizontal and vertical, and for the 150 pound force, it didn't change from the previous problem, so it's 129.9 on the run, and on the rise we have 75 pounds. Now the 100 pound is a little bit different. It's down, therefore minus 94, and to the left, therefore minus 34. So the horizontal is minus 34.2, and the vertical is minus 94 pounds. So let me add these up. I have 129 minus 34.2 is 95.7 positive, and 75 minus 94 is minus 19 or 19 negative. So let me plot this resultant. This is the resultant and let me please remember that the question was the not resultant or the opposite of the resultant which is the equilibrium. We have to find the resultant in order to find the equilibrium. Let's cheat a little bit and draw that graphic method again which is draw two parallels And that should be my resultant right here. Let me see if the numbers give me that answer. Okay, so the resultant has a horizontal component of 95.7 and a vertical component of minus 19. So the resultant square is equal to 95.7 square plus minus 19 square, which gives me a resultant, if you take the square root of those two numbers, of 97.57 pounds. And let me draw the resultant. It looks like something like that. it has a vertical of minus 19. It has a horizontal of 95.7. Therefore, this number squared turns out to be 97.57 pounds, which is 19 squared plus 95.7 squared. Excellent. Now what angle does it make here with the horizontal? I'd like to know how much that angle is. So that angle has a rise of 19 or negative 19 and it has a run of 95.7. So the tangent of this angle is rise over run, which is equal to minus 19 divided by 95.7, which gives me minus, I have the answer here, 0.199 with a minus sign. So that's the tangent of some angle out there that I don't know. Make sure you're in degrees, not in radians or grades, which are other ways of measuring angles. Make sure your calculator is set to degrees. So theta is the inverse tangent of minus 0.199. Turns out that angle is 11.23 degrees. So this angle is 11.23 degrees negative. Let me clean up a little bit the diagram. So the resultant here turned out to be, how much was it, 97.57, with an angle here of minus 11 degrees, or 11.23 degrees in the clockwise direction. That's minus 11.23. So the resultant must be over here at 97.50. Sorry, the equilibrate must be over here. The resultant is the red one. The equilibrate is the green one. So the equilibrium is 97.57 and the angle of the equilibrium is that much. But this is 11.23 and this must be 180 degrees minus 11.23 which gives me how much 169? Okay, answer B.