Forces in 2D Legacy Problem #11 Guided Solution

This is a very difficult problem. Like any difficult problem, you need to employ the habits of an effective problem solver. Those habits involve reading the problem very carefully, writing down the known and the unknown information, and planning out a strategy before actually using your calculator. So here, I'm going to use the free body diagram that is provided for me, and I'm going to record information on the diagram. Now, what makes the problem difficult is that there's not a lot of information given to you. Some of it is actually implied, and others of it is just kind of hidden within the problem. So, I'm going to begin by writing down the mass of the broom that the custodian is pushing on. That's the object that's moving. So, m equal 1.1 kilograms. And I'm going to write down the angle that the broom makes with the horizontal. That's an angle of 41 degrees. That's theta in the diagram. Now, I'm looking to calculate the applied force, the force with which Chuck is pushing downward and rightward upon the broom. I know that it's a constant speed motion, and so that would tell me that the acceleration in that force is zero. That's probably one of the most important phrases in this problem. So, F net equals zero, and A equals zero. From that, I would reason that all the forces balance each other out. Now, there's one more given quantity, and that's the coefficient of friction. Mu is in the equation F rigged equals mu F norm. Mu equals 0.45. Now, if I were to analyze the situation for the horizontal forces, what I would know is that the horizontal forces balance. So, Fx equals F friction. Now, here's where you need to be writing some things down. If F rigged equals Fx, then what I can say is that mu times F norm equals F times the cosine of 41, where F is my unknown, and it's simply the force with which Chuck is pushing on the broom. So, I'm going to say that again. You've got to write it down. 0.45, that's mu, times F norm equals F times cosine 41. Just leave it as that, and know that the F is an unknown quantity. The F norm, we don't know. We'll have to figure that out using the vertical information. So, analyzing vertical forces here, you'll notice that there's three. There's an Fy of y component of the applied force. There's an F graph, and there's an F norm. Now, the F graph can actually be calculated by going 1.1 times g. It comes out to be 10.78 newtons. So, I can say that the two down forces, F graph and Fy, is equal to one up force. Now, here's where you need to write something down. Applying that idea, I would say 10.78 newtons, that's F graph, plus Fy equals F norm. Now, I'm going to rewrite it again. I'm going to say 10.78 newtons plus F times sine of 41, that's Fy, equals F norm. I'm going to say that again. 10.78 plus F times sine of 41 equals F norm. Now, you'll notice you have two equations. This one just stated, and the previous one, 0.45 F norm equals F cosine of 41. Two equations for two unknowns, and we can solve for our two unknowns. The one we're interested in is this F, the applied force. So, to determine the value, I need to solve for two equations, two unknowns. And the way I typically do this is I have, for my vertical analysis, I have an expression for F norm written in terms of F. So, I usually take that expression for F norm and I plug it into the other equation. The 10.78 plus the F times sine of 41, since that's equal to F norm, I can plug it in for F norm in my horizontal equation. So, that horizontal equation would become 0.45 times the quantity 10.78 plus F sine 41 equal F times cosine 41 degrees. Now, I have that horizontal equation becoming a one equation, one unknown situation, and I should be able to use careful algebra skills to solve for F. I'll need to get the two F terms by themselves on the same side of the equation, group them together, and then divide through by its coefficient in order to solve for F. And it comes out to be 10.5569 newtons. I should round that to the proper number of significant digits. Here, that's two digits, so the answer becomes 11 newtons.