Forces in 2D Legacy Problem #10 Guided Solution

A good problem solver reads the problem carefully and identifies the known information and the unknown information, plots out a strategy as to how to get from the known to the unknown. In a problem such as this, the best method of organizing the known and unknown information, as well as helping you to visualize what's going on, is to construct a free body diagram. And so you'll notice that on this little audio help page, there's a free body diagram structure provided for you. I would use this very structure, or one very much like it, and organize your work on this structure. So what I know about our little rescue operation here is that the basket that the occupants are in has a mass of 621 kilograms. So I'll just write that down, M equal 621 kilograms. And I know there's a cable connected to it, and that cable's applying an applied force or tension force of 4980 newtons. So F applied or F tension equal 4980 newtons, and that's a force that's at an angle. The angle to the horizontal that that force makes is 78.2 degrees. Now that's a nasty force for us, and it's not of much usefulness as it's given there. We need to resolve it into horizontal and vertical components. Eventually we're going to need to calculate an acceleration, and to calculate an acceleration, we have to add up all the forces. You'll notice all the other forces, the F grab, and the F norm, and the F friction, are going up, down, or left and right. And so we like to take this 4980 newton force at 78.2 degrees, and to break it up or resolve it into horizontal and vertical components. So you'll notice that in the diagram, there's a little sketch of a force triangle with 4980 newtons being the hypotenuse of that force triangle, and if we wish to find the horizontal component, the Fx value, it's the side adjacent to the 78.2 degrees inside that force triangle. So to find Fx, we simply go the 4980 newtons multiplied by the cosine of 78.2 degrees, and when I do that, I find that the Fx is 1018.39 newtons, and I'm going to record that, and you should as well, and I'm going to write it down to several insignificant digits. I'll round when I finally get to my final answer. Now I can do a similar thing for the vertical component of that applied or tension force. It's the side opposite the 78.2 degrees in our force triangle. So to determine its value, I have to go hypotenuse times the sine of 78.2 degrees. 4980 newtons multiplied by the sine of 78.2 degrees gives me a Fy value, and we should write it down, of 4874.76 newtons. Now that Fy will be important because together with the F norm, it balances the F gravity force. The reason we need to know the F norm is we'll eventually have to determine the F friction force, and F friction depends on F norm. So now we're going to go about the process of determining F norm, and that will begin by first determining the down force, the F gravity force, and the F gravity force is simply M times G, where the M is 621 kilograms, and the G is 9.8 newtons per kilogram. So when I go MG, I get for my F graph, 6085.8 newtons. We should write that down, and together that F graph is equal to Fy plus F normal. So to find F normal, simply subtract Fy from both sides of that equation, and you would get for F normal, 6085.80 minus 4874.76. That comes out to be 1211.04 newtons. Now you know the normal force. The normal force is important because you use it to calculate F friction. F friction is mu times F norm, and the mu here is stated as 0.568. That's the coefficient of friction between the basket and the rooftop. So go 0.568 multiplied by this F norm value, and you get F friction of 687.87 newtons. Now what do I know? I know all of the forces, and I can add them up. The verticals are easy to add to zero since there's no vertical acceleration here. The acceleration is across the rooftop, and so to determine the acceleration, we need to find the F net in the horizontal direction. That's a matter of taking Fx and subtracting from it F friction. So I go 1018.39 newtons, my Fx value, minus this F friction I just calculated, and I get 330.52 newtons. If you divide that by 621 kilograms, the mass, you get the acceleration. You can round it to three significant digits, and it becomes 0.532 meters per second per second.