Forces in 2D Legacy Problem #13 Guided Solution

Here we have a situation in which a sign is being hung in equilibrium. The weight of the sign is being supported by two cables which make an angle of 34.5 degrees with the horizontal. These two cables are pulling upward and rightward, or leftward, up on the sign. The mass of the sign is 43.1 kilograms and we can use this mass to determine the F graph. If we know the F graph then we ought to be able to determine the vertical pole of the two tension cables. These vertical poles together have to balance out the down force of gravity up on the sign. So if we know one of the vertical poles we can use a little force triangle and the angle of 34.5 degrees to find the force with which each cable pulls vertically and horizontally, or at an angle to the horizontal. So that's my strategy. I'm going to begin by drawing a free body diagram. I make a little dot or a square, however you wish, and I draw three forces. One of them is down and that's the force of gravity. The other two are up and to the right and up and to the left. Those are the cables which are balancing the weight of the sign. Now I can find the down force of gravity by going M times G, 43.1, times 9.8. And if I do that I get 422.38 Newtons. That's the down force of gravity. This down force is going to be balanced by the two vertical poles. So I can find the FY value, the vertical pole, in either one of the signs by simply taking this force of gravity and dividing by two. When I do so I get 211.19 Newtons. This 211.19 Newtons is the force on the cable on the left as well as the force on the right. I now need to focus on either one of the cables. So focusing on the cable to the right, what I know is that it makes a 34.5 degree angle with the horizontal. And I want to find the hypotenuse of a force triangle, the tension force. And I know the sign opposite 34.5 degrees is equal to this 211.19 Newtons. So I say the sign of 34.5 degrees is equal to 211.19 Newtons divided by the hypotenuse or F tension. Now doing careful algebra on this equation I can solve for F tension and it comes out to be 372.86 Newtons. And I can round that to three significant digits. That's 373 Newtons of tension on the cable on the left and 373 Newtons on the cable on the right. And that's your answer.