646 CHAPTER 9 Polar Coordinates; Vectors Now Work PROBLEM 89 Analyzing an Object in Static Equilibrium A box of supplies that weighs 1200 pounds is suspended by two cables attached to the ceiling, as shown in Figure 67(a). What are the tensions in the two cables? EXAMPLE 10 Draw a force diagram using the vectors as shown in Figure 67(b). The tensions in the cables are the magnitudes F1 and F2 of the force vectors F1 and F .2 The magnitude of the force vector F3 equals 1200 pounds, the weight of the box. Now write each force vector in terms of the unit vectors i and j. For F1 and F ,2 use α α ( ) = + v v i j cos sin , where α is the angle between the vector and the positive x-axis. ( ) = ° + ° = − + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = − + F F i j F i j F i F j cos150 sin150 3 2 1 2 3 2 1 2 1 1 1 1 1 ↑ v v cosi sinj α α ( ) = + ( ) = ° + ° = + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + F F i j F i j F i F j cos45 sin 45 2 2 2 2 2 2 2 2 2 2 2 2 2 ↑ α α ( ) = + v v cosi sinj = − F j 1200 3 For static equilibrium, the sum of the force vectors must equal the zero vector. + + = − + + + − = F F F F i F j F i F j j 3 2 1 2 2 2 2 2 1200 0 1 2 3 1 1 2 2 The i component and j component must both equal zero. This results in the two equations − + = F F 3 2 2 2 0 1 2 (9) + − = F F 1 2 2 2 1200 0 1 2 (10) Solve equation (9) for F2 to obtain = F F 3 2 2 1 (11) Substituting into equation (10) and solving for F1 yields + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − = F F 1 2 2 2 3 2 1200 0 1 1 + − = F F 1 2 3 2 1200 0 1 1 + = F 1 3 2 1200 1 = + ≈ F 2400 1 3 878.5 pounds 1 Substituting this value into equation (11) gives F . 2 = = ⋅ + ≈ F F 3 2 3 2 2400 1 3 1075.9 pounds 2 1 The left cable has tension of approximately 878.5 pounds, and the right cable has tension of approximately 1075.9 pounds. 308 308 458 458 1200 pounds (a) Figure 67 Force diagram y 308 x F1 F2 F3 458 1508 (b) Solution
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