Arbolbinario

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PROBLEM 2.104
Three cables are used to tether a balloon as shown. Determine the vertical force P exerted by the balloon at A knowing that the tension in cable AC is 100 lb.

SOLUTION
See Problem2.103 for the figure and the analysis leading to the linear algebraic Equations (1), (2), and (3) below: −0.6TAB + 0.3242TAC = 0 −0.8TAB − 0.75676TAC − 0.8615TAD + P = 0 0.56757TAC − 0.50769TAD = 0Substituting TAC = 100 lb in Equations (1), (2), and (3) above, and solving the resulting set of equations using conventional algorithms gives TAB = 54 lb TAD = 112 lb P = 215 lb (1) (2) (3)

111 PROBLEM 2.105
The crate shown in Figure P2.105 and P2.108 is supported by three cables. Determine the weight of the crate knowing that the tension in cable AB is 3 kN.

SOLUTION
The forces appliedat A are: TAB , TAC , TAD and P where P = Pj . To express the other forces in terms of the unit vectors i, j, k, we write AB = − ( 0.72 m ) i + (1.2 m ) j − ( 0.54 m ) k , AC = (1.2 m ) j + ( 0.64 m) k , AD = ( 0.8 m ) i + (1.2 m ) j − ( 0.54 m ) k , and TAB = TABλ AB = TAB TAC = TAC λ AC = TAC TAD = TADλ AD = TAD AB = 1.5 m AC = 1.36 m AD = 1.54 m

AB = ( −0.48i + 0.8 j − 0.36k ) TAB AB

AC= ( 0.88235j + 0.47059k ) TAC AC

AD = ( 0.51948i + 0.77922 j − 0.35065k ) TAD AD

Equilibrium Condition with W = −Wj ΣF = 0: TAB + TAC + TAD − Wj = 0 Substituting the expressions obtained forTAB , TAC , and TAD and factoring i, j, and k:

( −0.48TAB + 0.51948TAD ) i + ( 0.8TAB + 0.88235TAC

+ 0.77922TAD − W ) j

+ ( −0.36TAB + 0.47059TAC − 0.35065TAD ) k = 0

112

PROBLEM 2.105CONTINUED
Equating to zero the coefficients of i, j, k: −0.48TAB + 0.51948TAD = 0 0.8TAB + 0.88235TAC + 0.77922TAD − W = 0 −0.36TAB + 0.47059TAC − 0.35065TAD = 0 Substituting TAB = 3 kN in Equations(1), (2) and (3) and solving the resulting set of equations, using conventional algorithms for solving linear algebraic equations, gives TAC = 4.3605 kN TAD = 2.7720 kN W = 8.41 kN

113

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