PROBLEM 2.99

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 AB is 259 N.

SOLUTION

The forces applied at A are:

where P = Pj. To express the other forces in terms of the unit vectors i, j, k, we write

AB =-(4.20 m)i - (5.60 m)j AB=7.00m

AC =(2.40 m)i - (5.60 m)j + (4.20 m)k AC=7.40 m

AD = -(5.60 m)j - (3.30 m)k AD=6.50m

and TAB =TABAAB =TAB AB =(-0.6i-0.8j)TABAB

TAC =TAC AAC =TAC AC =(0.32432 - 0.75676j +0.56757k)~c AC

TAD =TADAAD =TAD AD =(-0.86l54j-0.50769k)TADAD

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101

PROBLEM 2.99 (Continued)

Equilibrium condition IF =0: TAB +TAC + TAD + Pj =0

Substituting the expressions obtained for TAB' TAc , and TAD and factoring i, j, and k:

(-0.6TAB + 0.32432TAC )i + (-0.8TAB - 0.75676TAC - 0.86154TAD + P)j

+(0.56757TAC - 0.50769TAD )k = 0

Equating to zero the coefficients of i, j, k:

-0.6TAB + 0.32432TAC = 0 (1)

-0.8TAB -0.75676TAC -0.86154TAD +P=O (2)

0.56757TAC - 0.50769TAD = 0 (3) II Setting TAB = 259 N in (1) and (2), and solving the resulting set of equations gives

!__ TAc=479.15N TAD = 535.66 N P =1031 Nt ~

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