ch2_104_107

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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 Problem 2.103 for the figure and the analysis leading to the linear algebraic Equations (1), (2), and (3) 
below: 
 0.6 0.3242 0AB ACT T− + = (1) 
 0.8 0.75676 0.8615 0AB AC ADT T T P− − − + = (2) 
 0.56757 0.50769 0AC ADT T− = (3) 
Substituting 100 lbACT = in Equations (1), (2), and (3) above, and solving the resulting set of equations 
using conventional algorithms gives 
54 lbABT = 
112 lbADT = 
 215 lb=P W 
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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 applied at A are: 
, , and AB AC ADT T T P 
where P=P j . To express the other forces in terms of the unit vectors 
i, j, k, we write 
 ( ) ( ) ( )0.72 m 1.2 m 0.54 m ,AB = − + −i j kJJJG 1.5 mAB = 
 ( ) ( )1.2 m 0.64 m ,AC = +j kJJJG 1.36 mAC = 
 ( ) ( ) ( )0.8 m 1.2 m 0.54 m ,AD = + −i j kJJJG 1.54 mAD = 
and ( )0.48 0.8 0.36AB AB AB AB ABABT T TAB= = = − + −T i j k
JJJG
λ 
( )0.88235 0.47059AC AC AC AC ACACT T TAC= = = +T j k
JJJG
λ 
( )0.51948 0.77922 0.35065AD AD AD AD ADADT T TAD= = = + −T i j k
JJJG
λ 
Equilibrium Condition with W= −W j 
0: 0AB AC ADF WΣ = + + − =T T T j 
Substituting the expressions obtained for , , and AB AC ADT T T and 
factoring i, j, and k: 
( ) ( )0.48 0.51948 0.8 0.88235 0.77922AB AD AB AC ADT T T T T W− + + + + −i j
 ( )0.36 0.47059 0.35065 0AB AC ADT T T+ − + − =k 
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PROBLEM 2.105 CONTINUED 
Equating to zero the coefficients of i, j, k: 
0.48 0.51948 0AB ADT T− + = 
0.8 0.88235 0.77922 0AB AC ADT T T W+ + − = 
 0.36 0.47059 0.35065 0AB AC ADT T T− + − = 
Substituting 3 kNABT = in Equations (1), (2) and (3) and solving the 
resulting set of equations, using conventional algorithms for solving 
linear algebraic equations, gives 
4.3605 kNACT = 
2.7720 kNADT = 
 8.41 kNW = W 
 
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PROBLEM 2.106 
For the crate of Problem 2.105, determine the weight of the crate 
knowing that the tension in cable AD is 2.8 kN. 
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 
See Problem 2.105 for the figure and the analysis leading to the linear algebraic Equations (1), (2), and (3) 
below: 
 0.48 0.51948 0AB ADT T− + = 
 0.8 0.88235 0.77922 0AB AC ADT T T W+ + − = 
 0.36 0.47059 0.35065 0AB AC ADT T T− + − = 
Substituting 2.8 kNADT = in Equations (1), (2), and (3) above, and solving the resulting set of equations 
using conventional algorithms, gives 
3.03 kNABT = 
4.40 kNACT = 
 8.49 kNW = W 
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PROBLEM 2.107 
For the crate of Problem 2.105, determine the weight of the crate 
knowing that the tension in cable AC is 2.4 kN. 
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 
See Problem 2.105 for the figure and the analysis leading to the linear algebraic Equations (1), (2), and (3) 
below: 
 0.48 0.51948 0AB ADT T− + = 
 0.8 0.88235 0.77922 0AB AC ADT T T W+ + − = 
 0.36 0.47059 0.35065 0AB AC ADT T T− + − = 
Substituting 2.4 kNACT = in Equations (1), (2), and (3) above, and solving the resulting set of equations 
using conventional algorithms, gives 
1.651 kNABT = 
1.526 kNADT = 
 4.63 kNW = W 
115