A crate is supported by three cables as shown. Determine the weight of the crate knowing that the tension in cable AB is 750 lb.

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The forces applied at A are:

[latex]\mathbf{T}_{A B}, \mathbf{T}_{A C}, \mathbf{T}_{A D} ext { and } \mathbf{W}[/latex]

where [latex]\mathbf{P}=P \mathbf{j}[/latex]. To express the other forces in terms of the unit vectors [latex]\mathbf{i}, \mathbf{j}, \mathbf{k}[/latex], we write

[latex]\begin{aligned}& \overrightarrow{A B}=-(36 ext { in. }) \mathbf{i}+(60 ext { in. }) \mathbf{j}-(27 ext { in. }) \mathbf{k} \& A B=75 ext { in. } \& \overrightarrow{A C}=(60 ext { in. }) \mathbf{j}+(32 ext { in. }) \mathbf{k} \& A C=68 ext { in. } \& \overrightarrow{A D}=(40 ext { in. }) \mathbf{i}+(60 ext { in. }) \mathbf{j}-(27 ext { in. }) \mathbf{k} \& A D=77 ext { in. }\end{aligned}[/latex]

and

[latex]\begin{aligned}\mathbf{T}_{A B} & =T_{A B} \lambda_{A B}=T_{A B} \frac{\overrightarrow{A B}}{A B} \& =(-0.48 \mathbf{i}+0.8 \mathbf{j}-0.36 \mathbf{k}) T_{A B} \\mathbf{T}_{A C} & =T_{A C} \lambda_{A C}=T_{A C} \frac{\overrightarrow{A C}}{A C} \& =(0.88235 \mathbf{j}+0.47059 \mathbf{k}) T_{A C} \\mathbf{T}_{A D} & =T_{A D} \lambda_{A D}=T_{A D} \frac{\overrightarrow{A D}}{A D} \& =(0.51948 \mathbf{i}+0.77922 \mathbf{j}-0.35065 \mathbf{k}) T_{A D}\end{aligned}[/latex]

Equilibrium Condition with [latex]\qquad \mathbf{W}=-W \mathbf{j}[/latex]

[latex]\Sigma F=0: \quad \mathbf{T}_{A B}+\mathbf{T}_{A C}+\mathbf{T}_{A D}-W \mathbf{j}=0[/latex]

Substituting the expressions obtained for [latex]\mathbf{T}_{A B}, \mathbf{T}_{A C}[/latex], and [latex]\mathbf{T}_{A D}[/latex]and factoring [latex]\mathbf{i}, \mathbf{j}[/latex], and [latex]\mathbf{k}[/latex] :

[latex]\begin{gathered}\left(-0.48 T_{A B}+0.51948 T_{A D} ight) \mathbf{i}+\left(0.8 T_{A B}+0.88235 T_{A C}+0.77922 T_{A D}-W ight) \mathbf{j} \+\left(-0.36 T_{A B}+0.47059 T_{A C}-0.35065 T_{A D} ight) \mathbf{k}=0\end{gathered}[/latex]

Equating to zero the coefficients of [latex]\mathbf{i}, \mathbf{j}, \mathbf{k}[/latex] :

[latex]\begin{array}{r}-0.48 T_{A B}+0.51948 T_{A D}=0 \0.8 T_{A B}+0.88235 T_{A C}+0.77922 T_{A D}-W=0 \-0.36 T_{A B}+0.47059 T_{A C}-0.35065 T_{A D}=0\end{array}[/latex]

Substituting [latex]T_{A B}=750 \,\mathrm{lb}[/latex] in Equations (1), (2), and (3) and solving the resulting set of equations, using conventional algorithms for solving linear algebraic equations, gives:

[latex]\begin{aligned}& T_{A C}=1090.1 \,\mathrm{lb} \& T_{A D}=693 \,\mathrm{lb} & W=2100 \,\mathrm{lb}\blacktriangleleft\end{aligned}[/latex]

Question 2.103: A crate is supported by three cables as shown. Determine the...

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