Determining Scalar Components The crane in Fig. 2.27 exerts a 600-lb force F on the caisson. The angle between F and the x axis is 54°, and the angle between F and the y axis is 40°. The z component of F is positive. Express F in terms of components. Strategy Only two of the angles between the

Question

Determining Scalar Components

The crane in Fig. 2.27 exerts a 600-lb force F on the caisson. The angle between F and the x axis is 54°, and the angle between F and the y axis is 40°. The z component of F is positive. Express F in terms of components.

Strategy
Only two of the angles between the vector and the positive coordinate axes are given, but we can use Eq. (2.16) to determine the third angle. Then we can determine the components of F by using Eqs. (2. 15).

[latex]U_{x} = \left|U\right|\cos θ_{x}, U_{y} = \left|U\right|\cos θ_{y}, U_{z} = \left|U\right|\cos θ_{z}. [/latex] (2.15)

[latex] \cos ^{2} θ_{x} + \cos ^{2} θ_{y} + \cos ^{2} θ_{z} = 1. [/latex] (2.16)

Accepted Answer

The angles between F and the positive coordinate axes are related by

[latex] \cos ^{2}θ_{x} + \cos ^{2}θ_{y} + \cos ^{2}θ_{z} = (\cos 54°)^{2} + (\cos 40°)^{2} + \cos ^{2}θ_{z} = 1 [/latex] .

Solving this equation for [latex]\cos θ_{z}[/latex], we obtain the two solutions [latex]\cos θ_{z} = 0.260[/latex] and [latex]\cos θ_{z} = -0.260[/latex], which tells us that [latex]\cos θ_{z} = 74.9° [/latex] or [latex]\cos θ_{z} = 105.1°[/latex]. The z component of the vector F is positive, so the angle between F and the positive z axis is less than 90°. Therefore [latex]\cos θ_{z} = 74.9° [/latex].

The components of F are

[latex]F_{x} = \left|F\right| \cos θ_{x} = 600\cos 54° = 353 lb, [/latex]

[latex]F_{y} = \left|F\right| \cos θ_{y} = 600\cos 40° = 460 lb, [/latex]

[latex]F_{z} = \left|F\right| \cos θ_{z} = 600\cos 74.9° = 156 lb. [/latex]

Question 2.8: Determining Scalar Components The crane in Fig. 2.27 exerts ...

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