Determining Scalar Components The tether of the balloon in Fig. 2.28 exerts an 800-N force F on the hook at O. The vertical line AB intersects the x-z plane at point A. The angle between the z axis and the line OA is 60°, and the angle between the line OA and F is 45°. Express F in terms of

Question

Determining Scalar Components

The tether of the balloon in Fig. 2.28 exerts an 800-N force F on the hook at O. The vertical line AB intersects the x-z plane at point A. The angle between the z axis and the line OA is 60°, and the angle between the line OA and F is 45°. Express F in terms of components.

Strategy
We can determine the components of F from the given geometric information in two steps. First, we resolve F into two vector components parallel to the lines OA and AB. The component parallel to AB is the vector component [latex] \pmb{F}_{y} [/latex]. Then we can resolve the component parallel to OA to determine the vector components [latex] \pmb{{F}}_{x} [/latex] and [latex] \pmb{F}_{z} [/latex] .

Accepted Answer

In Fig. a, we resolve F into its y component [latex] \pmb{F}_{y} [/latex], and the component [latex]\pmb{ F}_{h} [/latex] parallel to OA. The magnitude of [latex] \pmb{F}_{y} [/latex] is

[latex]\left| \pmb{F}_{y}\right| = \left|\pmb{F}\right| \sin 45° = 800 \sin 45° = 566 N , [/latex]

and the magnitude of [latex] \pmb{F}_{h} [/latex], is
[latex]\left| \pmb{F}_{h}\right| = \left|\pmb{F}\right|\cos 45° = 800 \cos 45° = 566 N , [/latex]

In Fig. b, we resolve [latex] \pmb{F}_{h} [/latex] into the vector components [latex] \pmb{F}_{x} [/latex] and [latex] \pmb{F}_{z} [/latex] . The magnitude of [latex] \pmb{F}_{x} [/latex] is

[latex]\left| \pmb{F}_{x}\right| = \left| \pmb{F}_{h}\right| \sin 60° = 566 \sin 60 ° = 490 N , [/latex]

and the magnitude of [latex] \pmb{F}_{z} [/latex] is

[latex]\left| \pmb{F}_{z}\right| = \left| \pmb{F}_{h}\right|\cos 60° = 566 \cos 60 ° = 283 N , [/latex]

The vector components [latex] \pmb{F}_{x} [/latex] , [latex] \pmb{F}_{y} [/latex] , and [latex] \pmb{F}_{z} [/latex] , all point in the positive axis directions, so the scalar components of F are positive:

F = 490i + 566j + 283k (N).

Question 2.9: Determining Scalar Components The tether of the balloon in F...

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