Resolving a Vector into Components The force F in Fig. 2.12 lies in the plane defined by the intersecting lines LA and LB . Its magnitude is 400 lb. Suppose that you want to resolve F into vector components parallel to LA and LB. Determine the magnitudes of the vector components (a) graphically and

Question

Resolving a Vector into Components

The force F in Fig. 2.12 lies in the plane defined by the intersecting lines [latex] L_{A} [/latex] and [latex]L_{B} [/latex] . Its magnitude is 400 lb. Suppose that you want to resolve F into vector components parallel to [latex] L_{A} [/latex]and [latex] L_{B} [/latex]. Determine the magnitudes of the vector components (a) graphically and (b) by using trigonometry.

Strategy
The parallelogram rule (Fig. 2.4e) clearly indicates how we can resolve F into components parallel to [latex]L_{A} [/latex] and [latex]L_{B} [/latex] .

Accepted Answer

(a) We draw dashed lines from the head of F parallel to [latex]L_{A} [/latex] and [latex]L_{B} [/latex] to construct the vector components, which we denote [latex]F_{A} [/latex] and [latex]F_{B} [/latex] (Fig. a). By measuring . the figure, we estimate their magnitudes to be [latex]\left|F_{A} \right| = 540 lb[/latex] and [latex]\left|F_{B} \right| = 610 lb[/latex] .

(b) Consider the force F and the vector components [latex]F_{A}[/latex] and [latex]F_{B}[/latex] (Fig. b). Since α + 80° + 60° = 180°, the angle α = 40°. By applying the law of sines to triangle 1,

[latex]\frac{\sin 60°}{\left|F_{A}\right| } = \frac{sin \alpha }{\left|F\right| }[/latex]

we obtain the magnitude of [latex]F_{A}[/latex] :

[latex]\left|F_{A} \right| = \frac{\left|F\right| \sin 60° } {\sin \alpha } = \frac{400 \sin 60° }{\sin 40°} = 539 lb[/latex] ,

Then by applying the law of sines to triangle 2,

[latex]\frac{\sin 80°}{\left|F_{B} \right| } = \frac{\sin \alpha }{\left|F\right| }[/latex] ,

we obtain the magnitude of [latex]F_{B}[/latex] :

[latex]\left|F_{B} \right|= \frac{\left|F\right|\sin 80° }{\sin \alpha } = \frac{400 \sin 80° }{\sin 40°} = 613 lb[/latex]

Question 2.2: Resolving a Vector into Components The force F in Fig. 2.12 ...

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