Adding Vectors Figure 2.11 is an initial design sketch of part of the roof of a sports stadium that is to be supported by the cables AB and AC. The forces the cables exert on the pylon to which they are attached are represented by the vectors FAg and FAC. The magnitudes of the forces are |FAB|= 100

Question

Adding Vectors
Figure 2.11 is an initial design sketch of part of the roof of a sports stadium that is to be supported by the cables AB and AC. The forces the cables exert on the pylon to which they are attached are represented by the vectors [latex]F_{AB}[/latex] and [latex]F_{AC}[/latex]. The magnitudes of the forces are [latex]\left|F_{AB}\right| = 100 kN[/latex] and [latex]\left|F_{AC} \right| = 60 kN[/latex]. Determine the magnitude and direction of the sum of the forces exerted on the pylon by the cables (a) graphically and (b) by using trigonometry .

Strategy
(a) By drawing the parallelogram rule for adding the two forces with the vectors drawn to scale, we can measure the magnitude and direction of their sum.

(b) We will calculate the magnitude and direction of the sum of the forces by applying the laws of sines and cosines (Appendix A, Section A. 2) to the triangles formed by the parallelogram rule.

Accepted Answer

(a) We graphically construct the parallelogram rule for obtaining the sum of the two forces with the lengths of [latex]F_{AB}[/latex] and [latex]F_{AC}[/latex] proportional to their magnitudes (Fig. a). By measuring the figure, we estimate the magnitude of the vector [latex]F_{AB} + F_{AC}[/latex] to be 155 kN and its direction to be 19° above the horizontal.

(b) Consider the parallelogram rule for obtaining the sum of the two forces (Fig. b). Since α+ 30° = 180°, the angle α = 150°. By applying the law of cosines to the shaded triangle,

[latex]\left|F_{AB} + F_{AC}\right| ^{2} = \left|F_{AB}\right| ^{2} + \left|F_{AC}\right| ^{2} - 2\left|F_{AB}\right|\left|F_{AC}\right|\cos \alpha [/latex]

=(100)² + (60)² - 2(100)(60) cos 150° .

we determine that the magnitude [latex]\left|F_{AB} + F_{AC}\right| = 155 kN[/latex] .

To determine the angle ß between the vector [latex]F_{AB} + F_{AC}[/latex] and the horizontal, we apply the law of sines to the shaded triangle:

[latex]\frac{\sin \beta }{\left|F_{AB}\right|} = \frac{\sin \alpha }{\left|F_{AB} + F_{AC}\right|}[/latex]

The solution is

[latex]\beta = arcsin \left(\frac{\left|F_{AB} \right| sin \alpha }{\left|F_{AB} + F_{AC}\right|} \right) = arcsin \left(\frac{100 sin 150°}{155} \right)= 18.8°[/latex]

Discussion
Engineering applications of vectors usually require the precision of analytical solutions, but experience with graphical solutions can help you understand vector operations. Carrying out a graphical solution can also help you formulate an analytical solution.

Question 2.1: Adding Vectors Figure 2.11 is an initial design sketch of pa...

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