Centroid of a Composite Line Determine the centroid of the line in Fig. 7.25. The quarter-circular arc lies in the y-z plane.

Question

Centroid of a Composite Line

Determine the centroid of the line in Fig. 7.25. The quarter-circular arc lies in the y-z plane.

Accepted Answer

Choose the Parts The line consists of a quarter-circular arc and two straight segments, which we call parts 1, 2, and 3 (Fig. a).

Determine the Values for the Parts From Appendix B, the coordinates of the centroid of the quarter-circular arc are .[latex]\overline{x}_{1} = 0 , \overline{y}_{1} = \overline{z}_{1} = 2(2)/\pi m [/latex]. The centroids of the straight segments he at their midpoints. For segment 2, [latex]\overline{x_{2}} = 2 m , \overline{y}_{2} = 0 , and \overline{z}_{2} = 2 m , [/latex] The length of segment 3 is [latex]\overline{x}_{3} = 2 m , \overline{y}_{3} = 1 m , and \overline{z}_{3} = 2 m .[/latex] The length of segment 3 is [latex] \overline{L}_{3} = \sqrt{(4)^{2} + (2)^{2} + (2)^{2} } = 4.90 m . [/latex] This information is summarized in Table 7.6.

Table 7.6 Information for determining the centroid.

	[latex]\overline{x}_{i}[/latex]	[latex]\overline{y}_{i}[/latex]	[latex]\overline{z}_{i}[/latex]	[latex]\overline{L}_{i}[/latex]
Part 1	0	2(2)/π	2(2)π	π(2)/2
Part 2	2	0	2	4
Part 3	2	1	1	4.90

Calculate the Centroid The coordinates of the centroid of the composite line are

[latex]\overline{x} = \frac{\overline{x}_{1} L_{1} + \overline{x }_{2}L_{2} + \overline{x }_{3}L_{3}}{L_{1} + L_{2} + L_{3} } = \frac{0 + (2)(4) + (2)(4.90)}{\pi + 4 + 4.90} = 1.478 m .[/latex]

[latex]\overline{y} = \frac{\overline{y_{1} } L_{1} + \overline{y_{2} } L_{2} + \overline{y_{3} } L_{3}}{L_{1} + L_{2} + L_{3} } = \frac{\left[2(2)/\pi \right]\left[\pi (2)/2\right] + 0 + (1)(4.90) }{\pi + 4 + 4.90} = 0.739 m . [/latex]

[latex]\overline{z} = \frac{\overline{z}_{1} L_{1} + \overline{z}_{2}L_{2} + \overline{y }_{3}L_{3}}{L_{1} + L_{2} + L_{3} } = \frac{[2(2)/ \pi ][\pi (2)/2] + (2)(4) + (1)(4.90)}{\pi + 4 + 4.90} =1.404 m .[/latex]

Question 7.13: Centroid of a Composite Line Determine the centroid of the l...

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