Centroid of a Volume Containing a Cutout Determine the centroid of the volume in Fig. 7.24.

Question

Centroid of a Volume Containing a Cutout

Determine the centroid of the volume in Fig. 7.24.

Accepted Answer

Choose the Parts We can divide the volume into the five simple parts shown in Fig. a. Part 5 is the volume of the 20-mm-diameter hole.

Determine the Values for the Parts The centroids of parts 1 and 3 are located at the centroids of their semicircular cross sections (Fig. b). The information for determining the x coordinate of the centroid is summarized in in Table 7.5. Part 5 is a negative volume.

Table 7.5 Information for determining [latex] \overline{x} [/latex].

	[latex] \pmb{\overline{x}_{i} (mm)}[/latex]	[latex] \pmb{{V_{i} }(mm^{3} ) }[/latex]	[latex] \pmb{\overline{x}_{i} V_{i} (mm^{4} ) }[/latex]
part 1	[latex] -\frac{4(25)}{3\pi } [/latex]	[latex] \frac{\pi (25)^{2} }{2}(20)[/latex]	[latex] [-\frac{4(25)}{3\pi } ][\frac{\pi (25)^{2} }{2} (20)] [/latex]
part 2	100	(200)(50)(20)	(100)[(200)(50)(20)]
part 3	[latex] 200 + \frac{4(25)}{3\pi } [/latex]	[latex] \frac{\pi (25)^{2} }{2}(20)[/latex]	[latex][200 + \frac{4(25)}{3\pi } ][\frac{\pi (25)^{2} }{2} (20)] [/latex]
part 4	0	[latex] \pi (25)^{2}(40) [/latex]	0
part 5	200	[latex] -\pi (10)^{2}(20) [/latex]	[latex] -(200)[\pi (10)^{2} (20)] [/latex]

Calculate the Centroid The x coordinate of the centroid of the composite volume is

[latex]\overline{x} = \frac{\overline{x}_{1} V_{1} + \overline{x}_{2} V_{2 } + \overline{x}_{3} V_{3} + \overline{x}_{4} V_{4} + \overline{x}_{5} V_{5}}{V_{1} + V_{2} + V_{3} + V_{4} + V_{5}} \left[-\frac{4(25)}{3\pi } \right] \left[\frac{\pi (25)^{2} }{2} (20)\right] + (100)\left[(200)(50)(20)\right] = \frac{+\left[200 + \frac{4(25)}{3\pi } \right] \left[\frac{\pi (25)^{2} }{2} (20) \right] + 0 - (200)\left[\pi (10)^{2}(20) \right] }{\frac{\pi (25)^{2} }{2} (20) + (200)(50)(20) + \frac{\pi (25)^{2} }{2} (20) + \pi (25)^{2} (40) - \pi (10)^{2}(20)} = 72.77 mm .[/latex]

The z coordinates of the centroids of the parts are zero except [latex]\overline{z}_{4}= 30 mm [/latex]. Therefore the z coordinate of the centroid of the composite volume is

[latex]\overline{z} = \frac{\overline{z}_{4} V_{4} }{ V_{1} + V_{2} + V_{3} + V_{4} + V_{5} } = \frac{30 \left[\pi (25)^{2} (40) \right] }{\frac{\pi (25)^{2}}{2}(20) + (200)(50)(20) + \frac{\pi (25)^{2}}{2} (20) + \pi (25)^{2}(40) - \pi 10^{2}(20)} = 7.56 mm .[/latex]

Because of symmetry, [latex]\overline{y} = 0 .[/latex]

Question 7.12: Centroid of a Volume Containing a Cutout Determine the centr...

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