Centroid of a Composite Area Determine the centroid of the area in Fig. 7.8.

Question

Centroid of a Composite Area

Determine the centroid of the area in Fig. 7.8.

Accepted Answer

Choose the Parts We can divide the area into a triangle, a rectangle, and a semicircle, which we call parts 1, 2, and 3, respectively.

Determine the Values for the Parts The x coordinates of the centroids of the parts are shown in Fig. a. The x coordinates, the areas of the parts, and their products are summarized in Table 7.1.

Table 7.1 Information for determining the x coordinate of the centroid

	[latex]\overline{x}_{i} [/latex]	[latex] A_{i}[/latex]	[latex] \overline{x}_{i}A_{i} [/latex]
Part 1 (triangle)	[latex] \frac{2}{3}b[/latex]	[latex] \frac{1}{2}b(2R)[/latex]	[latex] (\frac{2}{3}b[\frac{1}{2}b(2R)] )[/latex]
Part 2 (rectangle)	[latex]b + \frac{1}{2} c [/latex]	[latex] c(2R)[/latex]	[latex](b + \frac{1}{2} c )[c(2R)] [/latex]
Part 3 (semicircle)	[latex] b + c + \frac{4R}{3\pi } [/latex]	[latex] \frac{1}{2} \pi R^{2} [/latex]	[latex] [ b + c + \frac{4R}{3\pi }]( \frac{1}{2} \pi R^{2})[/latex]

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

[latex]\overline{x} =\frac{\overline{x}_{1} A_{1} + \overline{x}_{2} A_{2} + \overline{x}_{3} A_{3} }{A_{1} + A_{2} + A_{3}} = \frac{(\frac{2}{3}b )[{\frac{1}{2}b(2R)] + (b + \frac{1}{2}c )[c(2R)] + \left(b + c + \frac{4R}{3\pi } \right) }(\frac{1}{2} \pi R^{2} )}{\frac{1}{2}b(2R)+ c(2R) + \frac{1}{2}\pi R^{2} }[/latex]

We repeat the last two steps to determine the v coordinate of the centroid. The y coordinates of the centroids of the parts are shown in Fig. b. Using the information summarized in Table 7.2, we obtain

[latex]\overline{y} =\frac{\overline{y}_{1} A_{1} + \overline{y}_{2} A_{2} + \overline{y}_{3} A_{3} }{A_{1} + A_{2} + A_{3}} = \frac{[\frac{1}{3}(2R)][{\frac{1}{2}b(2R)] + R[c(2R)] + }R(\frac{1}{2} \pi R^{2} )}{\frac{1}{2}b(2R)+ c(2R) + \frac{1}{2}\pi R^{2} }[/latex]

Table 7.2 Information for determining the y coordinate of the centroid

	[latex]\overline{y}_{i} [/latex]	[latex] A_{i}[/latex]	[latex] \overline{y}_{i}A_{i} [/latex]
Part 1 (triangle)	[latex] \frac{1}{3}2(b)[/latex]	[latex] \frac{1}{2}b(2R)[/latex]	[latex] [\frac{1}{3}2(b)][\frac{1}{2}b(2R)] )[/latex]
Part 2 (rectangle)	R	[latex] c(2R)[/latex]	[latex]R(b + \frac{1}{2} c )[c(2R)] [/latex]
Part 3 (semicircle)	R	[latex] \frac{1}{2} \pi R^{2} [/latex]	[latex] R ( \frac{1}{2} \pi R^{2})[/latex]

Question 7.3: Centroid of a Composite Area Determine the centroid of the a...

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