Question 6.2: Find the location of the centroid of the area shown in Figur...
Find the location of the centroid of the area shown in Figure 6–6.
Objective Compute the location of the centroid.
Given Shape shown in Figure 6–6
Analysis Because the shape has a vertical axis of symmetry, the centroid lies on that line. The vertical distance from the bottom of the shape to the centroid will be computed using Equation (6–2). The total area is divided into three rectangles, as shown in Figure 6–7.
Part 1 is the total lower large rectangle, 50 mm high and 40 mm wide. Part 2 is the 20 mm × 30 mm rectangle that is removed from the composite area. The area, A_{2} , is considered to be negative. Part 3 is the 10 mm × 60 mm rectangle on top. The distances to the centroids from the bottom of the shape are shown in Figure 6–7 as y_{1} , y_{2}, and y_{3} .
Results The following table facilitates the calculations for data required in Equation (6–2).
| Part | A_{i} | y_{i} | A_{i} y_{i} |
| 1 | 2000 mm² | 25 mm | 50 000 mm³ |
| 2 | -600 mm² | 15 mm | -9000 mm³ |
| 3 | 600 mm² | 55 mm | 33 000 mm³ |
| A_{T} = 2000 mm² | ∑ (A_{i}y_{i}) = 74 000 mm³ | ||
Then,
\bar{Y} = \frac{∑(A_{i}y_{i})}{A_{T}}= \frac{74 000 mm^{3}}{2000 mm^{2}} = 37.0 mm
Comment In summary, the centroid is on the vertical axis of symmetry at a distance of 37.0 mm up from the bottom of the shape.
