Question 6.1: Find the location of the centroid of the area shown in Figur...
Find the location of the centroid of the area shown in Figure 6–4.
Objective Compute the location of the centroid.
Given Shape shown in Figure 6–4
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 a rectangle (part 1) and a triangle (part 2), as shown in Figure 6–5. Each part is a simple shape for which the centroid is found using the data from Figure 6–2. The distances to the centroids from the bottom of the shape are shown in Figure 6–5 as y_{1} and y_{2} .
Results The following table facilitates the calculations for data required in Equation (6–2):
| Part | A_{i} | y_{i} | A_{i} y_{i} |
| 1 | 3200 mm² | 40 mm | 128 000 mm³ |
| 2 | 600 mm² | 90 mm | 54 000 mm³ |
| A_{T} = 3800 mm² | ∑ (A_{i}y_{i}) = 182 000 mm³ | ||
Now \bar{Y} can be computed:
\bar{Y} = \frac{∑(A_{i}y_{i})}{A_{T}}= \frac{182 000 mm^{3}}{3800 mm^{2}} = 47.9 mm
This locates the centroid as shown in Figure 6–5.
Comment In summary, the centroid is on the vertical axis of symmetry at a distance of 47.9 mm up from the bottom of the shape.
