Question 5.1: For the plane area shown, determine (a) the first moments wi...
For the plane area shown, determine (a) the first moments with respect to the x and y axes; (b) the location of the centroid.
STRATEGY: Break up the given area into simple components, find the centroid of each component, and then find the overall first moments and centroid.
MODELING: As shown in Fig. 1, you obtain the given area by adding a rectangle, a triangle, and a semicircle and then subtracting a circle. Using the coordinate axes shown, find the area and the coordinates of the centroid of each of the component areas. To keep track of the data, enter them in a table. The area of the circle is indicated as negative because it is subtracted from the other areas. The coordinate y of the centroid of the triangle is negative for the axes shown. Compute the first moments of the component areas with respect to the coordinate axes and enter them in your table.
ANALYSIS:
a. First Moments of the Area. Using Eqs. (5.8), you obtain
Q_y=\bar{X}\sum{A}=\sum{\bar{x} }A \quad \quad \quad Q_x=\bar{Y}\sum{A}=\sum{\bar{y} }A \quad \quad \quad \quad (5.8) \\ Q_x=\sum{\bar{y}}A=506.2 \times 10^3 \ \text{mm}^3 \quad \quad \quad Q_x=506 \times 10^3 \ \text{mm}^3 \\ Q_y=\sum{\bar{x}}A=757.7 \times 10^3 \ \text{mm}^3 \quad \quad \quad Q_y=758 \times 10^3 \ \text{mm}^3
b. Location of Centroid. Substituting the values given in the table into the equations defining the centroid of a composite area yields (Fig. 2)
\bar{X}\sum{A}=\sum{\bar{x}}A: \quad \quad \quad \bar{X}(13.828 \times 10^3 \ \text{mm}^2)=757.7 \times 10^3 \text{mm}^3\bar{X}=54.8 \ \text{mm}
\bar{Y}\sum{A}=\sum{\bar{y}}A: \quad \quad \quad \bar{Y}(13.828 \times 10^3 \ \text{mm}^2)=506.2 \times 10^3 \ \text{mm}^3\bar{Y}=36.6 \ \text{mm}
REFLECT and THINK: Given that the lower portion of the shape has more area to the left and that the upper portion has a hole, the location of the centroid seems reasonable upon visual inspection.
