Question A.03: Referring to the area A of Example A.02, we consider the hor...

We divide the area A^{\prime} into its components A_{1} and A_{3} (Fig. A.13). Recalling from Example A.02 that C is located 46 \mathrm{~mm} above the lower edge of A, we determine the ordinates \bar{y}_{1}^{\prime} and \bar{y}_{3}^{\prime} of A_{1} and A_{3} and express the first moment Q_{x^{\prime}}^{\prime} of A^{\prime} with respect to x^{\prime} as follows:

\begin{aligned} Q_{x^{\prime}}^{\prime} & =A_{1} \bar{y}_{1}^{\prime}+A_{3} \bar{y}_{3}^{\prime} \\ & =(20 \times 80)(24)+(14 \times 40)(7)=42.3 \times 10^{3} \mathrm{~mm}^{3} \end{aligned}

Alternative Solution. We first note that since the centroid C of A is located on the x^{\prime} axis, the first moment Q_{x^{\prime}} of the entire area A with respect to that axis is zero:

Q_{x^{\prime}}=A \bar{y}^{\prime}=A(0)=0

Denoting by A^{\prime \prime} the portion of A located below the x^{\prime} axis and by Q_{x^{\prime}}^{\prime \prime} its first moment with respect to that axis, we have therefore

Q_{x^{\prime}}=Q_{x^{\prime}}^{\prime}+Q_{x^{\prime}}^{\prime \prime}=0 \quad \text { or } \quad Q_{x^{\prime}}^{\prime}=-Q_{x^{\prime}}^{\prime \prime}

which shows that the first moments of A^{\prime} and A^{\prime \prime} have the same magnitude and opposite signs. Referring to Fig. A.14, we write

Q_{x^{\prime}}^{\prime \prime}=A_{4} \bar{y}_{4}^{\prime}=(40 \times 46)(-23)=-42.3 \times 10^{3} \mathrm{~mm}^{3}

and

Q_{x^{\prime}}^{\prime}=-Q_{x^{\prime}}^{\prime \prime}=+42.3 \times 10^{3} \mathrm{~mm}^{3}