Question A.4: For the rectangular area of Fig. A.12a, determine (a) the mo......
For the rectangular area of Fig. A.12a, determine (a) the moment of inertia I_x of the area with respect to the centroidal x axis, (b) the corresponding radius of gyration r_x.
a. Moment of Inertia \mathbf{I_x}. We choose to select a horizontal strip of length b and thickness dy (Fig. A.12b). Since all of the points within the strip are at the same distance y from the x axis, the moment of inertia of the strip with respect to that axis is
d I_x=y^2 d A=y^2(b d y)
Integrating from y=-h / 2 \text { to } y=+h / 2,
\begin{aligned}I_x=\int_A y^2 d A & =\int_{-h / 2}^{+h / 2} y^2(b d y)=\frac{1}{3} b\left[y^3\right]_{-h / 2}^{+h / 2} \\& =\frac{1}{3} b\left(\frac{h^3}{8}+\frac{h^3}{8}\right)\end{aligned}
or
I_x=\frac{1}{12} b h^3
b. Radius of Gyration \mathbf{r_x}. From Eq. (A.10),
I_x=r_x^2 A \quad \frac{1}{12} b h^3=r_x^2(b h)
and solving for r_x gives
r_x=h / \sqrt{12}