For the rectangular area of Fig. A.16, determine (a) the moment of inertia Ix of the area

Question

For the rectangular area of Fig. A.16, determine [latex](a)[/latex] the moment of inertia [latex]I_{x}[/latex] of the area with respect to the centroidal [latex]x[/latex] axis, [latex](b)[/latex] the corresponding radius of gyration [latex]r_{x}[/latex].

Accepted Answer

(a) Moment of Inertia [latex]I_{x}[/latex]. We select as an element of area a horizontal strip of length [latex]b[/latex] and thickness [latex]d y[/latex] (Fig. A.17). Since all the points within the strip are at the same distance [latex]y[/latex] from the [latex]x[/latex] axis, the moment of inertia of the strip with respect to that axis is

[latex]d I_{x}=y^{2} d A=y^{2}(b d y)[/latex]

Integrating from [latex]y=-h / 2[/latex] to [latex]y=+h / 2[/latex], we write

[latex] \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} ight]_{-h / 2}^{+h / 2} \ & =\frac{1}{3} b\left(\frac{h^{3}}{8}+\frac{h^{3}}{8} ight) \end{aligned} [/latex]

or

[latex]I_{x}=\frac{1}{12} b h^{3}[/latex]

(b) Radius of Gyration [latex]r_{x}[/latex]. From Eq. (A.10), we have

[latex]I_{x}=r_{x}^{2} A \quad \frac{1}{12} b h^{3}=r_{x}^{2}(b h)[/latex]

and, solving for [latex]r_{x}[/latex],

[latex]r_{x}=h / \sqrt{12}[/latex]

Question A.04: For the rectangular area of Fig. A.16, determine (a) the mom...

Related Answered Questions

Determine the moment of inertia Ix of the area shown with respect to the centroidal x axis ...

Verified Answer:

For the circular area of Fig. A.18, determine (a) the polar moment of inertia JO, (b) the ...

Verified Answer:

Referring to the area A of Example A.02, we consider the horizontal x′ axis through its ...

Verified Answer:

Locate the centroid C of the area A shown in Fig. A.10. ...

Verified Answer:

For the triangular area of Fig. A.7, determine (a) the first moment Qx of the area with ...

Verified Answer: