$m\={\int }_0^b{\int }_0^{c x\/b}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_b^a{\int }_0^{c \left(x-a\right)\/\left(b-a\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\=a c\/2$

$I_x\={\int }_0^b{\int }_0^{c x\/b}{y}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_b^a{\int }_0^{c \left(x-a\right)\/\left(b-a\right)}{y}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x\=a c^3\/12$

$I_y\={\int }_0^b{\int }_0^{c x\/b}{x}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_b^a{\int }_0^{c \left(x-a\right)\/\left(b-a\right)}{x}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x\=a c \left(a^2\+a b\+b^2\right)\/12$

$R_x\=\sqrt{\frac{I_y}{m}}\=\sqrt{\frac{a c \left(a^2\+a b\+b^2\right)\/12}{a c\/2}}$ = $\sqrt{\frac{a^2\+a b\+b^2}{6}}$ 

$R_y\=\sqrt{\frac{I_x}{m}}\=\sqrt{\frac{a c^3\/12}{a c\/2}}$ = $c\/\sqrt{6}$

Table 6.6.3(a)   Moments of inertia and radii of gyration

Obtain the equations of the edges of the triangle in Figure 6.6.3(a)

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$\left[0\,0\right]\,\left[b\,c\right]$$\stackrel{\text{equation of line}}{\to }$$y\=\frac{cx}{b}$

$\left[a\,0\right]\,\left[b\,c\right]$$\stackrel{\text{equation of line}}{\to }$$y\=-\frac{cx}{a-b}\+\frac{ca}{a-b}$$\stackrel{\text{simplify}}{\=}$$y\=\frac{c\left(a-x\right)}{a-b}$

Total mass

${\int }_0^b{\int }_0^{c x\/b}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_b^a{\int }_0^{c \left(x-a\right)\/\left(b-a\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x$

$\frac{1}{2}cb\+\frac{1}{2}\frac{c\left(a^2-b^2\right)}{b-a}-\frac{ca\left(a-b\right)}{b-a}$

$\stackrel{\text{simplify}}{\=}$

$\frac{1}{2}ca$

$\stackrel{\text{assign to a name}}{\to }$

$m$

Second Moments (Moments of Inertia)

${\int }_0^b{\int }_0^{c x\/b}{y}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_b^a{\int }_0^{c \left(x-a\right)\/\left(b-a\right)}{y}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$

$\frac{1}{12}{c}^{3}b\+\frac{1}{12}\frac{c^3\left(a^4-b^4\right)}{{\left(b-a\right)}^3}-\frac{1}{3}\frac{c^{3}a\left(a^3-b^3\right)}{{\left(b-a\right)}^3}\+\frac{1}{2}\frac{c^3{a}^2\left(a^2-b^2\right)}{{\left(b-a\right)}^3}-\frac{1}{3}\frac{c^3{a}^3\left(a-b\right)}{{\left(b-a\right)}^3}$

$\stackrel{\text{simplify}}{\=}$

$\frac{1}{12}{c}^{3}a$

$\stackrel{\text{assign to a name}}{\to }$

$\mathrm{Ix}$

${\int }_0^b{\int }_0^{c x\/b}{x}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_b^a{\int }_0^{c \left(x-a\right)\/\left(b-a\right)}{x}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$

$\frac{1}{4}c{b}^3\+\frac{1}{4}\frac{c\left(a^4-b^4\right)}{b-a}-\frac{1}{3}\frac{ca\left(a^3-b^3\right)}{b-a}$

$\stackrel{\text{simplify}}{\=}$

$\frac{1}{12}\left(a^2\+ab\+b^2\right)ca$

$\stackrel{\text{assign to a name}}{\to }$

$\mathrm{Iy}$

Radii of gyration

$R_x\=\sqrt{\mathrm{Iy}\/m}$

$R_x\=\frac{1}{6}\sqrt{6a^2\+6ab\+6b^2}$

$R_y\=\sqrt{\mathrm{Ix}\/m}$

$R_y\=\frac{1}{6}\sqrt{6}\sqrt{c^2}$

$\stackrel{\text{assuming positive}}{\to }$

$R_y\=\frac{1}{6}\sqrt{6}c$

Table 6.6.3(a)   Moments of inertia and radii of gyration for the triangle in Figure 6.6.3(a)

Initialize

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Obtain the total area of the triangle

$$\begin{gathered} \mathrm{MultiInt}\left(1\&comma;\left[x\&comma;y\right]\&equals;\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨a\&comma;0⟩\&comma;⟨b\&comma;c⟩\right)\&comma;\mathrm{output}\&equals;\mathrm{integral}\right) assuming b\&gt;0\&comma;b<a\&comma;c\&gt;0\&semi; \\ m≔\mathrm{simplify}\left(\mathrm{MultiInt}\left(1\&comma;\left[x\&comma;y\right]\&equals;\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨a\&comma;0⟩\&comma;⟨b\&comma;c⟩\right)\right)\right) assuming b\&gt;0\&comma;b<a\&comma;c\&gt;0 \end{gathered}$$

$\frac{1}{2}ac$

$I_x$ 

$$\begin{gathered} \mathrm{MultiInt}\left(y^2\&comma;\left[x\&comma;y\right]\&equals;\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨a\&comma;0⟩\&comma;⟨b\&comma;c⟩\right)\&comma;\mathrm{output}\&equals;\mathrm{integral}\right) assuming b\&gt;0\&comma;b<a\&comma;c\&gt;0\&semi; \\ \mathrm{Ix}≔\mathrm{simplify}\left(\mathrm{MultiInt}\left(y^2\&comma;\left[x\&comma;y\right]\&equals;\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨a\&comma;0⟩\&comma;⟨b\&comma;c⟩\right)\right)\right) assuming b\&gt;0\&comma;b<a\&comma;c\&gt;0 \end{gathered}$$

$\frac{1}{12}{c}^{3}a$

$I_y$ 

$$\begin{gathered} \mathrm{MultiInt}\left(x^2\&comma;\left[x\&comma;y\right]\&equals;\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨a\&comma;0⟩\&comma;⟨b\&comma;c⟩\right)\&comma;\mathrm{output}\&equals;\mathrm{integral}\right) assuming b\&gt;0\&comma;b<a\&comma;c\&gt;0\&semi; \\ \mathrm{Iy}≔\mathrm{simplify}\left(\mathrm{MultiInt}\left(x^2\&comma;\left[x\&comma;y\right]\&equals;\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨a\&comma;0⟩\&comma;⟨b\&comma;c⟩\right)\right)\right) assuming b\&gt;0\&comma;b<a\&comma;c\&gt;0 \end{gathered}$$

$\frac{1}{12}\left(a^2\&plus;ab\&plus;b^2\right)ca$

Radii of gyration

$R_x\&equals;\mathrm{sqrt}\left(\mathrm{Iy}\&sol;m\right)$

$R_x\&equals;\frac{1}{6}\sqrt{6a^2\&plus;6ab\&plus;6b^2}$

$R_y\&equals;\mathrm{simplify}\left(\mathrm{sqrt}\left(\mathrm{Ix}\&sol;m\right)\right) assuming c\&gt;0$

$R_y\&equals;\frac{1}{6}\sqrt{6}c$