直方体の慣性モーメント1

直方体の体積は ${\displaystyle V=abc}$ であるので，密度 ${\displaystyle \rho }$ は

${\displaystyle \rho =\frac{M}{V}=\frac{M}{abc}}$

である．図のような微小部分を考えると，微小部分の体積は ${\displaystyle dV=dxdydz}$ なので，微小部分の質量 ${\displaystyle dm}$ は

${\displaystyle dm=\rho dV=\frac{M}{abc}dxdydz}$

と表される．微小部分の回転半径は ${\displaystyle r=\sqrt{x^2+y^2}}$ である．積分範囲は， ${\displaystyle -\frac{a}{2}\le x\le \frac{a}{2}}$ ， ${\displaystyle -\frac{b}{2}\le y\le \frac{b}{2}}$ ， ${\displaystyle -\frac{c}{2}\le z\le \frac{c}{2}}$ である．よって， ${\displaystyle z}$ 軸まわりの慣性モーメントは

${\displaystyle I={\displaystyle {\int }_M{r}^{2}dm}={\displaystyle {\int }_V\left(x^2+y^2\right)\times \frac{M}{abc}dxdydz}}$

${\displaystyle \phantom{I}=\frac{M}{abc}{\displaystyle {\int }_{-c/2}^{c/2}\left\{{\displaystyle {\int }_{-b/2}^{b/2}\left({\displaystyle {\int }_{-a/2}^{a/2}\left(x^2+y^2\right)dx}\right)dy}\right\}}dz}$

${\displaystyle \phantom{I}=\frac{M}{abc}{\displaystyle {\int }_{-c/2}^{c/2}\left\{{\displaystyle {\int }_{-b/2}^{b/2}{\left[\frac{1}{3}{x}^3+x{y}^2\right]}_{-a/2}^{a/2}dy}\right\}}dz}$

${\displaystyle \phantom{I}=\frac{M}{abc}{\displaystyle {\int }_{-c/2}^{c/2}\left\{{\displaystyle {\int }_{-b/2}^{b/2}\left(\frac{1}{12}{a}^3+a{y}^2\right)dy}\right\}}dz}$ ${\displaystyle =\frac{M}{abc}{\displaystyle {\int }_{-c/2}^{c/2}{\left[\frac{1}{12}{a}^{3}y+\frac{1}{3}a{y}^3\right]}_{-b/2}^{b/2}}dz}$

${\displaystyle \phantom{I}=\frac{M}{abc}{\displaystyle {\int }_{-c/2}^{c/2}\left(\frac{1}{12}{a}^{3}b+\frac{1}{12}a{b}^3\right)}dz}$ ${\displaystyle =\frac{M}{abc}{\left[\frac{1}{12}{a}^{3}bz+\frac{1}{12}a{b}^{3}z\right]}_{-c/2}^{c/2}}$

${\displaystyle \phantom{I}=\frac{M}{abc}\left(\frac{1}{12}{a}^{3}bc+\frac{1}{12}a{b}^{3}c\right)}$ ${\displaystyle =\frac{M}{12}\left(a^2+b^2\right)}$

となる．

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