두꺼운 원기둥의 회전 관성

$$\title{Rotational Inertia of Ring Cylinder}$$ $$ \put \begin{cases} R_i:\text{Inner Radius}\\ R_o:\text{Outer Radius}\\ \end{cases} $$ $$ \rho=\frac{\dd m}{\dd v}=\frac{M}{V}=\frac{M}{AH}=\frac{M}{\pi H( {R_o}^2- {R_i}^2) } \taag1$$ $$ \begin{aligned} l&=r\theta,\\ \dd l&=r \dd \theta\\ \end{aligned} $$ $$ \begin{aligned} \dd a&=\dd l\cdot\dd r \\ &=(r \dd \theta)\cdot\dd r \\ &=r \cdot \dd r\dd \theta \taag2\\ \end{aligned} $$ $$ \begin{aligned} \dd v&=\dd a \cdot \dd h\\ &=(r \dd r \dd \theta) \cdot \dd h\\ &=r \cdot \dd r \dd \theta \dd h\taag3\\ \end{aligned} $$ $$ \begin{aligned} \dd m&=\rho\cdot\dd v\\ &=\rho\cdot (r \dd r \dd \theta \dd h)\\ &=\rho r\cdot\dd r \dd \theta \dd h\taag4\\ \end{aligned} $$ $$ \begin{aligned} \dd I&=r^2\cdot \dd m\\ &=r^2\cdot \(\rho r\dd r \dd \theta \dd h\)\\ &=\rho r^3\cdot\dd r \dd \theta \dd h\taag5\\ \end{aligned} $$ $$ \begin{aligned} I_{\text{Ring Cylinder}}&=\oiiint_V \dd I\\ &=\int_0^{H}\int_0^{2\pi}\int_{R_i}^{R_o} \rho r^3\dd r \dd \theta \dd h\\ &=\rho\cdot\(\int_{R_i}^{R_o} r^3\dd r\)\cdot\(\int_0^{2\pi} \dd \theta\)\cdot\(\int_0^{H} \dd h\)\\ &=\bra{\frac{M}{\pi H( {R_o}^2- {R_i}^2) }}\cdot\(\frac{{R_o}^4-{R_i}^4}{4}\)\cdot(2\pi)\cdot(H)\\ &=\frac{1}{2}M({R_i}^2+{R_o}^2)\\ \end{aligned} $$