微小輪要素の慣性モーメント:$dJ_z$ \[\begin{align} dJ_z &=\rho2\pi rdr t \times r^2\\ &=2\pi \rho tr^3\times dr \end{align}\] \[\begin{align} J_z =\int dJ_Z &=\int^{R}_{0}2\pi \rho t r^3 dr\\ &=2\pi \rho t\int^R_0r^3 dr\\ &=2\pi \rho t \left[\frac{r^4}{4} \right]^R_0\\ &=2\pi \rho t \times\frac{R^4}{4}\\ &=\pi R^2 \rho t \times\frac{R^2}{2}\\ &=m\times\frac{R^2}{2} \end{align}\] \[ \therefore m=\pi R^2 \rho t \] 直交軸の定理より ， \[ J_z=J_x+J_y \] 円板は対象なので， \[ J_x=J_y\ , \ J_z=2J_x=2J_y \] したがって， \[ J_x=J_y=\frac{J_z}{2}=\frac{1}{2}\times m\frac{R^2}{2}=m\times\frac{R^2}{4} \] \[ \therefore J_x=m\times \frac{R^2}{4}\\ \therefore J_y=m\times \frac{R^2}{4}\\ \therefore J_z=m\times \frac{R^2}{2}\\ \therefore 単位:\ kg\cdot m^2\\ \]