가는 원형 고리의 회전 관성

$$\title{Rotational Inertia of Hoop}$$ $$r=R,$$ $$ \lambda=\frac{\dd m}{\dd l}=\frac{M}{L}=\frac{M}{2\pi r}=\frac{M}{2\pi R} \taag1$$ $$ \begin{aligned} l&=r\theta,\\ \dd l&=r\dd\theta\\ &=R\dd\theta\taag2 \end{aligned} $$ $$$$ $$ \begin{aligned} \dd m&=\lambda\cdot\dd l\\ &=\lambda\cdot (R\dd\theta)\\ &=\lambda R\cdot\dd\theta\taag3\\ \end{aligned} $$ $$ \begin{aligned} \dd I&=r^2\cdot \dd m\\ &=R^2\cdot \(\lambda R\dd\theta\)\\ &=\lambda R^3\cdot\dd\theta\taag4\\ \end{aligned} $$ $$ \begin{aligned} I_{\text{Hoop}}&=\oint_L \dd I\\ &=\int_0^{2\pi} \lambda R^3\cdot\dd\theta\\ &=\lambda \cdot R^3\cdot\int_0^{2\pi} \dd\theta\\ &=\(\frac{M}{2\pi R} \)\cdot R^3\cdot(2\pi)\\ &=MR^2\\ \end{aligned} $$