솔루션피아

$$\title{Perpendicular Axis Theorem}$$$$ \begin{aligned}I_z&=\int r^2\dd m\\&=\int \br{x^2+y^2}\dd m\\&=\int {x^2}\dd m+\int {y^2}\dd m\\&=I_x+I_y\\\end{aligned} $$

$$ \begin{aligned} \theta&=\omega t, \end{aligned} $$$$ \begin{aligned} \vec r&=r\cos\theta\i+r\sin\theta\j\\ &=r\cos(\omega t)\i+r\sin(\omega t)\j\taag1\\ \end{aligned} $$$$ \begin{aligned} \vec v&=\dyt{\vec r}\\ &=\dt\bra{r\cos(\omega t)\i+r\sin(\omega t)\j}\\ &=-r\omega \sin(\omega t)\i+r\omega \cos(\omega t)\j\taag2\\ \end{aligned} $$$$ \begin{aligned} \vec \omega &= \omega \k\taag3 \end{ali..

$$\title{Rotational Inertia of Ring Cylinder}$$ $$ \put \begin{cases} R_i:\text{Inner Radius}\\ R_o:\text{Outer Radius}\\ \end{cases} $$ $$ \rho=\frac{\dd m}{\dd v}=\frac{M}{V}=\frac{M}{AH}=\frac{M}{\pi H( {R_o}^2- {R_i}^2) } \taag1$$ $$ \begin{aligned} l&=r\theta,\\ \dd l&=r \dd \theta\\ \end{aligned} $$ $$ \begin{aligned} \dd a&=\dd l\cdot\dd r \\ &=(r \dd \theta)\cdot\dd r \\ &=r \cdot \dd r\..

$$\title{Rotational Inertia of Ring}$$$$ \put \begin{cases} R_i:\text{Inner Radius}\\ R_o:\text{Outer Radius} \end{cases} $$$$ \sigma=\frac{\dd m}{\dd a}=\frac{M}{A}=\frac{M}{\pi {R_o}^2-\pi {R_i}^2} \taag1$$$$ \begin{aligned} l&=r\theta,\\ \dd l&=r \dd \theta\\ \end{aligned} $$$$ \begin{aligned} \dd a&=\dd r \cdot \dd l\\ &=\dd r \cdot (r \dd\theta)\\ &=r\cdot\dd r \dd\theta\taag2\\ \end{aligne..

$$\title{Rotational Inertia of Solid Sphere}$$$$\put \begin{cases} x&=\text{Latitude Line}\\ y&=\text{Longitude Line}\\ \end{cases} $$$$ \begin{cases} r_x&=r\sin\theta\\r_y&=r\end{cases} $$$$ \begin{cases} x&=r_x \phi=(r\sin\theta)\phi\\y&=r_y \theta=r \theta\end{cases} $$$$ \begin{cases} \dd x&=r\sin\theta \dd \phi\\\dd y&=r \dd \theta\end{cases} $$$$r=r_x,$$$$ \rho=\frac{\dd m}{\dd v}=\frac{M}..

$$\title{Rotational Inertia of Hollow Sphere}$$ $$\put \begin{cases} x&=\text{Latitude Line}\\ y&=\text{Longitude Line}\\ \end{cases} $$ $$ \begin{cases} r_x&=R\sin\theta\\ r_y&=R \end{cases} $$ $$ \begin{cases} x&=r_x \phi=(R\sin\theta)\phi\\ y&=r_y \theta=R\theta \end{cases} $$ $$ \begin{cases} \dd x&=R\sin\theta \dd \phi\\ \dd y&=R \dd \theta \end{cases} $$ $$r=r_x,$$ $$ \sigma=\frac{\dd m}{\..

$$\title{Rotational Inertia of Cuboid}$$ $$ r^2=x^2+ y^2,$$ $$ \rho=\frac{\dd m}{\dd v}=\frac{M}{V}=\frac{M}{X Y H}\taag1$$ $$ \dd v=\dd x \dd y \dd h\taag2 $$ $$ \begin{aligned} \dd m&=\rho\cdot\dd v\\ &=\rho\cdot(\dd x \dd y \dd h)\\ &=\rho\cdot\dd x \dd y \dd h\taag3\\ \end{aligned} $$ $$ \begin{aligned} \dd I&={r}^2\cdot \dd m\\ &=(x^2+ y^2)\cdot(\rho\cdot\dd x \dd y \dd h)\\ &=\rho(x^2+ y^2..

$$\title{Rotational Inertia of Rectangle}$$ $$ r^2=x^2+ y^2,$$ $$ \sigma=\frac{\dd m}{\dd a}=\frac{M}{A}=\frac{M}{X Y}\taag1$$ $$ \dd a=\dd x \dd y\taag2 $$ $$ \begin{aligned} \dd m&=\sigma\cdot\dd a\\ &=\sigma\cdot(\dd x \dd y)\\ &=\sigma\cdot\dd x \dd y\taag3\\ \end{aligned} $$ $$ \begin{aligned} \dd I&={r}^2\cdot \dd m\\ &=(x^2+ y^2)\cdot(\sigma\cdot\dd x \dd y)\\ &=\sigma(x^2+ y^2)\cdot\dd x..

$$\title{Rotational Inertia of Rod}$$$$ \lambda=\frac{\dd m}{\dd l}=\frac{M}{L}\taag1$$$$ \dd m=\lambda\cdot\dd l $$$$ \begin{aligned} \dd I&={r}^2\cdot \dd m\\&={l}^2\cdot (\lambda\cdot\dd l)\\&=\lambda{l}^2\cdot \dd l\taag2\\\end{aligned} $$$$ \begin{aligned} I_{\text{Rod}}&=\int_L \dd I\\&=\int_{-\frac{L}{2}}^{\frac{L}{2}} \lambda{l}^2 \dd l\\&=2\cdot\lambda\cdot\int_{0}^{\frac{L}{2}} {l}^2 \..

$$\title{Rotational Inertia of Solid Cylinder}$$ $$ \rho=\frac{\dd m}{\dd v}=\frac{M}{V}=\frac{M}{AH}=\frac{M}{\pi R^2 H} \taag1$$ $$ \begin{aligned} l&=r\theta,\\ \dd l&=r \dd \theta\\ \end{aligned} $$ $$ \begin{aligned} \dd a&=\dd l\cdot\dd r \\ &=(r \dd \theta)\cdot\dd r \\ &=r \cdot \dd r\dd \theta \taag2\\ \end{aligned} $$ $$ \begin{aligned} \dd v&=\dd a \cdot \dd h\\ &=(r \dd r \dd \theta)..