11. 강체

11.1 서론

강체:

  • 구성 입자의 상대적 거리가 고정된 물체.
(1)
\begin{align} \vec{v}_\alpha = \vec{V}_\mathrm{CM} + \vec{v}_r + \vec{\omega} \times \vec{r}_\alpha \end{align}

강체계에서는 $\vec{v}_r = 0$

(2)
\begin{align} T_\alpha & = {1 \over 2 } m_\alpha {v_\alpha}^2 = {1 \over 2} \sum_\alpha m_\alpha ( \vec{V} + \vec{\omega} \times \vec{r}_n )^2 \\ & = {1 \over 2} \sum_\alpha m_\alpha V^2 + \sum_\alpha m_\alpha \vec{V} \vec{\omega} \times \vec{r} + {1 \over 2} \sum_\alpha m_\alpha ( \vec{\omega} \times \vec{r}_\alpha )^2 \\ & = T_\mathrm{fixed} + T_\mathrm{rot} \\ & = {1 \over 2} M V^2 + {1 \over 2} \sum_\alpha m_\alpha ( \vec{\omega} \times \vec{r}_\alpha )^2 \end{align}

$T_\mathrm{rot}$가 강체계에서의 main 에너지

(3)
\begin{align} {1 \over 2} \sum_\alpha m_\alpha ( \vec{\omega} \times \vec{r}_\alpha )^2 & = {1 \over 2} \sum_\alpha \left[ \omega^2 {r_\alpha}^2 - ( \vec{\omega} \cdot \vec{r}_\alpha )^2 \right] \\ & = {1 \over 2} \sum_\alpha m_\alpha \left[ \left( \sum_i {\omega_i}^2 \right) \left( \sum_i x_{\alpha, i} \right) - \left( \sum_i \omega_i x_{\alpha, i} \right) \left( \sum_j \omega_j x_{\alpha, j} \right) \right] \\ & = {1 \over 2} \sum_\alpha \sum_{ij} m_\alpha \left[ \omega_i \omega_j \delta_{ij} \left( \sum_i x_{ij} \right) - \omega_i \omega_j x_{\alpha, i} x_{\alpha, j} \right] \\ & = {1 \over 2} \sum_{ij} \omega_i \omega_j \sum_\alpha m_\alpha ( \delta_{ij} \sum_{k} {x_{\alpha, k}}^2 - x_{\alpha, i} x_{\alpha, j} ) \end{align}
(4)
\begin{align} T & = {1 \over 2} \cdot I \omega^2 \\ I_{ij} & = \sum_\alpha m_\alpha ( \delta_{ij} \sum_k {x_{\alpha, k} }^2 - x_{\alpha, i} x_{\alpha, j} ) \end{align}
(5)
\begin{align} T_\mathrm{rot} = {1 \over 2} \sum_{ij} I_{ij} \omega_i \omega_j \end{align}
(6)
\begin{align} \vec{L} & = \sum_\alpha \vec{r}_\alpha \times \vec{p}_\alpha \\ & \qquad \vec{p}_\alpha = m_\alpha \vec{v}_\alpha = m_\alpha \vec{\omega} \times \vec{r}_\alpha \\ & = \sum_\alpha m_\alpha \vec{r}_\alpha \times ( \vec{\omega} \times \vec{r}_\alpha ) \\ & = \sum_\alpha m_\alpha \left[ {r_\alpha}^2 \vec{\omega} - \vec{r}_\alpha ( \vec{r}_\alpha \vec{\omega}) \right] \\ L_i & = \sum_\alpha m_\alpha \left( \omega_i \sum_k {x_{\alpha, k}}^2 - x_{\alpha, i} \sum_{j} x_{ij} \omega_{j} \right) \\ & = \sum_j \omega_j \sum_\alpha m_\alpha \left( \delta_{ij} \sum_k {x_{\alpha, k}}^2 - x_{\alpha, i} x _{\alpha, j} \right) \end{align}
(7)
\begin{align} \longrightarrow \begin{cases} \vec{L} & = \left\{ \vec{I} \right\} \vec{\omega} \\ L_i & = \sum_j I_{ij} \omega_j \end{cases} \end{align}
(8)
\begin{align} T_\mathrm{rot} & = {1 \over 2} \sum_{i,j} I_{ij} \omega_i \omega_j \\ & = {1 \over 2} \sum_i \omega_i L_i \\ & \implies \begin{cases} T_\mathrm{rot} & = {1 \over 2} \vec{\omega} \cdot \vec{L} = {1 \over 2} \vec{\omega} \left\{ \vec{I} \right\} \vec{\omega} \\ \vec{L} & = \left\{ \vec{I} \right\} \vec{L}_i \end{cases} \end{align}

예제 11.4:

11-5.png
(9)
\begin{align} \vec{\omega} = \omega_3 \hat{e}_3 = \dot{\theta} \hat{e}_3 = \begin{pmatrix} 0 \\ 0 \\ \omega_3 \end{pmatrix} \end{align}
(10)
\begin{align} I_{ij} = m_1 ( \delta_{ij} {x_{1, 1}}^2 - x_{1,i} x_{1, j} ) + m_2 ( \delta_{ij} {x_{2, 1}}^2 - x_{2,i} x_{2,j} ) \end{align}

11.2 단순평면운동

11.3 관성텐서

11.4 각운동량

11.5 관성 주축

11.6 다른 강체좌표계에 대한 관성모멘트

11.7 관성 텐서의 기타 성질

11.8 오일러 각

11.9 강체에 대한 오일러 방정식

11.10 대칭 팽이의 자유운동

11.11 고정점 주위의 대칭 팽이의 운동

11.12 강체 회전의 안정성