Coriolis力

導出

2次元での導出

$ \bm e_x'=\cos\theta\bm e_x+\sin\theta\bm e_y

$ \bm e_y'=-\sin\theta\bm e_x+\cos\theta\bm e_y

$ \frac{\mathrm d}{\mathrm dt}(m\dot\bm r)=\bm F

$ \dot\bm e_x'=\dot\theta\bm e_y'

$ \dot\bm e_y'=-\dot\theta\bm e_x'

座標変換

$ [\bm r]^{\sf E'}=[\bm I]^{\sf E'E}[\bm r]^{\sf E}

$ \dot{[\bm I]^{\sf E'E}}=\frac{\mathrm d}{\mathrm dt}\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}

$ =\dot\theta\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}

$ =\dot\theta\begin{pmatrix}0&1\\-1&0\end{pmatrix}[\bm I]^{\sf E'E}

$ m\ddot{[\bm r]^{\sf E}}=[\bm F]^{\sf E}\quad\because\bm e_\bullet=\rm const.

$ \iff m[\bm I]^{\sf E'E}\left(\ddot{[\bm I]^{\sf EE'}}[\bm r]^{\sf E'}+2\dot{[\bm I]^{\sf EE'}}\dot{[\bm r]^{\sf E'}}+[\bm I]^{\sf EE'}\ddot{[\bm r]^{\sf E'}}\right)=[\bm F]^{\sf E'}

$ \iff m[\bm I]^{\sf E'E}\left(\ddot\theta[\bm I]^{\sf E'E}\begin{pmatrix}0&-1\\1&0\end{pmatrix}[\bm r]^{\sf E'}-\dot\theta^2[\bm I]^{\sf E'E}[\bm r]^{\sf E'}+2\dot\theta[\bm I]^{\sf EE'}\begin{pmatrix}0&-1\\1&0\end{pmatrix}\dot{[\bm r]^{\sf E'}}+[\bm I]^{\sf EE'}\ddot{[\bm r]^{\sf E'}}\right)=[\bm F]^{\sf E'}

$ \iff m\left(\ddot\theta\begin{pmatrix}0&-1\\1&0\end{pmatrix}[\bm r]^{\sf E'}-\dot\theta^2[\bm r]^{\sf E'}+2\dot\theta\begin{pmatrix}0&-1\\1&0\end{pmatrix}\dot{[\bm r]^{\sf E'}}+\ddot{[\bm r]^{\sf E'}}\right)=[\bm F]^{\sf E'}

$ \iff m\ddot{[\bm r]^{\sf E'}}=[\bm F]^{\sf E'}+\underbrace{m\dot\theta^2[\bm r]^{\sf E'}}_\text{遠心力}+\underbrace{2m\dot\theta\begin{pmatrix}0&1\\-1&0\end{pmatrix}\dot{[\bm r]^{\sf E'}}}_\text{Coriolis力}+\underbrace{m\ddot\theta\begin{pmatrix}0&1\\-1&0\end{pmatrix}[\bm r]^{\sf E'}}_\text{Euler力}

回転速度を上げると、後ろに引っ張られる

$ \iff \dot m[\bm I]^{\sf E'E}(\dot{[\bm I]^{\sf EE'}}[\bm r]^{\sf E'}+[\bm I]^{\sf EE'}\dot{[\bm r]^{\sf E'}})+m[\bm I]^{\sf E'E}\left(\ddot{[\bm I]^{\sf EE'}}[\bm r]^{\sf E'}+2\dot{[\bm I]^{\sf EE'}}\dot{[\bm r]^{\sf E'}}+[\bm I]^{\sf EE'}\ddot{[\bm r]^{\sf E'}}\right)=[\bm F]^{\sf E'}

$ \iff \frac{\mathrm d}{\mathrm dt}\left(m\dot{[\bm r]^{\sf E'}}\right)=[\bm F]^{\sf E'}+\underbrace{m\dot\theta^2[\bm r]^{\sf E'}}_\text{遠心力}+\underbrace{2m\dot\theta\begin{pmatrix}0&1\\-1&0\end{pmatrix}\dot{[\bm r]^{\sf E'}}}_\text{Coriolis力}+\underbrace{\frac{\mathrm d}{\mathrm dt}\left(m\dot\theta\right)\begin{pmatrix}0&1\\-1&0\end{pmatrix}[\bm r]^{\sf E'}}_\text{Euler力}

References