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加速度の大きさ

作成: 2022-10-12 更新: 2022-10-12

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加速度 \( {\bf{a}} = ( a_x,\ a_y,\ a_z ) \) は， \[ a_x = ( - \frac{v^2}{R} + R \Omega^2 )\ cos( P_a ) \\ a_y = ( - \frac{v^2}{R} + R \Omega^2 )\ cos( S_a )\ sin( P_a ) \\ a_z = - \frac{v^2}{R}\ sin( S_a )\ sin( P_a ) \\ \] よって， \[ \begin{align} | {\bf{a}} |^2 = & ( - \frac{v^2}{R} + R \Omega^2 )^2\ cos( P_a )^2 \\ & + ( - \frac{v^2}{R} + R \Omega^2 )^2\ cos( S_a )^2\ sin( P_a )^2 \\ & + ( \frac{v^2}{R} )^2\ sin( S_a )^2\ sin( P_a )^2 \\ \end{align} \] ここで，はじめの２項の和は， \[ ( - \frac{v^2}{R} + R \Omega^2 )^2\ cos( P_a )^2 \\ + ( - \frac{v^2}{R} + R \Omega^2 )^2\ cos( S_a )^2\ sin( P_a )^2 \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( cos( P_a )^2 + cos( S_a )^2\ ( 1 - cos( P_a )^2 ) ) \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( cos( P_a )^2\ ( 1 - cos( S_a )^2 ) + cos( S_a )^2 ) \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( cos( P_a )^2\ sin( S_a )^2 + cos( S_a )^2 ) \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( cos( P_a )^2\ sin( S_a )^2 + ( 1 - sin( S_a )^2 ) ) \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( 1 - sin( S_a )^2 + cos( P_a )^2\ sin( S_a )^2 ) \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( 1 - sin( S_a )^2\ ( 1 - cos( P_a )^2 )) \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( 1 - sin( S_a )^2\ sin( P_a )^2 ) \\ \] よって， \[ | {\bf{a}} |^2 \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ cos( P_a )^2 \\ + ( - \frac{v^2}{R} + R \Omega^2 )^2\ cos( S_a )^2\ sin( P_a )^2 \\ + ( \frac{v^2}{R} )^2\ sin( S_a )^2\ sin( P_a )^2 \\ \ \\ \ \\ = ( - \frac{v^2}{R} + R \Omega^2 )^2\ ( 1 - sin( S_a )^2\ sin( P_a )^2 ) \\ \ \\ + ( \frac{v^2}{R} )^2\ sin( S_a )^2\ sin( P_a )^2 \\ \]