User:John Morrison~enwiki/My Code Store

$$\ h(t)= A_0 + \sum_{j=1}^m A_j\ cos\ \omega_j\ t\ +\ \sum_{j=1}^m B_j\ sin\ \omega_j\ t \,$$

$$\ [X] = \begin{bmatrix} 1 & cos \omega_1\ t_1 & sin \omega_1\ t_1 \\ 1 & cos \omega_1\ t_2 & sin \omega_1\ t_2 \\ 1 & cos \omega_1\ t_3 & sin \omega_1\ t_3 \\ \cdots & \cdots & \cdots \\ 1 & cos \omega_1\ t_n & sin \omega_1\ t_n \end{bmatrix} \,$$

$$\ [Y] = \begin{bmatrix} h_1 \\ h_2 \\ h_3 \\ \vdots \\ h_n \end{bmatrix} \,$$

$$\ [SSX] = [X]'[X] = \begin{bmatrix} n & \sum_{i=1}^n cos \omega_1\ t_i & \sum_{i=1}^n sin \omega_1\ t_i \\ \sum_{i=1}^n cos \omega_1\ t_i & \sum_{i=1}^n cos^2 \omega_1\ t_i & \sum_{i=1}^n sin \omega_1\ t_i cos \omega_1\ t_i \\ \sum_{i=1}^n sin \omega_1\ t_i & \sum_{i=1}^n sin \omega_1\ t_i cos \omega_1\ t_i & \sum_{i=1}^n sin^2 \omega_1\ t_i \end{bmatrix} \,$$

$$\ [SXY] = [X]'[Y] = \begin{bmatrix} \sum_{i=1}^n h_i \\ \sum_{i=1}^n h_i cos \omega_1\ t_i \\ \sum_{i=1}^n h_i sin \omega_1\ t_i \end{bmatrix} \,$$

$$\ [A] = \begin{bmatrix} A_0 \\ A_j \\ B_j \\ \vdots \\ A_m \\ B_m \end{bmatrix} \,$$

$$\ RV=\frac{\sum [h(t) - h_0]^2}{\sum [h_t - h_0]^2}$$

$$\ SSQ= \sum_{}^N [h_t - h(t)]^2 $$

$$\ RMS= \frac{\sqrt{\sum_{}^N [h_t - h(t)]^2}}{N}$$

$$\ h(t)= A_0 + \sum_{j=1}^m R_j\ cos\ ( \omega_j t\ - \phi_j\ ) $$ $$\ A_0 $$ $$\ A_j $$ $$\ \omega_j\ $$ $$\ m $$ $$\ \phi_j\ $$ $$\ h_t $$ $$\ t $$ $$\ h(t) $$

$$\ [A]=[SSX]^{-1} [SXY] $$

$$\ R_j = \sqrt ( A_j ^2 + B_j ^2 ) $$ $$\ \phi_j\ = arctan (B_j / A_j ) $$

$$\ F = G\frac{m_1 m_2}{R^2} $$

$$\ F_M $$ $$\ F_E $$ $$\ m_M $$ $$\ m_E $$ $$\ R_M $$ $$\ R_E $$

$$\ F_M = G\frac{m_M}{ R_M ^2} $$ $$\ F_E = G\frac{m_E}{R_E ^2} $$ $$\ F_M = F_E\frac{m_M R_E ^ 2}{m_E R_M ^2} $$ $$\ F_E = g $$ $$\ F = ma $$ $$\ F_M = g\frac{m_M R_E ^2} {m_E R_M ^2} $$

$$\ F_M = g(\frac{1}{85})(\frac{1}{60.3}) $$

$$\ =0.0000034g $$

$$\ =3.3 \cdot 10^-5 \ N $$

$$\ F_M = 3.4 \cdot 10^{-6} g $$

$$\ F_M = (9.8)(3.4)\cdot 10^{-6} $$

$$\ =3.3 \cdot 10^{-5} kg \cdot ms^{-2} $$ $$\ =3.3 \cdot 10^{-5} $$

$$\ F_Y = F_M sin \theta $$ $$\ y = R_M sin \theta = R_E sin \phi $$ $$\ F_Y = F_M (\frac {R_E}{R_M}) sin \phi $$ $$\ F_Y = g \frac {m_M R_E ^3} {m_E R_M ^3} sin \phi $$

$$\ F $$ $$\ R^2 $$ $$\ m_1 $$ $$\ m_2 $$ $$\ G $$ $$\ t $$ $$\ \omega_j\ $$

$$\ F_C $$ $$\ h(t) = A_j\ cos\ \omega_j\ t\ $$

$$\ x $$ $$\ \phi\ $$ $$\ R_E $$ $$\ y $$ $$\ P $$ $$\ F_M $$ $$\ R_M $$ $$\ \theta\ $$ $$\ F_Y $$ $$\ F_X $$ $$\ perigee $$ $$\ apogee $$ $$\ spring $$ $$\ neap $$ $$\ g $$ $$\ h_0 $$ $$\ R_j $$ $$\ \phi_j\ $$ $$\ A_0,A_1,B_1,\cdots\,A_m,B_m $$ /n

$$\ 3.1 $$ $$\ 3.2 $$ $$\ 3.3 $$ $$\ 3.4 $$ $$\ 3.5 $$ $$\ 3.6 $$ $$\ 3.7 $$ $$\ 3.8 $$

$$\ N $$ $$\ ms^{-2} $$

$$\ F_M = 3.4 \cdot 10^{-6} g $$

$$\ F_M = (9.8)(3.4)\cdot 10^{-6} $$

$$\ =3.3 \cdot 10^{-5} kg \cdot ms^{-2} $$ $$\ =3.3 \cdot 10^{-5} N $$

$$\ 4.1 $$ $$\ 4.2 $$ $$\ 4.3 $$ $$\ 4.4 $$

$$\ 5.1 $$ $$\ 6.1 $$ $$\ 6.2 $$ $$\ 6.3 $$ $$\ 7.1 $$ $$\ 7.2 $$ $$\ 8.1 $$ $$\ 8.2 $$ $$\ 8.3 $$ $$\ 8.4 $$ $$\ 8.5 $$ $$\ 8.6 $$ $$\ 8.7 $$ $$\ 8.8 $$

$$

\begin{matrix} &&&&&1\\ &&&&1&&1\\ &&&1&&2&&1\\ &&1&&3&&3&&1\\ &1&&4&&6&&4&&1\\ &&&&&&&\cdot\cdot\cdot \end{matrix} $$ $$\ 8.9 $$ $$\ 8.10 $$



\begin{matrix} \qquad\quad\;\, x^2 \; - 9x \quad - 27\\ \qquad\quad x-3\overline{\vert x^3 - 12x^2 + 0x - 42}\\ \;\; \underline{\;\;x^3 - \;\;3x^2}\\ \qquad\qquad\quad\; -9x^2 + 0x\\ \qquad\qquad\quad\; \underline{-9x^2 + 27x}\\ \qquad\qquad\qquad\qquad\qquad -27x - 42\\ \qquad\qquad\qquad\qquad\qquad \underline{-27x + 81}\\ \qquad\qquad\qquad\qquad\qquad\qquad\;\; -123 \end{matrix} $$