$$\newcommand\v[1]{ \vec{#1} } \newcommand\pos[2]{\vec{r}^{\mathrm{#1}/\mathrm{#2}}} \newcommand\vel[2]{ {}^{\mathrm{#2}} \vec{v}^{\mathrm{#1}}} \newcommand\acc[2]{ {}^{\mathrm{#2}} \vec{a}^{\mathrm{#1}}} \newcommand\avel[2]{ {}^{\mathrm{#2}} \vec{\omega}^{\mathrm{#1}}} \newcommand\aacc[2]{ {}^{\mathrm{#2}} \vec{\alpha}^{\mathrm{#1}}} \newcommand\mass[1]{m_{\mathrm{#1}}} \newcommand\acctwo[4]{\acc{#3}{#2} + \aacc{#4}{#2} \times \pos{#1}{#3} + \avel{#4}{#2} \times (\avel{#4}{#2} \times \pos{#1}{#3})} \newcommand\aveladd[3]{ \avel{#2}{#1} + \avel{#3}{#2} } \newcommand\aaccadd[3]{ \aacc{#2}{#1} + \aacc{#3}{#2} + \avel{#2}{#1} \times \avel{#3}{#2} } \newcommand\expr[2]{ [#1]_{#2} } \newcommand\rot[2]{ {}^{\mathrm{#2}} R^{\mathrm{#1}} } \newcommand\q[1]{q_{\mathrm{#1}}} \newcommand\qd[1]{\dot{q}_{\mathrm{#1}}} \newcommand\qdd[1]{\ddot{q}_{\mathrm{#1}}} \newcommand\u[2]{ \hat{#1}_{\mathrm{#2}} } \newcommand\deriv[2]{ \frac{{}^{#1}d}{dt} #2}$$


In [1]:

from IPython.display import display, Math, Latex



%%latex \begin{align} \v{F} &= \mass{S} \acc{Scm}{N} \\ \v{R}^A + \v{F}^A + \v{F}^B + \v{F}^C &= \mass{A} \acc{Acm}{N} + \mass{B} \acc{Bcm}{N} + \mass{C} \acc{Ccm}{N} \\ \end{align}

A

%%latex \begin{align} \acc{Acm}{N} &= \acctwo{Acm}{N}{Ao}{A} \\ \acc{Ao}{N} &= \acc{No}{N} = \vec{0} \\ \avel{A}{N} &= \qd{A} \u{a}{z}\\ \aacc{A}{N} &= \deriv{N}{\avel{A}{N}} = \qdd{A} \u{a}{z}\\ \end{align}

B

%%latex \begin{align} \acc{Bcm}{N} &= \acctwo{Bcm}{N}{Bo}{B} \\ \acc{Bo}{N} &= \acctwo{Bo}{N}{Ao}{A} \\ \aacc{B}{N} &= \aaccadd{N}{A}{B} \\ \avel{B}{A} &= \qd{B} \u{b}{z} \\ \aacc{B}{A} &= \qdd{B} \u{b}{z} \\ \avel{B}{N} &= \aveladd{N}{A}{B} \\ \end{align}

C

%%latex \begin{align} \acc{Ccm}{N} &= \acctwo{Ccm}{N}{Co}{C} \\ \aacc{C}{N} &= \aaccadd{N}{B}{C} \\ \avel{C}{N} &= \aveladd{N}{B}{C} \\ \end{align}

# Position sweep

%%latex \begin{align} \expr{\pos{Acm}{Ao}}{N} = \rot{A}{N} \expr{\pos{Acm}{Ao}}{A} \end{align}

# Velocity sweep

%%latex \begin{align} \avel{A}{N} = \qd{A} \u{a}{z} \\ \avel{B}{A} = \qd{B} \u{b}{z} \\ \avel{C}{B} = \qd{C} \u{c}{z} \\ \end{align}