Circular motion

[ilmath]\newcommand{\Vecc}[2]{\left[\begin{array}{c} {#1} \\ {#2} \end{array}\right]} [/ilmath][ilmath]\newcommand{\dd}[3]{\frac{\mathrm{d} }{\mathrm{d}{#1} }\left[{#2}\right]\Big\vert_{#3} } [/ilmath]

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This and pendulum stuff

Notes

General polar motion

For convenience we will denote:

• [math]\dd{t}{\theta(t)}{t} [/math] as [ilmath]\theta'(t)[/ilmath] and [math]\dd{t}{\theta'(t)}{t} [/math] as [ilmath]\theta''(t)[/ilmath]
• [math]\dd{t}{r(t)}{t} [/math] as [ilmath]r'(t)[/ilmath] and [math]\dd{t}{r'(t)}{t} [/math] as [ilmath]r''(t)[/ilmath]

Let:

• [ilmath]p(t):\eq\Vecc{r(t)\ \!\cos(\theta(t))}{r(t)\ \!\sin(\theta(t))} [/ilmath] be used for [ilmath]p[/ilmath]osition - with respect to time
• [math]v(t):\eq\dd{t}{p(t)}{t} [/math]
  • [math]v(t)\eq\dd{t}{r(t)}{t}\cdot\Vecc{\cos(\theta(t))}{\sin(\theta(t))}\ +\ \dd{t}{\theta(t)}{t}\cdot r(t)\cdot \Vecc{-\sin(\theta(t))}{\cos(\theta(t))} [/math]

    [math]\eq r'(t)\Vecc{\cos(\theta(t))}{\sin(\theta(t))}+\theta'(t)\cdot r(t)\cdot\Vecc{-\sin(\theta(t))}{\cos(\theta(t))} [/math]

• [ilmath]a(t):\eq\dd{t}{v(t)}{t} [/ilmath]
  • [ilmath]a(t)\eq r''(t)\Vecc{\cos(\theta(t))}{\sin(\theta(t))} +2\theta'(t)r'(t)\Vecc{-\sin(\theta(t))}{\cos(\theta(t))} + \theta''(t)r(t)\Vecc{-\sin(\theta(t))}{\cos(\theta(t))} - \big(\theta'(t)\big)^2\Vecc{r(t)\cos(\theta(t))}{r(t)\sin(\theta(t))} [/ilmath]

    [math]\eq\frac{r''(t)}{r(t)}p(t)+\Vecc{-\sin(\theta(t))}{\cos(\theta(t))}\Big[\theta''(t)r(t)+2\theta'(t)r'(t)\Big] - \big(\theta'(t)\big)^2 p(t) [/math]

    [math]\eq p(t)\left[\frac{r''(t)}{r(t)} - \big(\theta'(t)\big)^2\right]+\Vecc{-\sin(\theta(t))}{\cos(\theta(t))}\Big[\theta''(t)r(t)+2\theta'(t)r'(t)\Big] [/math]

Work to do:

1. Reduce to circular case first by setting [ilmath]r(t)\eq c[/ilmath] for some constant [ilmath]c>0[/ilmath] and handle [ilmath]r(t)\eq 0[/ilmath] special cases.
2. Define [ilmath]\omega(t):\eq \theta'(t)[/ilmath]
   • Get to [ilmath]p''(t)\eq -\big(\omega(t)\big)^2\ \!p(t)[/ilmath]