View source for Circular motion/Notes

==Acceleration==
* {{MM|a(t)\eq p(t)\cdot\left(\frac{r' '(t)}{r(t)}-(\theta'(t))^2\right)+\big(\theta' '(t)\cdot r(t)+2\theta'(t)\cdot r'(t)\big)\cdot\left[\begin{array}{r}-\sin(\theta(t)) \\ \cos(\theta(t))\end{array}\right]}}, or:
** substituting in {{M|p(t)}} by it's definition:
*** {{MM|a(t)\eq \left(\frac{r' '(t)}{r(t)}-(\theta'(t))^2\right)\cdot\left[\begin{array}{r}r(t)\cdot\cos(\theta(t)) \\ r(t)\cdot\sin(\theta(t))\end{array}\right]+\big(\theta' '(t)\cdot r(t)+2\theta'(t)\cdot r'(t)\big)\cdot\left[\begin{array}{r}-\sin(\theta(t)) \\ \cos(\theta(t))\end{array}\right]}}
*** However in many special cases it is useful to consider the first form with {{M|p(t)}} in it.

===Special cases===
# unchanging radius, {{M|r(t):\eq r_0\in\mathbb{R}_{>0} }}
#* obviously, now {{M|r'(t)\eq 0}} and {{M|r' '(t)\eq 0}}, thus:
#** {{M|a(t)\eq -(\theta' '(t))^2\cdot p(t)+\big(\theta' '(t)\cdot r(t)\big)\cdot\left[\begin{array}{r}-\sin(\theta(t)) \\ \cos(\theta(t))\end{array}\right]}}
#**: {{M|\eq \theta' '(t)\left(r(t)\cdot\left[\begin{array}{r}-\sin(\theta(t)) \\ \cos(\theta(t))\end{array}\right]-\theta' '(t)\cdot p(t)\right)}}
#** But notice:
#*** {{M|a(t)\eq \theta' '(t)\left(\left[\begin{array}{r}-r(t)\cdot\sin(\theta(t)) \\ r(t)\cdot\cos(\theta(t))\end{array}\right]-\theta' '(t)\cdot p(t)\right)}}
#***: {{M|\eq \theta' '(t)\left(\left[\begin{array}{r} -p_y(t)\\p_x(t)\end{array}\right]-\theta' '(t)\cdot p(t)\right)}}
There must be a geometric interpretation for this! As the vector here is {{M|p(t)}} reflected in the line {{M|x\eq 0}}!
* {{XXX|This is wrong actually}} as the {{M|x}} component comes from cos in {{M|p(t)}}, not sin. Which is not a reflection in {{M|x\eq 0}}, a matrix might be able to represent this linear combination better
** I've corrected the formula [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 18:56, 13 September 2018 (UTC)