Callisto
  • Callisto
How do the plane polar coordinates work?
OCW Scholar - Physics I: Classical Mechanics
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Callisto
  • Callisto
How do the plane polar coordinates work?
OCW Scholar - Physics I: Classical Mechanics
schrodinger
  • schrodinger
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experimentX
  • experimentX
you know polar coordinate system right!!|dw:1349364467374:dw|
Callisto
  • Callisto
Yes.. a little... But NOT in vectors!!
experimentX
  • experimentX
no they are just same.|dw:1349364550790:dw|

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Callisto
  • Callisto
|dw:1349364634178:dw|
experimentX
  • experimentX
we know that we write (x, y) as our point. In polar coordinate we write (r, theta) <--- this means this is a vector space. you have been using it unknowingly.
Callisto
  • Callisto
Since it's related to sine, and cosine, it's not linear translation, right?
experimentX
  • experimentX
|dw:1349364748021:dw|
experimentX
  • experimentX
yep. now let's go back to unit vector for a while. in cartesian how do you represent a vector?
Callisto
  • Callisto
xi + yj? i and j are unit vectors
experimentX
  • experimentX
yes ... and in cartesian coordinate?
Callisto
  • Callisto
Huh?! Are they different??
experimentX
  • experimentX
Woops sorry ... polar coordinate??
Callisto
  • Callisto
Well... Isn't it what I got stuck?
experimentX
  • experimentX
\[ r \hat r + \theta \hat \theta\]
experimentX
  • experimentX
now we need to find the relation between \( \hat r , \hat \theta \) and \( \hat i, \hat j \)
experimentX
  • experimentX
for that we know
experimentX
  • experimentX
|dw:1349365095537:dw|
experimentX
  • experimentX
|dw:1349365280570:dw|
experimentX
  • experimentX
|dw:1349365350758:dw|
experimentX
  • experimentX
how do we define unit vectors?
Callisto
  • Callisto
vector/length of vector?
experimentX
  • experimentX
yep ...
experimentX
  • experimentX
|dw:1349365586284:dw|
experimentX
  • experimentX
can you write the relation between \( \hat r \) and i,j
Callisto
  • Callisto
unit vector of r = (xi + yj) / (x^2 + y^2) ? I know it's strange...
experimentX
  • experimentX
|dw:1349365738630:dw|
Callisto
  • Callisto
Oh, should I express it in terms of theta?
experimentX
  • experimentX
yep
Callisto
  • Callisto
unit vector of r = (xi + yj) / (x^2 + y^2) \(= \frac{(rcos) \theta i +(rsin\theta)j}{r^2cos^2\theta+r^2 sin^2 \theta} \) \(= \frac{(rcos) \theta i +(rsin\theta)j}{r^2 } \) Hmm.. doesn't look good, sorry!!
experimentX
  • experimentX
|dw:1349366033743:dw|
experimentX
  • experimentX
can you check page no 14 again?
Callisto
  • Callisto
Yes....
experimentX
  • experimentX
you got your first relation right?
Callisto
  • Callisto
Eh.... I was about to say no, but then, yes :|
experimentX
  • experimentX
for second relation there are many tricks. this is my favourite. polar coordinate is orthogonal coordinate system. \[ \hat r \cdot \hat \theta = 0\] so change \( \theta = \theta + 90 \) you will get second relation.
Callisto
  • Callisto
Hmm, sorry!! OS is not loading properly here... And... please give me sometime to work it out first.. :(
experimentX
  • experimentX
sure till then i'll work out on the original method|dw:1349366641329:dw|
Callisto
  • Callisto
Sorry... May I know what that e is?? I meant he wrote \(\hat e_\theta\) or \(\hat e_r\) there.
experimentX
  • experimentX
|dw:1349366770653:dw|
experimentX
  • experimentX
|dw:1349367409678:dw|
Callisto
  • Callisto
So, they are the unit vectors?
experimentX
  • experimentX
yep!!
experimentX
  • experimentX
|dw:1349367720837:dw|
Callisto
  • Callisto
Right, your trick works :|
Callisto
  • Callisto
PQ as a vector? or...?
experimentX
  • experimentX
yep. PQ
experimentX
  • experimentX
|dw:1349367904332:dw|
experimentX
  • experimentX
i hope you are understanding.
Callisto
  • Callisto
PQ (vector) =\(r (cos(\theta +d\theta) -cos\theta )i + r (sin(\theta+d\theta) -sin\theta)j\) Sum-to-product formula?
experimentX
  • experimentX
yep .. .cos C - cos D .. and ...
Callisto
  • Callisto
\[PQ = r[cos(\theta+d\theta)-cos\theta] i + r[sin(\theta+d\theta)-sin\theta]j\] \[=-2rsin(\theta+\frac{d\theta}{2})sin\frac{d\theta}{2}i +2rcos(\theta+\frac{d\theta}{2})sin\frac{d\theta}{2}j\]
experimentX
  • experimentX
|dw:1349368740990:dw|
Callisto
  • Callisto
Isn't that |PQ| something horrible??
experimentX
  • experimentX
lol ... no. Looks like i forgot cos C - cos D
experimentX
  • experimentX
|dw:1349369192240:dw|
Callisto
  • Callisto
Enough hint already!! (shhh!!~)
experimentX
  • experimentX
|dw:1349369373117:dw|
experimentX
  • experimentX
|dw:1349369423357:dw|
experimentX
  • experimentX
|dw:1349369566965:dw|
experimentX
  • experimentX
|dw:1349369666573:dw|
experimentX
  • experimentX
**************************************************************************************************************************************************************
experimentX
  • experimentX
|dw:1349369888678:dw|
experimentX
  • experimentX
|dw:1349369925726:dw|
experimentX
  • experimentX
similarly for theta |dw:1349370057135:dw|
experimentX
  • experimentX
*************************************************************************************** A more general approach.
experimentX
  • experimentX
|dw:1349370210626:dw|