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Polar coordinates in mecanics?

  1. Oct 7, 2008 #1
    Ok, here is my problem. I haven't taken anything vector related since at least one year ago. And back then, I wasn't such a good student.. So now my past has come back to haunt me..
    I still have some basic notions, but other than that, I pretty much forgot things..

    http://img393.imageshack.us/img393/8378/mec1wi0.jpg
    Can someone please explain to me why ir = cosθ . i + sinθ . j and iθ = -sinθ.i + cosθ. j ?

    Why isn't for example iθ = -cosθ.i + sinθ.j?
    Our teacher told us that these two are formulas. Or do they vary in each case?

    (ir, iθ, i and j are vectors.)

    Thanks in advance for your help.
     
  2. jcsd
  3. Oct 7, 2008 #2

    CompuChip

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    Well, it depends on how you choose your coordinate system. First of all, you want the unit vectors in the r and theta direction to be perpendicular to each other, and the one in the r-direction to point along the line from the origin O to M(t).
    Now normally, one would choose the unit vector i along the x-axis, and j along the y-axis. Of course, for theta = 0, you see that the r-unit vector should just be i, and theta- unit vector should be along j. If you plug in theta = 0 in the sine and cosine, you will see that one of the two definitions you gave comes out wrong.

    Of course, you are free to choose your i and j vectors differently (for example, i along the y axis and j along the -x or +x-axis is a possibility, although it goes against the conventions) and you would find another expression for i_theta and i_r. That is why, before solving any mechanics problem, you should always draw a picture with all the relevant basis vectors in it, so no confusion may arise about how you set up the coordinate system.
     
  4. Oct 7, 2008 #3

    tiny-tim

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    Hi Patrick.Gh! :smile:
    ooh … scary! :eek:
    -cosθ.i + sinθ.j is the reflection of cosθ.i + sinθ.j in the x-axis.

    you want the perpendicular vector, so you want their dot-product to be zero … in this case, (cosθ . i + sinθ . j).(-sinθ.i + cosθ. j) = cosθsinθ - sinθcosθ = 0. :smile:
     
  5. Oct 7, 2008 #4
    I don't know why I have this feeling you are being sarcastic :P Maybe because I'm usually sarcastic, or I have been over dramatic in the post :D

    I see. So if iθ is in the opposite direction, it would still be -sinθ.i + cosθ. j since it's still perpendicular?
     
  6. Oct 7, 2008 #5

    HallsofIvy

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    No, tiny-tim is never sarcastic (he's much nicer than I am). He really meant it!

    He does, however, seem to me to be wrong about one point: [itex]-cos(\theta)i+ sin(
    \theta)j[/itex] is the reflection of [itex]cos(\theta)i+ sin(\theta)j[/itex] in the y-axis, not the x-axis. the y coordinate is the same, [itex]sin(\theta)[/itex] in both, only the x coordinate, [itex]cos(\theta)[/itex] is negated. That's a reflection in the y-axis.
     
  7. Oct 7, 2008 #6

    CompuChip

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    Which is consistent with the image you posted.
     
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