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Arc length in polar coordinates

  1. Feb 10, 2016 #1
    I know that an arc length in polar coordinates can be computed by integrating $$\int ds$$ using the formula ##ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta##. But, seeing that ##s=\rho\theta## and ##ds = \rho d\theta##, why is it wrong to calculate arc lengths with this expression for ##ds##?
     
  2. jcsd
  3. Feb 10, 2016 #2

    andrewkirk

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    The formula ds=ρdθ is not correct for arc length. It assigns a zero length to any radial line, such as the line from (in Cartesian coordinates) (0,0) to (0,1).
     
  4. Feb 10, 2016 #3
    Right. Thanks
     
  5. Feb 11, 2016 #4
    The case of motion along a circular arc is a limiting case. Take your expression

    ## ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta ##

    and ask what happens if we confine motion to a circular arc. In that specific case, ## \frac{dr}{d\theta} = 0 ## and you see the familiar formula for the length of arc along a circular arc emerges, ## ds = \rho d\theta ##.
     
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