1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Length of curve in Polar coordinate system

  1. Aug 4, 2009 #1
    I want to caculate length of curve in Polar coordinate system like this: if r=r(a)
    then length of the curve is ∫r(a)da Is this right? if not ,why ?
    What's the right one ?
    I konw the way in rectangular coordinate system,I just want to do it in Polar coordinate system .
     
  2. jcsd
  3. Aug 4, 2009 #2
    You can think of it like an infinitesimal form of the Euclidean distance formula. For a function [itex]f(t)=\langle x_1(t),x_2(t),x_3(t),\ldots\rangle[/itex]

    [tex]\sum_a^b \sqrt { \Delta x_1^2 + \Delta x_2^2+\Delta x_3^2+\ldots } \longrightarrow s=\int_{a}^{b} \sqrt { dx_1^2 + dx_2^2+dx_3^2+\ldots} = \int_{a}^{b} \sqrt { \left(\frac{dx_1}{dt}\right)^2 + \left(\frac{dx_2}{dt}\right)^2+\left(\frac{dx_3}{dt}\right)^2+\ldots}\text{ } dt[/tex]
     
  4. Aug 6, 2009 #3
    Is this ∫r(a)da wrong? Why?
     
  5. Aug 6, 2009 #4

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    You need to know what the appropriate infinitesemal length segments are!

    Now, if you do polar coordinates, you may decompose a stretch of a curve into two parts:
    1. The change in the radial position from the initial point on the curve to the final point.
    Infinitesemally, this has length dr.

    2. Here's the tricky part: The tiny arc by which the curve segment can be approximated by a circular arc, supported by a tiny angular change between the first point and the final point on the curve.
    Clearly, that circular arc lies AT a radius of the value "r", and setting the angular change as [itex]d\theta[/tex], we get the expression [itex]rd\theta[/tex] for that length segment.

    3. Now, we apply the Pythogorean theorem to these two length segment to gain the proper curve segment ds:
    [tex]ds=\sqrt{(dr)^{2}+(rd\theta)^{2}}[/tex]

    4. Assuming that the radial position of the point of the curve is describable as a function of the angular variable, we may rewrite this as:
    [tex]ds=\sqrt{(\frac{dr}{d\theta})^{2}+r^{2}}d\theta[/tex]

    5. This is then the proper infinitesemal form of the lengthn segment, and the length s of the curve can then be calculated as:
    [tex]s=\int_{\theta_{0}}^{\theta_{1}}\sqrt{(\frac{dr}{d\theta})^{2}+r^{2}}d\theta[/tex]

    6. Note that in the case of a CIRCLE, where r is a constant function of the angle, this reduces to:
    [tex]s=\int_{\theta_{0}}^{\theta_{1}}rd\theta[/tex]
    as it should do.
     
  6. Aug 6, 2009 #5
    Thank you ! Especially you,Arildno.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Length of curve in Polar coordinate system
  1. Length of a curve (Replies: 4)

  2. Coordinate system (Replies: 3)

Loading...