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Analytic mechanics in polar coordinates

  1. Oct 14, 2008 #1
    1. The problem statement, all variables and given/known data
    A mass follows the path of a cardioid r=1+sinφ with given speed, what is its period?

    2. Relevant equations



    3. The attempt at a solution

    I attempt to do an integral on polar coordinates to find the distance covered by the mass first.
    The integral I derived is
    [tex]\int_0^{2\pi} \sqrt{r^{2}+(\frac{dr}{d\varphi})^2}d\varphi[/tex]
    A mistake here? Or there are other methods to find the time?
    Any help is appreciated.

    UPDATE: I figure out the integral, but it is zero (ok, I know it's correct ya), but certainly not something I want.
     
    Last edited: Oct 14, 2008
  2. jcsd
  3. Oct 15, 2008 #2

    siddharth

    User Avatar
    Homework Helper
    Gold Member

    The infinitesimal arc length in polar coordinates [itex]dl[/itex] is what you integrated over in your expression. ie,

    [tex] dl= \sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta [/tex]

    If you integrate this, it gives you the total arc-length. If the speed is constant, what is the time required to travel a distance dl? From this, you can get the period.

    Hint: Remember that

    [tex]\left(\sin\frac{\theta}{2} + \cos\frac{\theta}{2}\right)^2 = 1+\sin{\theta}[/tex]
     
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