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Time derivatives of polar motion

  1. Sep 9, 2015 #1
    1. The problem statement, all variables and given/known data
    A particle moves with v=constant along the curve
    $$r = k(1+\cos \theta)$$Find ##\mathbf{a}##

    2. Relevant equations
    $$ \mathbf{r} = r\mathbf{e_r}$$ $$ \mathbf{v} = \frac{\partial}{\partial t}(r\mathbf{e_r}) $$ $$ \mathbf{a} = \frac{\partial \mathbf{v}}{\partial t} $$ $$\mathbf{\dot e_r}=\dot \theta\mathbf{e_\theta} $$ $$\mathbf{\dot e_\theta}= -\dot \theta\mathbf{e_r} $$

    3. The attempt at a solution
    $$\mathbf{v} = \dot r\mathbf{e_r}+r\mathbf{\dot e_r} = \dot r\mathbf{e_r} + r\dot \theta\mathbf{e_\theta}$$ $$\mathbf{a} = \ddot r\mathbf{e_r}+\dot r\mathbf{\dot e_r}+\dot r\dot \theta\mathbf{e_\theta}+r\ddot \theta\mathbf{e_\theta}+r\dot \theta\mathbf{\dot e_\theta} $$ $$\mathbf{a} = \ddot r\mathbf{e_r}+\dot r\dot \theta\mathbf{e_\theta}+\dot r \dot \theta\mathbf{e_\theta}+r\ddot \theta\mathbf{e_\theta}-r\dot \theta ^2\mathbf{e_r}$$ $$\mathbf{a} = (\ddot r - r\dot \theta ^2)\mathbf{e_r} + (2\dot r\dot \theta + r\ddot \theta)\mathbf{e_\theta}$$
    From here I am having some trouble. I cannot figure out how to get equations for ##r## and ##\theta## in terms of ##t##. I would guess that it has something to do with v being constant. Because r is a cardioid I don't think I can just use ##\theta=t##.
     
  2. jcsd
  3. Sep 9, 2015 #2

    andrewkirk

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    You have not yet used the fact that speed is constant. The first equation in your attempted solution gives the velocity. Use that to develop a formula for the speed and then set that equal to a constant. That should give you something to substitute into your last equation to get rid of some stuff.
     
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