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Determining possible trajectories (Classical Mechanics)

  1. Feb 13, 2014 #1
    1. The problem statement, all variables and given/known data

    Determine possible trajectories for particle with constant magnitude of velocity |[itex]\dot{\vec{r}}[/itex]| = v0 and constant angular momentum [itex]\vec{L}[/itex] = [itex]\vec{L}[/itex]0

    2. Relevant equations

    |[itex]\dot{\vec{r}}[/itex]| = v0
    [itex]\vec{L}[/itex] = [itex]\vec{L}[/itex]0


    3. The attempt at a solution

    I know that L dot is zero and thereby the torque is zero. My intuition tells me that the possibly trajectories would be circles but mathematically, I am not sure where to start.
     
  2. jcsd
  3. Feb 14, 2014 #2

    BvU

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    Look at he definition of L. Perhaps the time derivative of that tells you something
     
  4. Feb 14, 2014 #3
    Well, as I said, if [itex]\dot{\vec{L}}[/itex] = 0, then the net torque and net force are zero. The constant velocity magnitude implies that the direction of the velocity vector can change but the speed would remain constant.

    If F is zero, then you have

    m[itex]\ddot{x}[/itex] = 0

    Which implies that x(t) is of some linear form:

    x(t) = at + b, where a = v[itex]_{0}[/itex] and b = x[itex]_{0}[/itex].

    I'm assuming the question requires the graphing of the trajectories in phase space and hence would require some form of mathematical development in terms of the energy of the system.
     
  5. Feb 14, 2014 #4

    vela

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    I think BvU meant for you to differentiate ##\vec{L} = \vec{r}\times\vec{F}## and to interpret the result.

    By the way, your inference that ##\dot{\vec{L}} = 0## implies ##\vec{F}=0## is not correct.
     
  6. Feb 14, 2014 #5

    BvU

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    I meant: differentiate ##\vec L = \vec r \times \vec p## wrt time, knowing ##\vec p = m\dot{\vec r}##. Don't venture into x and y because you have everything you need in polar coordinates.
     
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