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I Acceleration only due to conservation of angular momentum

  1. Apr 9, 2016 #1
    I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force.

    Consider for istance planetary motion. The angular momentum [itex]\vec{L}[/itex] of the planets is conserved and that means [itex]\mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta}[/itex] is conserved too.

    Consider the acceleration in polar coordinates
    $$
    \left( \ddot r - r\dot\theta^2 \right) \hat{\mathbf r} + \left( r\ddot\theta+ 2\dot r \dot\theta\right) \hat{\boldsymbol{\theta}} \ $$

    The second term is zero since [itex]\vec{L}[/itex] is constant. Infact the second term can be rewritten as [itex]
    a_{\theta}=\frac{1}{r}[\frac{d}{dt}(r^2 \dot{\theta})]=\frac{1}{r}[\frac{d}{dt}(\frac{L}{m})][/itex].
    This means that there is no acceleration in the direction of [itex] \hat{\boldsymbol{\theta}} [/itex], which is clear since the gravitational force is a central froce.

    But if the distance [itex]r[/itex] decreases [itex]v_{\theta}[/itex] (i.e. the velocity in the direction of [itex] \hat{\boldsymbol{\theta}} [/itex]) must increase in order to keep [itex]\mid\vec{L} \mid[/itex] constant.

    How can [itex]v_{\theta}[/itex] increase if there is no acceleration in the direction of [itex] \hat{\boldsymbol{\theta}} [/itex]?

    I understood that it happens because of the conservation of angular momentum but if there is an acceleration, necessarily a force is needed. I don't see where do this force come from.
     
  2. jcsd
  3. Apr 9, 2016 #2

    A.T.

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    Science Advisor
    Gold Member

    Speed changes because the force is not perpedicular to velocity, when the distance changess.
     
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